QMReview.dvi

A Review of Quantum Mechanics

Chun Wa Wong

Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547

∗

(Dated: September 21, 2006)

Review notes prepared for students of an undergraduate course in quantum mechanics at UCLA,

Fall, 2005 – Spring, 2006. Copyright

2006 Chun Wa Wong

I. The Rise of Quantum Mechanics

1.1 Physics & quantum physics:

Physics = objective description of recurrent physi-

cal phenomena

Quantum physics = “quantized” systems with both

wave and particle properties

1.2 Scientiﬁc discoveries:

Facts → Theory/Postulates → Facts

1.3 Planck’s black body radiation:

Failure of classical physics:

Emissivity =

uc, λ

max

T = const,

energy density = u = ρ

hEi,

hEi

exp

6= k

T, (1)

(ρ

= density of states in frequency space).

Planck hypothesis of energy quantization (E

nhν) explains observation:

hEi =

hν

∞

n=0

−nx

∞

n=0

−nx

hν

hν/k

− 1

, (2)

where x = hν/(k

T ).

1.4 Einstein’s photoelectric eﬀect:

Photoelectrons ejected from a metal by absorbed

light of frequency ν have maximum energy:

E = hν −W (work function) (3)

that depends on the frequency ν of light and not

its intensity.

1.5 Bohr’s atomic model:

Electron of mass m in a classically stable circular

orbit of radius r = a and velocity v around the

atomic nucleus:

, or T = −

, (4)

∗

Electronic address: cwong@physics.ucla.edu

= e

/4πǫ

in Gaussian units). Hence

E = T + U =

= −T. (5)

Bohr’s two quantum postulates (L quantization and

quantum jump on photon emission):

L = mva = n~

~, n = 1, 2, ...,

hν = E

− E

. (6)

For ground state:

= −

= −13.6 eV

= −T

= −

2ma

a =

= 0.53

A (Bohr radius), (7)

For excited states:

= n

a, E

. (8)

1.6 Compton scattering:

Energy-momentum conservation for a photon of lab

energy E = hc/λ scattered from an electron (of

mass m) initially at rest in the lab (before=LHS,

after=RHS; 1=photon, 2= recoiling e):

p = p

+ p

E + mc

= E

+ p

. (9)

Photon scattered into lab angle θ has longer wave-

length λ

− λ = λ

(1 − cos θ), where

= 2.4 pm (10)

is the electron Compton wavelength.

1.7 de Broglie’s matter wave:

Massive matter and massless light satisfy the same

energy-momentum and momentum-wavelength re-

lations

= p

+ m

p = ~k =

. (11)

Thus moving matter is postulated to be a wave with

a motional de Broglie wavelength λ = h/p, leading

to the diﬀraction maxima for both matter and light

2-sided formula: 2d sin θ = nλ,

1-sided formula: D sin θ = nλ.

II. Waves and quantum waves

Classical and quantum waves have identical mathemat-

ical properties.

2.1 Waves in physics:

Waves rise and fall, travel, have coherence in space

and in time, interfere, and diﬀract.

Two classes of waves:

(a) Inertial waves in massive media: ocean waves,

sound waves, vibrations of violin strings;

(b) Noninertial waves in vacuum: EM waves,

matter waves.

2.2 Mathematical description of waves:

The 1-dimensional wave equation (1DWEq)



∂

∂x

−

∂

∂t



Ψ(x, t) = 0 (12)

is an equation of state that gives a wave function

Ψ(x, t) describing an unfolding event in spacetime.

The wave equation is a partial diﬀerential equation

(PDE) because it depends on more than one vari-

able, here the two variables x and t.

In contrast, the solution x(t) of an equation of mo-

tion such as a Newton equation describes how a

single point x(t) on an object evolves in time. Sim-

ilarly, a snapshot of an object is not necessarily a

wave. A wave function Ψ(x, t) is needed to describe

a wave’s coherent structure in both space and time.

However, a wave (any wave) can have particle prop-

erties if a point x can be deﬁned on it such that one

can describe how its position changes in time.

The wave equation (12) describes linear waves sat-

isfying the superposition principle that a sum

of solutions is also a solution :

Ψ = a

+ a

. (13)

Traveling waves:

(a) f (x−vt)= a wave traveling to the right (+x),

(b) g(x + vt)= a wave traveling to the left (-x),

f(x −vt) + a

g(x + vt)

is also a wave.

LDEs (linear DEs) with constant coeﬃcients

have exponential solutions:

ikx

= ike

ikx

−iωt

= −iωe

−iωt

. (14)

These DE are called eigenvalue equations be-

cause the constant ik is called an eigenvalue of the

diﬀerential operator d/dx, while the solution e

ikx

called its eigenfunction belonging to its eigenvalue

ik.

The 1DWEq (12) has four distinct (i.e., linearly

independent) eigenfunctions:

Ψ(x, t) = e

±i(kx±ωt)

. (15)

Energy in both classical and quantum waves is car-

ried by the intensity or energy ﬂux

I ∝ |Ψ|

. (16)

Its quadratic dependence on Ψ is responsible for the

interference between two traveling waves Ψ

iθ

with real A

|Ψ

+ Ψ

= |Ψ

+ |Ψ

+ 2Re(Ψ

∗

)

= A

+ A

+ 2A

cos(∆s),

where ∆s = ∆k



x −



∆ω

∆k





. (17)

In the limit ∆k → 0,

dω

= v

= group velocity

6= v =

= wave velocity. (18)

Examples: Certain waves satisfy the relation ω =

. Then v

/v = p. Speciﬁc examples are p = 2

for transverse vibrations of a thick bar, p = 3/2

for short ripples moving under surface tension, and

p = 1/2 for long waves on deep sea.

The result (17) is similar in structure to the squared

length of a sum of two vectors in space that inter-

sect at an angle θ:

+ A

= A

+ A

+ 2A

cos θ, (19)

Two waves of the same amplitude traveling in op-

posite directions interfere form a standing wave:

Ψ = A[cos(kx − ωt) + cos(kx + ωt)]

= 2A cos(kx) cos(ωt) (20)

with factorized space and time dependences.

2.3 Matter-wave quantization:

Light wave of momentum p and energy E = pc is

described by the wave function:

i(kx−ωt)

= e

i[(p/~)x−(E/~)t]

. (21)

de Broglie and Schr¨odinger suggested that the RHS

expression holds also for m atter waves so that:



∂

∂x



i(p/~)x

= ˆp e

i(p/~)x

= p e

i(p/~)x



∂

∂t



−i(E/~)t

H e

−i(E/~)t

= E e

−i(E/~)t

.(22)

The diﬀerential operators that appear are called

the momentum operator ˆp and the Hamiltonian (or

energy) operator

H, respectively.

Commutation relations: The rules of diﬀeren-

tial calculus dictates that a diﬀerential operator

does not commute with its own variable:

∂

∂x

x = 1 + x

∂

∂x

, or



∂

∂x

, x



= 1;

∂

∂t

t = 1 + t

∂

∂t

, or



∂

∂t

, t



= 1. (23)

These results give the fundamental commuta-

tors of wave mechanics

[ˆp, x] =

, [

H, t] = i~. (24)

2.4 Wave packet and the uncertainty principle:

A wave packet is a superposition of waves of the

type (21)

f(x, t) =

√

2π

∞

−∞

g(k)e

i[kx−ω(k)t]

dk, (25)

with a linear superposition function g(k) sometimes

called a spectral amplitude.

Expectation value (= mean value) of a property

A(x, t) of the wave:

hA(x, t)i ≡

∞

−∞

(x, t)A(x, t)dx, where

(x, t) =

|f(x, t)|

∞

−∞

|f(x, t)|

(26)

is the normalized probablity density of ﬁnding the

wave at the spacetime point (x, t).

The expectation value of a time-independent prop-

erty B(k) of the wave can be calculated by using

the spectral amplitude g(k):

hB(k)i ≡

∞

−∞

˜ρ

(k)B(k)dk, where

˜ρ

(k) =

|g(k)|

∞

−∞

|g(k)|

(27)

is the normalized probablity density of ﬁnding the

wave in “k space”. (We shall not consider more

complicated properties that also depend on the

time t or the energy E.)

Uncertainty relations: A wave of any kind sat-

isﬁes the uncertainty relation

∆x∆k ≥

, where

(∆x)

= h(x − ¯x)

i, ¯x = hxi,

(∆k)

= h(k −

k = hki. (28)

The expression for matter waves is usually written

as the Heisenberg uncertainty principle:

∆x∆p ≥

. (29)

Spreading wave packets: A Gaussian wave

packet has the spectral amplitude

g(k) = e

−αk

, with ∆k =

√

. (30)

It gives rise to a wave function at time t = 0 of

f(x, t = 0) ∝ e

−x

/(4α)

, with ∆x =

√

α. (31)

Hence the wave packet at time t = 0 has the mini-

mal uncertainty product of

∆x∆k =

. (32)

For t 6= 0, however, the uncertainty product could

be greater than the minimal value of 1/2. This hap-

pens if the system is dispersive, meaning a nonzero

second Taylor coeﬃcient β in the following expan-

sion:

ω(k) ≈ ω(k = 0) + k

dω



= ω(0) + kv

β. (33)

The resulting approximate wave function at any

time t can then be evaluated to the closed form

f(x, t) ∝ e

−x

/(4α

)

, with

= x − v

t, α

= α +

iβ

t. (34)

Thus the position x of the wave moves in time with

the group velocity

dω



. (35)

Its position uncertainty increases in time to

∆x(t) =

|α

√

1 +



βt

2α



1/2

. (36)

For a massive particle in free space E

= p

. Hence

β = ~

= ~

6= 0. (37)

Thus the spatial width of the Gaussian packet of

a massive particle spreads out in both positive and

negative times.

III. Wave mechanics

Wave mechanics describes physics by using the

Schr¨odinger wave eq. and its wave functions.

3.1 Schr¨odinger wave equation (SchWEq) :

First quantization: The classical E-p relation

= p

is quantized into a wave equation

by replacing E, p by the diﬀerential operators

E →

H = i~

∂

∂t

p → ˆp =

∂

∂x

: (38)



∂

∂t



Ψ(x, t) =



∂

∂x



+ m

Ψ, (39)

where the wave function Ψ(x, t) has to be added for

the operators to operate on. In this way, a classical

state of motion “spreads out” into a wave structure.

Nonrelativistic limit: Likewise, the NR energy

E = T + V = p

/2m + V (not containing the rest

energy mc

) is quantized into the SchWEq:



∂

∂t



Ψ(x, t) =



∂

∂x



+ V (x)

Ψ, (40)

Commutation relations: A diﬀerential operator

does not commute with its variable. Hence

[ˆp, x] =

, [

H, t] = i~. (41)

Group velocity: E

= p

+ m

⇒ EdE =

pdp. Hence





= c

. (42)

3.2 Born’s interpretation of |Ψ|

The squared wave function gives the probabililty

density of ﬁnding system at spacetime location

(x, t):

|Ψ(x, t)|

= ρ

(x, t) (43)

in the sense that the probability for ﬁnding the

state somewhere in space at time t is

P (t) =

∞

−∞

(x, t)dx

∞

−∞

|Ψ(x, t)|

dx = 1. (44)

By choosing P (t) = 1, the wave function is said to

be normalized.

3.3 Conservation of |Ψ|

The probability density can be moved around, but

cannot be created or destroyed in the absence of

explicit creative or destructive physical processes:

0 =

(x, t) =



∂

∂t

∂

∂x



(x, t)

∂

∂t

(x, t) +

∂

∂x

(x, t). (45)

This is called the continuity equation for the con-

served probability current density

(x, t) = v ρ

(x, t), (46)

where v = dx/dt.

IV. Wave mechanics in one spatial di-

mension

Wave mechanics is applied to simple one-dimensional

problems to illustrate how the SchWEq is solved and how

its wave character aﬀects the properties of a system.

4.1 Solving the Schr¨odinger wave equation

(SchWEq) :

Separation of varables: Since solutions are

unique up to a superposition, we may look for a

solution of the factorized form

Ψ(x, t) = ψ(x)φ(t). (47)

Then

ψ(x)

Hφ(t)

ψ(x) φ(t)

φ(t)



ˆp

+ V



ψ(x)

φ(t) ψ(x)

= E. (48)

On simpliﬁcation, the ﬁrst expression is clearly a

function of t only, while the middle expression is a

function of x only. Hence the separation constant

E must be a constant independent of t or x.

The PDE can thus be broken up into two ordinary

DEs:

Hφ(t) =



∂

∂t



φ(t) = Eφ(t),



∂

∂x



+ V (x)

ψ(x) = Eψ(x), (49)

x = 0

- V

BCs: Ψ



= Ψ



Ψ '



= Ψ '



FIG. 1: Wave numbers and wave functions at a potential step.

where E appears in both ODEs as the energy

eigenvalue. The ﬁrst ODE gives the time-

dependent factor

(t) = e

−i(E/~)t

, (50)

but the second equation is a second-order ODE

whose solution requires the imposition of two

Boundary conditions at a suitable boundary x =

) = ψ

′

) ≡

(x)



= ψ

′

). (51)

Note that if the ﬁrst BC is violated, ψ

′

) becomes

inﬁnite and the DE (diﬀerential eq.) cannot be sat-

isﬁed. If the second BC is violated, ψ

′′

) becomes

inﬁnite. The DE also cannot be satisﬁed, except for

the special case where V (x

) has a compensating

inﬁnity at x

4.2 Particle in a box of length L:

With the two BCs ψ(0) = ψ(L) = 0, the solutions

turn out to be the same as those describing classical

vibrations of a violin string of length L:

(x) =

sin



nπ



(52)

for a normalized wave function.

4.3 Potential step:

Figure 1 shows a particle of mass m incident from

the left with energy E. The wave numbers k, q are

deﬁned as follows:

E =

, E − V =

; (53)

or k

= q

− k

, where k

= 2mV

. The parti-

cle’s matter wave is partially reﬂected at the bound-

ary x = 0, and partially transmitted across it (with

no further reﬂection downstream):

x < 0 : ψ

(x) = e

ikx

+ Be

−ikx

x > 0 : ψ

(x) = A

iqx

. (54)

x = 0 x = L

FIG. 2: Wave numbers for a rectangular potential barrier.

Its probability current density is conserved across

the boundary:



1 − |B|



= j

. (55)

Its reﬂection and transmission coeﬃcients at the

boundary are:

R = |B|

, T = 1 − R. (56)

4.4 Potential barrier:

A particle of incident energy E higher than the

height V

of a rectangular potential barrier is par-

tially reﬂected and partially transmitted by a potet-

nial barrier. The wave functions for x < 0 before

the barrier and for x > L after the barrier remain

the same as Eq.(54), but the presence of the sec-

ond boundary at x = L causes reﬂected waves to

be present in the barrier region

0 < x < L : ψ

(x) = Ce

ikx

+ De

−ikx

. (57)

The two boundary conditions at the second bound-

ary x = L are enough to determine the two extra

coeﬃcients C and D. The reﬂection coeﬃcient re-

mains R = |B|

, but its value is changed. The

transmission coeﬃcient is

T = 1 − R =

1 +



−q

2kq



sin

(qL)

. (58)

Quantum tunneling: The formula for T works

even when the energy E falls below the height V

the potential barrier, as shown in Fig. 2. Then q

−κ

< 0, and q = iκ becomes purely imaginary.

The function [sin(qL)/q]

in the formula is analyti-

cally continued to the function [sinh(κL)/κ]

. The

resulting transmission coeﬃcient T across the ﬁnite

barrier is shown in Fig. 3. The fact that T does not

vanish for E < E

, as it does in classical mechanics,

is called quantum tunneling. Note that tunneling

can occur only when the barrier thickness is ﬁnite.

1 2 3 4 5

EV

0.2

0.4

0.6

0.8

FIG. 3: Transmission coeﬃcient across a rectangular potential

barrier.

1 2 3

rR

-0.5

0.5

Alpha

emission Fusion

FIG. 4: Tunneling across the Coulomb barrier in α decay and

in nuclear fusion.

4.5 Applications of quantum tunneli ng:

Alpha decay: An α particle inside a nucleus feels

an attractive nuclear potential. As the α particle

leaves the nucleus in alpha decay, it sees the repul-

sive Coulomb potential outside the nucleus. These

two potentials combine to form a potential barrier

that the α particle must tunnel through on its way

out, as illustrated in Fig. 4. The decay probablility

is thus fT , the product of the frequency f of the

α particle’s hitting the inside wall of the potential

barrier and the transmission coeﬃcient T through

the potential barrier (that gives the tunneling prob-

ability per attempt).

Nuclear fusion: is just the time reversed situation

of emission. Here a charged particle, say a proton,

is trying to enter a nucleus from the outside by tun-

neling through its very high and thick Coulomb po-

tential barrier. The transmission coeﬃcient is typi-

cally very small, but increases very rapidly with the

energy E. If a number of atomic nuclei are exposed

to a gas of protons of increasing temperature, the

high-energy tail of the Boltzmann energy distribu-

tion of the proton gas will allow more and more

protons to get into atomic nuclei to initiate nuclear

fusion reactions. For example, the pp reaction

p + p → d + e

+ ν (59)

(where e

is the positron and ν is a particle called

HaL

-W

Sample

External

field

HbL

-W

Sample

Scanning

tip

FIG. 5: Tunneling through a ﬁnite potential barrier (a) in

cold emission, and (b ) in a scanning tunneling microscope.

the electron neutrino) powers the conversion of

mass into solar energy in the interior of the Sun.

The reaction rate depends on the temperature T

as T

The luminosity of a star usually increases as M

its mass M, or T

of its central temperature. Such

an unusually strong dependence led Bethe in 1938

to look for another source of hydrogen burning. He

discovered that this alternative route is a certain

CNO cycle that involves certain carbon, nitrogen

and oxgen isotopes in intermediate steps.

Cold emission: Electrons are bound in metals by

their work function W . However, they can be in-

duced to tunnel out of the metal by applying an

external electric ﬁeld to change the potential to

a triangular penetrable potential, as illustrated in

Fig. 5(a).

In s canning tunneling microscopes, the tunnel-

ing is induced by the proximity of the scanning tip,

as shown in Fig. 5(b). Since the tunneling current

increases greatly as the thickness of the potential

barrier decreases, one has a very sensitive device for

mapping the geometrical shape of a surface down

to atomic dimensions.

4.6 Bound states in a potential well:

If V (x) → constant as x → ±∞, the lower of these

two contant values is usually taken to be the zero

of the energy scale. Two situations can then be

realized:

(a) A system with E > 0 has continuous energy

spectrum with wave functions that are not lo-

calized or square-integrable, while

(b) a system with E < 0 has a discrete energy

spectrum with localized wave functions that

are square-integrable. The resulting physical

states are called bound states in a potential

well.

The wave numbers associated with a bound state in

an attractive one-dimensional square-well potential

x = -a x = a

-V

V=0

FIG. 6: Wave numbers associated with a bound state in an

attractive squ are-well potential.

are summarized in Fig. 6, where

|E| = −E =

, E −V =

. (60)

Such an attractive one-dimensional potential has

at least one bound state, the ground state of

even parity whose wave function is proportional to

cos qx. This is because such a wave function al-

ready “bends” over at the boundaries, and can al-

ways be matched to a decreasing exponential func-

tion outside the potential well.

V. The mathematical structure of quan-

tum mechanics

Physical states in quantum mechanics are linear waves.

Both quantum states and classical linear waves can be

superposed to form other waves, in the same way that

ordinary vectors in space can be added to form other

vectors.

5.1 Fourier transform (FT):

The superposition principle for linear wave func-

tions (solutions of linear wave equations) is formal-

ized in the methods of Fourier series and Fourier

transforms (FT). Both methods and their general-

izations are used in quantum mechanics.

The FT of a wave function f(x) in one spatial vari-

able x is deﬁned by the integral

F{f(x)} ≡ g(k) =

√

2π

∞

−∞

f(x)e

−ikx

dx. (61)

For example,



f(x)



= ikg(k) →





g(k). (62)

Thus the momentum operator ˆp = (~/i)∂/∂x yields

the momentum value p = ~k associated with the

spectral amplitude g(k).

Fo uri er inversion formula:

−1

{g(k)} ≡ f(x) =

√

2π

∞

−∞

g(k)e

ikx

dk. (63)

This is how a wave packet is constructed, namely

by taking a linear superposition of waves. Also

−1



g(k)



= −ixf(x). (64)

Thus a position operator ˆx = i∂/∂k gives the value

of the position x associated with the wave function

f(x).

The FT shows that waves can be described either

by wave functions f (x) in space, or by their spectral

amplitudes g(k) in k-space. Functions of one space

can be constructed from those in the other space.

Hence wave properties are completely speciﬁed ei-

ther in ordinary space or in k-space, a “simpliﬁca-

tion” that follows from a wave’s coherent structure.

Table I gives some simple examples of FTs.

Dirac δ-function: Since

−1

{F{f(x)}} = f(x), (65)

we ﬁnd (by writing out the two integral transforms

on the LHS in two steps) that

∞

−∞

δ(x − x

′

)f(x

′

)dx

′

= f(x), (66)

where

δ(x − x

′

) =

2π

∞

−∞

ik(x−x

′

)

dk (67)

TABLE I: Some Fourier t ransforms.

Property If f(x) is then F{f(x)} is

f(x) g(k)

−αx

√

2α

−

4α

unit step function

−

√

2π

sin ka

Derivatives (

)

f(x) (ik)

g(k)

Derivatives (− ix)

f(x) (

)

g(k)

Translation f(x − a) e

−ika

g(k)

Attenuation f(x)e

−αx

g(k −iα)

Real g

∗

(k ) = g(−k)

Real and even g

∗

(k ) = g(−k) = g(k)

Real and odd g

∗

(k ) = g(−k) = −g(k)

from −a to a

Complex conjugation

is called a Dirac δ-function.

Dirac has given a simple intuitive construction of

his δ-function as a rectangular function of width ǫ

and height 1/ǫ with a unit area under it:

δ(x − x

′

) = lim

ǫ→0



1/ǫ, when |x − x

′

| ≤ ǫ/2,

0, otherwise.

(68)

5.2 Operators and their eigenfunctions:

When a diﬀerential operator operates on its eigen-

function, it gives a function that is proportional

to the eigenfunction itself:

ˆp e

ikx



∂

∂x



ikx

= ~k e

ikx

ˆx e

−i(x/~)p



∂

∂p



−ixp/~

= x e

−ixp/~

. (69)

The proportionality constant is called its eigen-

value.

Two functions or eigenfunctions can be combined

into an inner product:

(φ, ψ) ≡

∞

−∞

∗

(x)ψ(x)dx

∞

−∞

∗

(k)g

(k)dk.

Dirac denotes this inner product by the bracket

symbol hφ|ψi.

5.3 Dirac notation:

Functions in function space and states of a physi-

cal system in quantum mechanics are mathemati-

cally similar to vectors in space. Dirac emphasizes

this similarity using a mathematical notation that

works for vectors, functions and quantum states.

A vector (function, state) can be expanded in terms

of basis vectors (functions, states). Dirac distin-

guishes between two kinds of vectors (functions,

states) – column vectors (functions, ket states)

and row vectors (complex-conjugated functions,

bra states), where the name bra and ket comes from

the two parts of the word “bracket”:

|ψi = ket = ψ

|1i + ψ

|2i





= column vector

hφ| = bra = φ

∗

h1| + φ

∗

h2|

= (φ

∗

, φ

∗

) = row vector

= |φi

†

, (70)

For simple vectors, the basis states |1i and |2i are

just the unit basis vectors usually denoted e

and

, respectively. The

†

symbol is called an adjoint

operation. It combines the matrix operation of

transposition with the complex conjugation of all

“matrix elements”.

The Dirac bracket hφ|ψi is a scalar product of

vectors or an inner product of functions

hφ|ψi = (φ

∗

, φ

∗

)





= φ

∗

+ φ

∗

, (71)

where the complex conjugation on the bra com-

ponents is needed to keep the self scalar product

hψ|ψi nonnegative. For two orthonormal vectors

or states:

hi|ji = δ

. (72)

Eigenbras and ei genkets: One goes beyond sim-

ple vector spaces by using the eigenfunctions of op-

erators such as position ˆx and momentum ˆp. In

fact, we immediately associate their eigenfunctions

with abstract eigenstates in the following eigen-

bra or eigenket form:

ˆx|xi = x|xi, and hx|ˆx = x

∗

hx| = xhx|,

ˆp|pi = p|pi, and hp|ˆp = p

∗

hp| = php|, (73)

where the eigenvalues x and p are real, as is true for

all physical observables, i.e., properties that can be

measured experimentally. The eigenvalues x and p

normally range continuously from −∞ to ∞.

Projection and quantum destructive mea-

surement: A simple 2-dimensional vector |Ai can

be decomposed into its components A

= hi|Ai with

respect to the basis |ii:

|Ai = |1iA

+ |2iA

i=1

|iihi|

|Ai. (74)

In a similar way, an N-dimensional vector space is

made up of the N one-dimensional subspaces pro-

jected out by the projection operators

= M

= |iihi| (75)

in the sense that

|Ai = A

|ii (76)

leaves only that part of the vector that is in this

ith subspace. P

is also called a measurement sym-

bol (hence M

). This measurement is not a classi-

cal nondestructive measurement but a quantum de-

structive measurement that destroys all other com-

ponents of the original vector |Ai. The projection

collapses the state |Ai into just the part A

|ii in

the ith subspace.

Completeness relations: The original vector |Ai

can be restored by adding up all its projections

|ii. Hence an identity operator can be con-

structed for this linear vector space

1 =

i=1

|iihi| (77)

for discrete basis states that satisfy the orthonor-

mality relation hi|ji = δ

The idea of completeness or closure holds also for

eigenstates of operators such as ˆx and ˆp that have

continuous eigenvalues:

1 =

∞

−∞

|xihx|dx

∞

−∞

|pihp|dp. (78)

The resulting basis states |xi or |pi are said to be

continuous basis states.

Representations: A discrete representation is a

speciﬁcation of a vector (or state) |Ai by its com-

ponents A

in a discrete basis. A continuous rep-

resentation uses the continuous basis states |xi to

give the components A(x) = hx|Ai in the expansion

|Ai =

∞

−∞

dx |xiA(x). (79)

Wave functions: The (position) wave function of

a state or wave |ψi is just its component ψ(x) =

hx|ψi in the position representation. Its compo-

nent ψ(p) = hp|ψi in the momentum representa-

tion is called its momentum wave function. A wave

function can be expanded into an arbitrary discrete

representation of basis states |ii

hx|ψi ≡ ψ(x) =

i=1

hx|iihi|ψi

i=1

(x)ψ

, (80)

where φ

(x) = hx|ii is the wave function of the basis

state |ii. Incidentally, N must be inﬁnite to match

the number of points on the straight line.

A wave function can also be expanded in another

continuous representation, say the momentum rep-

resentation:

hx|ψi = ψ(x) =

∞

−∞

dp hx|pihp|ψi

√

2π

∞

−∞

i(p/~)x

ψ(p). (81)

This is just the inverse Fourier transform relation.

Dirac δ-function: The equivalence

ψ(x) =

∞

−∞

δ(x − x

′

)ψ(x

′

)dx

′

= hx|ψi =

∞

−∞

hx|x

′

ihx

′

|ψidx

′

(82)

shows that

hx|x

′

i = δ(x − x

′

). (83)

It immediately follows that the Dirac δ-function

can be represented in inﬁnitely many ways. For

example,

hx|x

′

i =

i=1

hx|iihi|x

′

i =

i=1

(x)φ

∗

′

). (84)

Quantum measurement and wave-function

collapse: Electrons emitted from a source pass

through a single slit and fall on a distant observing

screen on the other side of the slit: As a coherent

matter wave, each electron forms a diﬀraction pat-

tern on the observing screen in the direction y of

the diﬀraction pattern. (Diﬀraction = spreading of

a wave into the geometrical shadow regions.)

A detector of acceptance ∆y placed at position

y = d on the observing screen detects electrons

from signals of their passage through the detec-

tor. If ψ(y

′

) = hy

′

|ψi is the wave function of self-

diﬀracting electrons normalized in some way, the

part that enters the detector is

∆ψ(d) =

d+∆y/2

d−∆y/2

′

ihy

′

|ψi. (85)

The probability of an electron entering the detector

∆P (d) =

d+∆y/2

d−∆y/2

′

|ψ(y

′

. (86)

Note that the electron described by the wave func-

tion ψ(y) can be anywhere on the screen before the

detection. However, once it is known to have gone

into the detector, e.g., by the detector signal it gen-

erates, its wave function has collapsed into ∆ψ(d).

This is how a quantum measurement can modify

the state itself.

5.4 Operators and their matrix elem ents:

Matrix mechanics

Operators as matrices: Operators can be visu-

alized as matrices because they satisfy matrix op-

erations. Hence quantum mechanics is also called

matrix mechanics. For example, an operator

acting on a state always gives another state:

A|ψi = c|φi (87)

in much the same way that a square matrix multi-

plying a column vector always gives another column

vector. It can happen occasionally that the result

is actually proportional to the original state |ψi

A|ai = a|ai. (88)

These special states are called the eigenstates of

A. The proportionality constants a are called their

eigenvalues. Note how the eigenvalue a is used to

label the eigenstate |ai itself. This operator equa-

tion is called an eigenvalue equation.

Noncommuting operators: Operators, like ma-

trices, do not generally commute. For example,

[ˆp, ˆx] =

. (89)

In matrix mechanics, commutators like this specify

the fundamental wave nature of the matter waves

described by these operators. It is the starting

point from which other results can be derived. In

contrast, the wave property of matter is described

in wave mechanics by postulating the quantization

rule

ˆp =

∂

∂x

, (90)

and the wave function ψ(x) on which ˆp operates.

Operator matrix elements: Operators like ma-

trices have matrix elements. The matrix elements

of a matrix or operator

M can be expressed conve-

niently in the Dirac notation as

= hi|

M|ji, (91)

a two-sided representation in terms of a suitable

basis |ii. A 2 × 2 matrix or operator can thus be

written in operator form as

M = M

|1ih1| + M

|1ih2|

+ M

|2ih1| + M

|2ih2|. (92)

The ket-bra combinations on the RHS are them-

self operators or square matrices. The oﬀ-diagonal

operators are the transition operators

|ψihφ| =





(φ

∗

) =



∗



, (93)

while the diagonal operators (with φ = ψ) are pro-

jection operators.

An operator in the representation of its own eigen-

states is a diagonal matrix. Thus

hx|ˆx|x

′

i = hx|x

′

= x

′

δ(x − x

′

)

hx|ˆp|x

′

i = hx|x

′



′



= δ(x − x

′

)



′



.(94)

Note how the operator ˆp becomes a diagonal diﬀer-

ential operator in the x-representation, and how it

comes out to the right of the Dirac bracket to avoid

operating on it. The Hamiltonian operator too has

interesting diagonal representations:

hE|

H|E

′

i =



hE|E

′

= δ(E − E

′

, if continuous

hn|n

′

= δ

′

, if discrete.

(95)

Eigenvalue eq. as wave eq.: Matrix mechanics

can be shown to be equivalent to wave mechanics by

writing the operator eigenvalue equation

H|Ψ

i =

E|Ψ

i as a Schr¨odinger wave equation:

Ehx|Ψ

i = hx|

H|Ψ

∞

−∞

′

hx|



ˆp

+ V (ˆx)



′

iΨ

′

)

∞

−∞

′

δ(x − x

′

)



′



+ V (x

′

)

× Ψ

′

)





+ V (x)

(x). (96)

5.5 Hermitian operators and physical observ-

ables:

Hermitian operator: is one for which

†

T ∗

H (itself). (97)

It has the important properties that

(a) Eigenvalues are real.

(b) Eigenstates can be orthonormalized.

Physical observables can have only real values.

Hence they are described by Hermitian operators

in quantum mechanics.

Dirac notation: For any operator

hφ|

Aψi =

∞

−∞

†

(x)[

Aψ](x)dx

∞

−∞

{[

†

φ](x)}

†

ψ(x)dx

= h

†

φ|ψi. (98)

The Hermiticity of an operator

H is conﬁrmed by

the equality hφ|

H|ψi = h

Hφ|ψi of the two distinct

integrals involved. Note that if φ(x) is a column

wave function (called a spinor), the transposition

part of the adjoint operator will come into play to

change it into a row spinor.

5.6 One-dimensional harmonic oscillator:

The 1DHO problem can be solved elegantly in ma-

trix mechanics. First write its Hamiltonian

H =

ˆp

mω

ˆx

(99)

in the compact form

h =

~ω



ˆq

+ ˆy



where ˆy =

ˆx

ˆq =

ˆp =

. (100)

Here the oscillator length x

~/mω is deﬁned

by the equation

~ω = mω

. (101)

The dimensionless position and momentum opera-

tors do not commute:

[ˆq, ˆy] = 1/i. (102)

The dimensionless Hamiltonian

h can be factorized:

h = ˆa

†

ˆa +

where ˆa =

√

(ˆy + iˆq)

ˆa

†

√

(ˆy − iˆq) (103)

are called the step-down and step-up operators, re-

spectively. They do not commute



ˆa, ˆa

†



= 1. (104)

Number operator and its eigenstates: The

number operator

N = ˆa

†

ˆa is Hermitian. It has real

eigenvalues n that can be used to label its eigen-

states

N|ni = n|ni. (105)

It then follows that |ni is also an energy eigenstate:

H|ni = (n +

)~ω|ni. (106)

The eigenvalue n of the number operator is a non-

negative number or integer 0, 1, 2, ... This im-

portant quantization property follows from non-

negative nature of its Hamiltonian (meaning that

Hi ≥ 0) and from the nonzero commutators

[

N, ˆa

†

] = ˆa

†

, [

N, ˆa] = −ˆa. (107)

The commutators show that ˆa

†

and ˆa change the

state |ni as follows:

ˆa

†

|ni =

√

n + 1|n + 1i, ˆa|ni =

√

n|n − 1i. (108)

Oscillator quanta: The quantum number n (qu.

no. = any number used to label a quantum state)

gives the number of oscillator energy quanta, each

of energy ~ω, that can be emitted from that state.

Thus ˆa

†

increases the number of energy quanta in

the state by 1, and is called a creation operator,

while the destruction operator ˆa decreases n by

1. Physical processes that causes the absorption

or emission of energy quanta can be expressed in

terms of these operators.

The ground state which by deﬁnition cannot emit

any energy has the qu. no. n = 0. It has a

zero-point energy ~ω/2 that comes from the wave

spreading of its wave function from the classical

point x = 0 of stable equilibrium.

Wave functions: The impossibility of energy

emission from the ground state is stated by the op-

erator equation

ˆa|0i = 0. (109)

Its y-representaion gives the ﬁrst-order diﬀerential

equation for the ground-state (GS) wave function:



y +



(y) = 0. (110)

The resulting normalized GS wave function is

(y) = α

−y

, where α

1/4

. (111)

Excited states can be created by adding energy

quanta to the GS:

|ni =



(ˆa

†

)

√



|0i. (112)

Consequently, its wave function can be constructed

by successive diﬀerentiations of the GS wave func-

tion:

(y) = hy|



(ˆa

†

)

√



|0i

√



y −



−y

. (113)

Operators in the number representation: In

the number representation |ni, the only nonzero

matrix elements of ˆa and ˆa

†

lie along the diagonal

one line away from the main diagonal of the matrix:

ˆa =







√

1 0 .

0 0

√

2 .

0 0 0 .

. . . .







, ˆa

†







0 0 0 .

√

1 0 0 .

√

2 0 .

. . . .







. (114)

Hence the dimensionless position and momentum

operators are the square matrices

ˆy =

√

(ˆa + ˆa

†

) =

√







√

1 0 .

√

1 0

√

2 .

√

2 0 .

. . . .







ˆq =

√

(ˆa − ˆa

†

) =

√







√

1 0 .

−

√

1 0

√

2 .

0 −

√

2 0 .

. . . .







(115)

5.7 Uncertainty principle:

The quintessential wave property described by the

Heisenberg uncertainty principle takes on a sur-

prisingly elegant form in matrix mechanics: If the

square uncertainty of the expectation value of a

physical observable

A in the quantum state |ψi is

deﬁned as

(∆A)

= h(

A −

, (116)

then the product of uncertainties of two physical

observables

A and

B in the same quantum state

satisﬁes the inequality

(∆A)(∆B) ≥



B] i



. (117)

Simultaneous eigenstates: |a, bi exist for com-

muting operators

A and

A|a, bi = a|a, bi,

B|a, bi = b|a, bi. (118)

Then ∆A = 0 = ∆B.

5.8 Classical correspondence:

Bohr’s correspondence principle: Quantum

mechanics reproduces classical mechanics in the

limit of large quantum numbers. Example: The

probability density of quantum state |ni of the

1DHO agrees with the classical value:

lim

n→∞



(x)



→ ρ

(x). (119)

For a periodic system of period T , the classical

probability density can be deﬁned in terms of the

probablity dP (x) of ﬁnding the classical point mass

within a width dx of the position x during a mo-

ment dt in time t:

dP (x) = ρ

(x)dx =

dt, or

(x) =

v(x)T

, (120)

where v(x) = dx/dt is the velocity at position x.

See Fig. 4-18, 19 of Bransden/Joachain for a com-

parison between quantum and classical ρs for the

1DHO.

Ehrenfest’s theorem: The expectation value of

a physical observable satisﬁes a classical equation

of motion. The expectation value of an operator

A(t) in the time-dependent state

|Ψ(t)i = e

−i

Ht/~

|Ψ(0)i is (121)

A(t)i

≡ hΨ(t)|

A(t)|Ψ(t)i

= hΨ(0)|e

Ht/~

A(t)e

−i

Ht/~

|Ψ(0)i. (122)

A time diﬀerentiation of the three time factors gen-

erates three terms:

A(t)i

= h

∂

A(t)

∂t

A(t)] i

. (123)

Examples:

hˆxi

H, ˆx] i

hˆpi

;

hˆpi

H, ˆp] i

= −h

dˆx

. (124)

VI. Quantum mechanics in three spatial

dimensions

Diﬀerent directions in space are independent of one

another in quantum mechanics in the sense that

[ˆx

, ˆx

] = [ˆp

, ˆp

] = [ˆx

, ˆp

] = 0, if i 6= j. (125)

QM problems in three-dimensional space can be solved

by using methods of both wave and matrix mechanics.

6.1 Separation of variables in rectangular coor-

dinates:

If a Hamiltonian

H =

is separable

into terms referring to diﬀerent dimensions in rect-

angular coordinates, its energy eigenstates can be

factorized into the product states |n = n

i =

i|n

i, one in each spatial dimension:

H|n

i = (E

+ E

)|n

i. (126)

The method works for other operators too:

ˆp

i = (p

+ p

)|p

i. (127)

Examples:

(a) 3-dim HO: E

= ~ω(n+3/2), n = n

(b) Free particle: p

= p

+ p

in the state

: The discrete mo-

mentum spectrum is deﬁned by the quantum

numbers n

= k

/π = 1, 2, ....

6.2 Separation in spherical coordinates:

Kinetic energy: In the spherical coordinates r =

(r, θ, φ),

separates into a radial part and an an-

gular part:

= ˆp

where

L =

r ×

p =



ˆx ˆy ˆz

ˆp



(128)

is the orbital angular momentum operator. Note

that the position and momentum operators in a

typical term of

L such as e

ˆyˆp

commute because

they refer to diﬀerent directions in space. However,

L is quite a diﬀerent type of operator from

r or

because its components

do not commute with

one another:

[

] = i~

, (i, j, k in RH order). (129)

On the other hand,

[

] = 0, for any component i. (130)

The two commuting operators

can be used

to deﬁne the simultaneous eigenstate |ℓmi:

|ℓmi = ~

ℓ(ℓ + 1)|ℓmi,

|ℓmi = ~m|ℓmi, (131)

where ℓ and m are called the orbital (angular mo-

mentum) and magnetic quantum numbers, respec-

tively. This eigenstate describes completely an an-

gular state in spherical coordinates. The resulting

angular wave function

hθφ|ℓmi = Y

ℓm

(θ, φ) (132)

is called a spherical harmonic.

In the position represention, the

equation can

be written as the diﬀerential equation



∂

∂φ



ℓm

(θ, φ) = mY

ℓm

(θ, φ). (133)

The solutions in the variable φ are

ℓm

(θ, φ) = f(θ)e

imφ

. (134)

These solutions are periodic in φ with a period of

2π:

im(φ+2π)

= e

imφ

. (135)

Hence m must be an integer (0, ± 1, ±2, ...). It will

turn out that ℓ is a nonnegative integer bounded

from below by |m|.

The kinetic energy operator in spherical coordi-

nates is

T =

ˆp

, (136)

Central po te ntial: Central potentials are spheri-

cally symmetric and depend only on the radial dis-

tance from the center of potential (or force):

V = V (r) 6= f (θ, φ). (137)

For central potentials, the Hamiltonian separates

in the radial and angular coordinates

H = (

V ) +

. (138)

As a result, the wave function of deﬁnite energy E

factorizes as follows:

Eℓm

(r) = hr|Eℓmi = R

Eℓ

(r)Y

ℓm

(θ, φ). (139)

The radial factor satisﬁes the Schr¨odinger equation





ˆp

ℓ(ℓ + 1)



+ V (r)



Eℓ

(r)

= ER

Eℓ

(r), (140)

where

ˆp

= −~

∇

= −~





= −~





r. (141)

Note that the wave function R

Eℓ

(r) depends on

ℓ as well as the energy eigenvalue E. An en-

ergy eigenstate in three-dimensional space has only

three state labels.

Laplace equation: of electrostatics is also im-

portant in QM because its solutions describe the

quantum states of a static or resting free particle

(V = 0, p = 0 and E = 0):

∇

ψ(r) = 0. (142)

The solutions regular at the origin r = 0 are the

solid spherical harmonics (or harmonic polynomi-

als)

ℓm

(r) = r

ℓ

ℓm

(θ, φ). (143)

Solutions of the Laplace equations are sometimes

called harmonic functions.

The solutions in rectangular coordinates are sim-

pler. They are all rectangular polynomials with

terms of the form x

having the same de-

gree n = n

+ n

. All rectangular polynomials

TABLE II: Rectangular polyn omials and solid spherical har-

monics.

n Rect. Poly. Harmonic? Number Y

ℓm

0 1 Y 1 Y

1 z Y 3 Y

1m=0

x ±iy Y m = ±1

2 2z

− x

− y

Y 5 Y

2m=0

z(x ± iy) Y m = ±1

(x ± iy)

Y m = ±2

+ y

+ z

N 1 —

can be written in the spherical form with deﬁnite

ℓ, m:

ℓm

(θ, φ) =



harmonic, if ℓ = n

non − harmonic, if ℓ < n



, (144)

but only those where the power n of r is equal to

the degree ℓ of Y

ℓm

are solutions of the Laplace

equation.

Because ∇

is simple in rectangular coordinates,

the result

∇

= n

− 1)x

−2

(145)

can be used to determine if any rectangular poly-

nomial is a harmonic function. Examples are given

in Table II. Functions with deﬁnite m include:

m = ±1 : x ± iy = r sin θe

±iφ

m = 0 : z, x

+ y

. (146)

All solid spherical harmonics can be built from

these two classes of functions.

6.3 Three-dimensional harmonic oscillator:

Rectangular coordinates: The Hamiltonian of

the 3DHO is separable in rectangular coordinates:

H =

mω

. (147)

Consequently, its energy eigenstates are factorable:

i =

(ˆa

†

)

√

(ˆa

†

)

√

(ˆa

†

)

√

|000i. (148)

All possible rectangular polynomials made up of

the components ˆa

†

of the vector operator

†

can

create a 3DHO state. Their energy eigenvalues are

= (n +

)~ω, where

n = n

+ n

= 0, 1, 2... (149)

Spherical coordinates: The energy eigenstates

are

|nℓmi ∝ (

†

)

2ν

ℓ=n−2ν,m

(

†

)|000i, ν = 0, 1, ...(150)

All the states shown in Table II with x replaced by

the operator ˆa

†

, etc., are valid 3DHO states. The

degeneracy of these states is thus

D(n) =

(n + 1)(n + 2) =

ℓ=even or odd

(2ℓ + 1). (151)

6.4 Plane wave in free space:

Three-dimensional plane waves have the product

wave functions

ik·r

= e

i(k

x+k

y+k

∞

ℓ=0

ℓ0

Eℓ

(r)Y

ℓ0

(θ, φ)

∞

ℓ=0

ℓ

(2ℓ + 1)j

ℓ

(kr)P

ℓ

(cos θ) (152)

where the choice k = ke

makes explicit the axial

symmetry of the wave function about the propaga-

tion direction e

The spherical Bessel function j

ℓ

(ρ = kr) that ap-

pears is the solution of the radial equation



−

dρ



dρ



ℓ(ℓ + 1)

− 1



ℓ

(ρ) = 0. (153)

It has simple properties when ρ → 0 and ∞. The

Legendre polynomials P

ℓ

(cos θ) are those used in

electrostatics.

6.5 Hydrogen atom:

Similarity with the Bohr model: In wave me-

chanics, the energy expectation values will turn out

to satisfy the Bohr-model relations

E = T + U =

= −T,

= −

, T

2ma

, (154)

where n is the principal (or Bohr) quantum num-

ber, and a is the Bohr radius.

Diﬀerences from the Bohr model: The single

circular orbit of radius a

= n

a of the Bohr model

spreads out into states of quantum numbers n, ℓ, m,

with 0 ≤ ℓ < n. This result is consistent with the

spherical symmetry of the electrostatic attraction

between the electron and the proton. The degen-

eracy of states is then

d(n) =

n−1

ℓ=0

(2ℓ + 1) = n

. (155)

Wave functions: are

nℓm

(r) = R

nℓ

(r)Y

ℓm

(θ, φ). (156)

The radial functions R

nℓ

are functions of

ρ = κr, where

= −E. (157)

Then

nℓ

(ρ) =

nℓ

(ρ)

= ρ

ℓ

−ρ

v(ρ) (158)

can be written in two other forms with each func-

tion satisfying a diﬀerent diﬀerential equation:

∇

nℓ

(r) =



ℓ(ℓ + 1)

−





+ κ



nℓ

′′

nℓ

(ρ) =



ℓ(ℓ + 1)

−

+ 1



nℓ

ρv

′′

(ρ) + 2(ℓ + 1 − ρ)v

′

+ (ρ

− 2ℓ − 2)v = 0. (159)

where ρ

= 2/κa and v

′

(ρ) = dv(ρ)/dρ. All solu-

tions have

= 2n, giving E

, (160)

in agreement with the Bohr model.

Polynomials v

(ρ): The factor v(ρ) in Eq.(158)

is a polynomial of ρ:

(a) v

(ρ) = 1 gives solutions with ℓ = n − 1,

(b) v

(ρ) = ρ − c(a constant) gives solutions with

ℓ = n −2,

(ρ) =

j=0

, (161)

give solutions with ℓ = n − 1 − ν.

Hydrogen states: The properties of the states of

the hydrogen atom are summarized in Table III.

The spectroscopy notation used is:

[ℓ] = s, p, d, f, ... for ℓ = 0, 1, 2, 3, ... (162)

The spectroscopic symbol for a quantum state is

2s+1

[ℓ]

, giving the spin degeneracy 2s + 1, the

orbital angular momentum [ℓ] and the total spin

quantum number j.

VII. Spin and statistics

The study of orbital angular momentum states leads

us to intrinsic spins and quantum statistics.

TABLE III: States of the hydrogen atom. The radial wave

functions are expressed in terms of the dimensionsless distance

˜r = r/a.

n E

ℓ m Spect.Symbol R

nℓ

(˜r)

1 E

0 0 1s e

−˜r

2 E

/4 1 0, ±1 2p ˜re

−˜r/2

0 0 2s (1 − ˜r/2)e

−˜r /2

n E

ℓ

max

= n − 1 n[ℓ

max

] ˜r

ℓ

max

−˜r/n

ℓ n[ℓ]

7.1 Angular momentum:

The orbital angular momentum operator

L =

r ×

p =



ˆx ˆy ˆz

ˆp



(163)

is very diﬀerent from

x and

p because its compo-

nents do not commute with one another:

[

] = i~

, or

L ×

L = i~

L. (164)

However, every component

commutes with

. (165)

The angular momentum states |ℓmi are the simul-

taneous eigenstate of the two commuting operators

|ℓmi = ~

ℓ(ℓ + 1)|ℓmi,

|ℓmi = ~m|ℓmi, (166)

The nature of these states can be studied by using

the

Ladder o perators: for angular momentum states

± i

. (167)

These operators have the structure of

x ± iy = r sin θe

±iφ

(168)

that in wave functions have the magnetic quan-

tum numbers m = ±1. Hence

can be called

the m = ±1 components of

L in spherical coor-

dinates. Operators and states are both matrices,

though states are not square matrices. They can

be multiplied together. In spherical coordinates,

the m value of a product of functions or matrices

is the sum of the m values of the individual func-

tions or matrices. Hence we expect that

|ℓmi

to be a state of magnetic quantum number m ± 1,

respectively.

Before demonstrating this result, let us ﬁrst note

however that the ladder operators, like their con-

stituent rectangular components of

, com-

mute with

itself. Hence they do not change the

orbital quantum number ℓ when acting on the state

|ℓmi. However, they too do not commute with

[

] = ±~

. (169)

These commutators can be used to show explicitly

that

is the step-up operator for m, increasing

its value by 1, while the step-down operator

−

decreases m by 1:

|φ

i =

|ℓmi = A

ℓm

|ℓm ± 1i. (170)

ℓm

can be found by calculating the normalization

hφ

|φ

i using

−

∓

± ~

. (171)

The result is

|ℓmi = ~

ℓ(ℓ + 1) − m(m ± 1) |ℓm ± 1i. (172)

Possible ℓm values: m

is bounded because

i < h

i, or m

< ℓ(ℓ + 1). (173)

Hence m

min

≤ m ≤ m

max

. From Eq.(172):

|ℓm

max

i = 0 ⇒ m

max

= ℓ,

−

|ℓm

min

i = 0 ⇒ m

min

= −ℓ. (174)

So the total number of steps N on the ladder is

max

− m

min

= 2ℓ = N − 1. (175)

Assuming that ladders with any number of steps

exist in nature, one ﬁnds

Odd N = 2ℓ + 1 ⇒ ℓ =

N − 1

= integer,

Even N = 2s + 1 ⇒ s =

N − 1

= half integer.

(176)

The orbital quantum numbers ℓs are integers be-

cause they are also the integer degrees of the har-

monic polynomials Y

ℓm

. The half-integer s’s are

the spin quantum numbers of an intrinsic spin an-

gular momentum

S. The word “spin” is also used

for any angular momentum.

Angular-momentum representations:

and therefore also

are N × N matrices

for those states with N steps in their ladder.

Example: For spin s = 1/2, the ladder has only

two steps, with spin states “up” and “down”.

Then the spin operators are 2 × 2 matrices:

(3/4)~

diag(1, 1). The Pauli spin matrices are

the components of

σ = 2

S/~:



1 0

0 −1



, σ



0 1

1 0



, σ



0 −i

i 0



. (177)

7.2 Addition of angular momenta:

Product states: Two angular momentum opera-

tors

L and

S are independent if [

] = 0. The

simultaneous eigenstates of the commuting opera-

tors

are the product states

|ℓm

ℓ

i = |ℓm

ℓ

i|sm

i. (178)

They have the degeneracy

d(ℓ, s) = (2ℓ + 1)(2s + 1). (179)

Coupled states: Two independent angular mo-

menta

L and

S can be added to the total angular

momentum:

J =

L +

S. (180)

The four operators

can be shown to

commute among themselves. Hence their simulta-

neous eigenstates |ℓsjmi exist:

|ℓsjmi = ~

ℓ(ℓ + 1)|ℓsjmi,

|ℓsjmi = ~

s(s + 1)|ℓsjmi,

|ℓsjmi = ~

j(j + 1)|ℓsjmi,

|ℓsjmi = ~m|ℓsjmi. (181)

Clebsch-Gordan coeﬃcients: The two distinct

ways (product or coupled) of specifying the states

of two angular momenta are related by a change of

representation:

|ℓsjmi =

ℓ

|ℓm

ℓ

ihℓm

ℓ

|ℓsjmi,

|ℓm

ℓ

i =





|ℓsjmihℓsjm|





|ℓm

ℓ

i, (182)

where each sum involves only one summation in-

dex because m

ℓ

+ m

= m. The transformation

(Clebsch-Gordan) coeﬃcients are chosen real, and

therefore

hℓsjm|ℓm

ℓ

i = hℓm

ℓ

|ℓsjmi. (183)

Examples: Two spin 1/2 states can be coupled to

a total spin of S = 1 or 0:

SM = 11i = |↑↑i, |

1 − 1i = |↓↓i,

10i =

√

|↑↓+↓↑i;

00i =

√

|↑↓−↓↑i, (184)

where ↑ (↓) denotes the spin up (down) magnetic

state of the spin 1/2 system.

Vector model: The length J of the vector sum

J = L + S satisﬁes a triangle inequality:

|L − S| ≤ J ≤ L + S, but

min

= |ℓ − s| ≤ j ≤ j

max

= ℓ + s, (185)

for quantum angular momenta involves the quan-

tum numbers.

Construction of total angular momentum

states: The jth ladder of states has 2j + 1 steps.

The topmost state m = ℓ + s of the longest ladder

max

= ℓ + s is simply

|ℓs ℓ + s ℓ + si = |ℓℓssi. (186)

Then step down to ﬁnd the remaining states on the

ladder:

−

|ℓsjmi = ~

j(j + 1) − m(m −1) |ℓsjm − 1i

= (

−

)

ℓ

|ℓm

ℓ

× hℓm

ℓ

|ℓsjmi, (187)

where the RHS is a sum over two terms involving

−

|ℓm

ℓ

i and

−

|sm

i, respectively.

The topmost state of the next ladder with j =

ℓ + s − 1 is the remaining or unconstructed state

with m = ℓ + s − 1. It is constructed by orthonor-

malization from |ℓs ℓ + s ℓ + s −1i, using the prod-

uct representation. By convention, its overall phase

is taken to be the Wigner (aka Condon-Shortley)

phases. The other states on the ladder are found

by stepping down. This construction method works

for all the remaining j ladders.

7.3 Identical particles in quantum mechanics:

Spin-statistics theorem: Identical particles of

integer (half integer) spins satisfy the Bose-Einstein

(Fermi-Dirac) statistics. They are called bosons

(fermions).

Two identical particles are indistinguishable in

their probability densities

ρ(2, 1) = ρ(1, 2), (188)

but their wave functions may diﬀer by a negative

sign

Ψ(2, 1) = ±Ψ(1, 2) for



bosons

fermions

. (189)

Boson (Fermion) wave functions are symmetric (an-

tisymmetric) in the particle labels.

Examples: If α, β are single-particle (s.p.) states:

αβ

(1, 2) ∝ [ψ

(1)ψ

(2) ± ψ

(2)ψ

(1)] . (190)

TABLE IV: electronic structure of the elements. The quan-

tum numbers of some atomic ground states are also given as

the spectroscopic symbol

2S+1

[L]

Z Element Spatial Conﬁguration Ground State

1 H 1s

1/2

2 He (1s)

2 1

3 Li (He)2s

1/2

4 Be (He)(2s)

2 1

5 B (He)(2s)

1/2

6 C (He)( 2s)

(2p)

7 N (He)(2s)

(2p)

8 O (He)(2s)

(2p)

9 F (He)(2s)

(2p)

5 2

3/2

10 Ne (He)(2s)

(2p)

6 1

If β = α is the same s.p. state:

Bosons : Ψ

αα

(1, 2) = ψ

(1)ψ

(2) 6= 0, but

Fermions : Ψ

αα

(1, 2) = 0. (191)

These important results are called:

Bose-Eins tein condensation: The ground state

of a system of identical bosons is one where every

boson is in the lowest energy single-particle state.

Pauli exclusion principle: No two identical

fermions can populate the same single-particle

state.

Many identical particles: The wave function of

N identical bosons is symmetric under any per-

mutation of the N particle labels. For identical

fermions, the wave function changes sign under an

odd permutation of the N particle labels, but re-

mains unchanged under an even permutation.

7.4 The Periodic Table:

Electronic structure of the elements: is sum-

marized in Table IV:

Helium atom: Its two electrons with spins point-

ing ↑ and ↓ can populate the same 1s spatial state.

They complete the principal qu. no. n = 1 shell.

The shell n can accomodate 2n

electrons.

To get the lowest energy, both electrons in the He

ground state (GS) occupy the lowest s.p. spatial

state nℓm = 100. The total wave function

(1, 2) = ψ

100

)ψ

100

)χ

SM=00

(1, 2)

= −Ψ

(2, 1) (192)

must be antisymmetric in the two fermion labels 1

and 2. Hence the spin “function” χ (not a “state”)

for the total intrinsic spin must be the antisymmet-

ric function with S = 0.

The atomic Hamiltonian

H(1, 2) = −

−

(193)

gives the GS energy (with Z = 2)

(He) ≈ −Z

+ E

≈ −109 eV + 30 eV. (194)

where E

= 13.6 eV is the Rydberg energy.

Closed subshells: The m

value of the spin

state |sm

i changes when the quantization axiz e

changes direction. However, if all m

states are

ﬁlled by 2s + 1 particles, they will remain ﬁlled

when the arbitrarily chosen direction e

changes.

Hence the total M

is 0, and this unique state of

2s + 1 particles has S = 0. Similarly, the complete

ﬁlling of the 2ℓ + 1 states of diﬀerent m

ℓ

gives a

unique spherical symmetric spatial system of 2ℓ + 1

particles that has L = M

= 0. This is why the

ground states of He, Be and Ne are all

states.

Spin-orbit interaction: The atomic Hamilto-

nian contains a relativistic term A

L ·

S, A

0. It gives a negative (attractive) energy of

−(A

/2)(ℓ+1) for the s.p. state with the total an-

gular momentum j = ℓ −1/2. For the j = ℓ + 1/2

state on the other hand, the energy is repulsive,

namely (A

/2)ℓ. This explains why the B ground

state in the spatial 2p conﬁguration has the quan-

tum numbers

1/2

of the lowest-energy single elec-

tron state outside the closed n = 1 shell and the

closed 2s subshell.

The F ground state is one electron short of the

closed n = 2 shell. The lowest energy is obtained by

putting the missing electron or hole in the higher

j = 3/2 state to give a

3/2

ground state, i.e.,

leaving a higher state unpopulated by populating

all the lower-energy states in the 2p subshell.

7.5 Free Fermi gas:

Box norm al ization: For a particle extending over

all space, the plane-wave wave functions e

ik·r

are

not square-integrable. This feature causes awk-

wardness including the appearance of Dirac δ-

functions. The problem can be circumvented by

subdividing space into identical cubes of sides L

satisfying a periodic boundary condition (BC). The

wave functions in each identical cube can then be

normalized.

One-dimensional space: The periodic BC

(x + L) = e

(x+L)

= φ

(x) = e

(195)

gives the discrete spectrum

L = 2πn

, with n

= 0, ±1, ± 2, ... (196)

and the normalized wave functions

(x) =

√

. (197)

The corresponding states |n

i can be labeled by the

qu. no. n

. They form a discrete basis for wave

functions in the representative 1D box.

Three-dimensional s pace: The normalized 3D

wave functions in the representative cube are

hr|ni =

3/2

ik·r

, where k

2π

, (198)

and n = n

Fe rm i gas: For identical fermions, each spatial

state can be populated at most singly (the spin

degeneracy being treated separately). Then n is

called a (vector) occupation number. A free parti-

cle of mass m has energy E

(~k)

+ m

So the lowest energy state of a free Fermi gas of

N fermions in the cube is one where the states of

lowest k or occupation number n = |n| are popu-

lated. With spatial isotropy (spherical symmetry),

the occupation pattern is that of a sphere in n or

k space of radius k = k

N =

dN =

n =



2π







dk =



2π



4π

. (199)

Here



2π



4πk

(200)

is called the density of states in k-space. The radius

of the Fermi sphere in k-space

= (6π

)

1/3

, where ρ

, (201)

is called the Fermi momentum. It does not depend

on the size of the cube. For electrons with their spin

degeneracy d(s) = 2, the total electron number is

= 2N.

The energy per particle in a free Fermi gas is

= hE

gas





R





(3/5)ǫ

, if nonrelativistic;

(3/4)~ck

, if very relativistic.

(202)

The nonrelativistic energy

(203)

at the surface k = k

of the Fermi sphere is called

the (NR) Fermi energy.

Chandrasekhar limit: A white dwarf star con-

tains Z protons, Z electrons and A − Z neutrons.

The main contributions to its energy are the repul-

sive kinetic energy of the highly relativistic elec-

tron gas and the attractive gravitational potential

energy of its mass M . Both energies are inversely

proportional to R, the radius of the star, but they

have diﬀerent M dependences:

Tot

= b

4/3

− c







> 0, unbound,

= 0, critical,

< 0, collapse.

(204)

The white dwarf will collapse gravitationally if its

mass exceeds the critical mass





3/2

. (205)

VIII. Approximation methods for time-

independent problems

Few problems in quantum mechanics are exactly solv-

able. We ﬁrst consider some approximation methods for

Hamiltonians that are time-independent.

8.1 Time-independe nt perturbation theory:

2 × 2 matrices: can be diagonalized exactly. The

Hermitian matrix

H =



0 b

b a



, (206)

where both a and b are real, has the eigenvalues

and orthonormal eigenvectors

1,2

(a ∓ ∆E),

∆E =

+ 4b

≈ a + 2

;



cos θ

sin θ



, ψ



−sin θ

cos θ



tan θ =

≈ −

. (207)

Many problems of physical interest involving only

two dominant states can be solved approximately

by this method.

Perturbation theory: Let the Hamiltonian

H =

′

contains a main part

whose energy

eigenvalues E

(0)

, eigenfunctions φ

or eigenstates

|ni are already known. If a complicated but weak

perturbation

′

is now added to the system, the

eigenvalue equation

(

+ λ

′

)ψ

= E

(208)

can be solved by a systematic expansion in powers

of λ that can be set back to its numerical value of

1 at the end of the expansion.

This λ expansion is done in both E

and ψ

(

+ λ

′

)



(0)

+ λψ

(1)

+ ...





(0)

+ λE

(1)

+ ...



(0)

+ λψ

(1)

+ ...



. (209)

The equation is solved by equating terms of the

same power of λ on the two sides of the equation:

(0)

= E

(0)

⇒ ψ

(0)

= φ

: E

(1)

= H

′

(1)

= −

k6=n

′

(0)

− E

(0)

where H

′

= hk|

′

|ni. (210)

8.2 Calculation of E

(1)

Many perturbations are made up of sums of fac-

torable terms of the type

′

= Q(r)

A (211)

involving a radial factor Q(r) and an angle-spin fac-

tor

A for a system with intrinsic spin S. Then

(1)

= Q

nℓ

where Q

nℓ

∞

nℓ

(r)|

Q(r)r

dr,

α = ℓm

ℓ

or ℓsjm. (212)

Examples:

L ·

Si =

−

ℓsjm

+ 2

ℓm

ℓ

= (m

ℓ

+ 2m

)~,

ℓsjm

j(j + 1)

J ·

ℓsjm

. (213)

8.3 Variational method:

Variational principle: For any wave function ψ:

= h

≡

hψ|

H|ψi

hψ|ψi

≥ E

(GS). (214)

The approximate wave function ψ gives a higher

energy when it contains excited-state components

of higher energies.

Variational wave function: is any wave func-

tion carrying a number of parameters b that can be

varied in order to minimize the energy

E(b) =

hψ(b)|

H|ψ(b)i

hψ(b)|ψ(b)i

. (215)

The energy minimum E(b

) where dE(b)/db = 0 is

the best variational estimate because it is closest to

the true ground-state (GS) energy E

He atom: The ee repulsion between the two elec-

trons in the He atom can be eliminated approxi-

mately by reducing the actual nuclear charge Z = 2

to the eﬀective charge z ≈ 27/16.

Hartree mean-ﬁeld screening: In an atom with

Z electrons, the mutual ee interactions can be re-

placed by a mean ﬁeld experienced by each electron

characterized by an eﬀective nuclear charge of

z(r) =



Z, for r → 0

1, for r → ∞

(216)

8.4 WKB approximation:

Since any complex number can be written in the

amplitude-phase form Ae

iφ

, one can look for a so-

lution of the 1D Schr¨odinger wave equation in this

amplitude-phase form. The WKB approximation

is a semi-classical result used between the two clas-

sical turning points x

and x

of the potential V (x)

where the local wave number

k(x) =

[E −V (x)] (217)

is real. It is obtained by ignoring the A

′′

(x) =

A(x)/dx

term. Then

ψ(x) = A(x)e

iφ(x)

≈ ψ

W KB

(x) =

k(x)

±i

k(x

′

)dx

′

. (218)

WKB quantization: At a turning point where

the potential has a ﬁnite slope, the wave func-

tion can penetrate into the classically forbidden re-

gion where the local wave number becomes purely

imaginary k → iκ. Under this analytic continua-

tion, the amplitude factor becomes complex, A ∝

√

k → e

−iπ/4

√

κ, thereby contributing an addi-

tional phase of −π/4. The resulting WKB-modiﬁed

Bohr-Sommerfeld quantization condition is

k(x)dx =



n −



π, n = 1, 2, ..., (219)

where m is the number of turning points where the

potential is not an ∞ wall. The resulting WKB

energies are often quite good. They even agree

with the exact energies in the case of the one-

dimensional harmonic oscillator potentials.

IX. Approximation methods for time-

dependent problems

Time-dependent problems are more complicated partly

because one has to keep track of the time evolution of a

state, but mostly because the Hamiltonian itself might

be time-dependent.

9.1 The time-dependent two-state problem:

is exactly solvable for the time-independent Hamil-

tonian (206). Let a state be prepared initially as

|ψi ≡ |Ψ(t = 0)i = d

|ψ

i + d

|ψ

i, (220)

where |ψ

i are the time-independent eigenstates of

H. Then it evolves at time t into

|Ψ(t)i = e

−i

Ht/~

|ψi = d

|Ψ

(t)i + d

|Ψ

(t)i

= d

−iE

t/~

|ψ

i + d

−iE

t/~

|ψ

i. (221)

Unperturbed basis: It is often useful to express

|Ψ(t)i in terms of the time-dependent unperturbed

basis states

|Φ

(t)i = e

−iE

(0)

t/~





|Φ

(t)i = e

−iE

(0)

t/~





(222)

of energies E

(0)

= 0 and E

(0)

= a:

|Ψ(t)i = c

(t)|Φ

(t)i + c

(t)|Φ

(t)i. (223)

The probabilities |c

(t)|

can be found by using the

energy eigenstates



cos θ

sin θ



, ψ



−sin θ

cos θ



(224)

in Eq.(221):

(t)|



cos θ − d

−i∆ωt

sin θ



(t)|



sin θ + d

−i∆ωt

cos θ



, (225)

where ∆ω = (E

− E

)/~. These probabilities are

in general functions of t. Indeed, their time depen-

dences betray the fact that the φ

are not energy

eigenstates, for otherwise they will be stationary

states with time-indepedent probabilites.

The expansion (223) is particularly interesting

when the basis states |φ

i are themselves physically

observable, as happens in

Neutrino oscillations: Neutrinos come in three

distinct varieties or “ﬂavors”: ν

, ν

, associated

with the electron e, muon µ and tau τ, respec-

tively. Although they are physically distinct from

one another, they are not mass eigenstates (energy

eigenstates at zero momentum). The mass eigen-

states are linear combinations of these neutrino ba-

sis states of diﬀerent ﬂavors. For this reason, the

ﬂavor probabilities are not constants of motion, but

instead they oscillate in time.

Solar neutrino oscillations: The neutrinos ini-

tially produced in the solar interior where masses

are converted to energy are ν

= φ

, and not

= φ

, i.e., c

(t = 0) = d

sin θ + d

cos θ = 0,

or d

= cos θ, d

= −sin θ. Allowing only oscil-

lation into ν

for simplicity, the probabilities for

ﬁnding ν

and ν

at a later time t are

(t)|

= 1 − sin

2θ sin

(∆ωt/2),

(t)|

= sin

2θ sin

(∆ωt/2). (226)

The neutrinos may have very small masses, so that

≈ p

c +

. (227)

The the probability for ﬁnding ν

of energy E at a

distance L = ct from the Sun is

(t)|

= 1 − sin

2θ sin

[A(∆m

)L/E];

∆m

= |m

− m

A =

4~c

= 1.27 GeV(eV)

−2

−1

. (228)

Experimental data for solar neutrinos of energies

broadly distributed around a mean value of E =

3 MeV give

(∆m

= 7.3 × 10

−5

(eV)

, tan

θ = 0.41. (229)

9.2 Time-dependent perturbation theory:

Expansion in exact eigenstates: Suppose

H 6=

H(t), and its energy eigenvalues E

and eigenstates

|ψ

i are known. Then any arbitrary quantum state

at time t can be expanded in terms of these energy

eigenstates:

|Ψ(t)i =

j=1

|Ψ

(t)i =

−iE

t/~

|ψ

i. (230)

Expansion in basis states: This same state can

also be expanded in terms of a known set of basis

states |Φ

(t)i = e

−iE

(0)

|φ

i that are eigenstates of

a known Hamiltonian

of energies E

(0)

|Ψ(t)i =

(t)|Φ

(t)i =

(t)e

−iE

(0)

t/~

|φ

i.(231)

It happens occasionally that both |ψi and |φi states

are physically observable. Then it is of interest to

express one observational probabilities in terms of

the other expansion coeﬃcients. For example,

(t)|



j=1

−iE

t/~



, (232)

is a generalization of what is done for neutrino os-

cillations to higher dimensions. Here U, with

= hφ

|ψ

i, (233)

is an N ×N unitary matrix if both sets of states are

normalized. U is then made up of the eigenstates

|ψ

i stored columnwise.

TD perturbation: The expansion (231) can also

be used if

H =

′

contains a time-dependent

(TD) perturbation

′

. Unfortunately, the TD ex-

pansion coeﬃcients c

(t) must be obtained more

laboriously by actually solving the TDSchEq

∂

∂t

|Ψ(t)i = (

+ λ

′

)|Ψ(t)i. (234)

The resulting diﬀerential equations (DEs) of mo-

tion for c

(t),

i~ ˙c

(t) = λ

′

iω

(t), (235)

are a set of coupled ﬁrst-order DEs that can be

solved easily in a PC. Here ω

= (E

(0)

− E

(0)

)/~.

Systematic TD perturbation theory: is con-

cerned with the Taylor expansion in λ (eventually

set to 1):

(t) = c

(0)

(t) + λc

(1)

+ ... (236)

Eq.(235) can then be separated term by term into

separate equations for c

(n)

terms : i~ ˙c

(0)

(t) = 0 ⇒ c

(0)

(t) = const,

terms : i~ ˙c

(1)

(t) =

′

iω

(0)

, ... (237)

These ﬁrst-order DEs in time can be integrated di-

rectly, making it easy to calculate ﬁrst-order wave

functions in TD perturbation theory.

Sinusoidal perturbation: Let the sinusoidal per-

turbation

′

V cos ωt, 0 ≤ t ≤ t

only, (238)

be turned on for a ﬁnite period of time. After the

perturbation has ceased, a state initially in |φ

will ﬁnd itself in the state |φ

i, b 6= a, with the

probability



(1)

(t ≥ t

)



≈



(z)

, (239)

where z =

(ω

± ω)t

(z) = sin(z)/z,

= [E

(0)

− E

(0)

]/~. (240)

Thus most of the excitations are close to the two

states |φ

i with energies E

(0)

= E

(0)

± ~ω.

Golden rule: The transition rate during the in-

teraction lasting a time t

from a state a to a group

of states b is thus

≡

∞

−∞

(1)

d(~ω

)

≈

2π



, (241)

where ρ

= dN

/dE

is the density of ﬁnal states.

Example: The Bohr transition of an atomic elec-

tron from a higher atomic state b to a lower state

a with the emission of a photon of energy ~ω =

(0)

−E

(0)

is caused by the electric dipole interac-

tion

′

V cos(ωt). (242)

Here

V = −ǫ·

is the electric-dipole interaction

between the electric dipole operator

D = q

r of the

atomic electron and the electric ﬁeld E = ǫE

the emitted photon. The latter can be obtained

from the photon energy density in a cube of side L:

~ω

, (243)

where ǫ

is the permittivity of free space. When

′

is used in the Golden Rule, the resulting transi-

tion rate is the Einstein coeﬃcient for spontaneous

photon emission by the atom:

A = W





4πǫ



, (244)

9.3 Sudden, adiabatic and impulsive perturba-

tions:

In sudden perturbation, the Hamiltonian

changes suddenly by a ﬁnite amount to the ﬁnal

value

. If the change is so sudden that the state

|ψi at t = 0 has not changed, the state at times

t > 0 is just

|Ψ(t)i = e

−i

t/~

|ψi

−iE

(a)

t/~

|φ

ihφ

|ψi

|Φ

(t)ihφ

|ψi. (245)

Adiabatic perturbation: When a time-

dependent Hamiltonian

H(t) changes very slowly

in time, we expect the instantaneous eigenvalues

(t)

(t) and eigenstates |ψ

(t)

i deﬁned by the eigen-

value equation

H(t)|ψ

(t)

i = E

(t)

(t)|ψ

(t)

i (246)

to give a good approximation. That is

(t) ≈ E

(t)

(t), (247)

a result known as the adiabatic theorem.

If one watches the system for a suﬃciently long

time, however, sooner or later additional features

will appear. They can be described conveniently

by using the time-dependent basis |ψ

(t)

|Ψ(t)i =

(t)e

−(i/~)

′

)

′

)dt

′

|ψ

(t)

i. (248)

The new features include the appearance of a ge-

ometrical phase ϕ

geo

and of transitions to other

states k 6= n.

Impulsive perturbation: An impulsive pertur-

bation contains a δ-function in time:

′

(x, t) =

A(x)δ(t). (249)

The TDSchEQ can be solved exactly to show that

the δ-function makes the state discontinuous at t =

|Ψ(0

)i =

1 −

2i~

−1

1 +

2i~

|Ψ(0

−

)i, (250)

X. Applications

In this chapter, we see how quantum problems of great

physical interest have been solved exactly or approxi-

mately.

10.1 Solving special m atrix eigenvalue problems:

Special matrices of arbitrarily large dimensions of

great physical interest can sometimes be solved an-

alytically by symmetry considerations.

To begin, it is easy to verify by inspection the eigen-

values and eigenvectors of the Pauli matrix σ



0 1

1 0











0 1

1 0



−1



= −



−1



. (251)

Indeed, three simple rules often go a long way in

ﬁnding the eigenvectors and eigenvalues of certain

Hermitian matrices:

(a) The vector v

= (1, 1, ..., 1), where all compo-

nents are 1, is an eigenvector of a matrix where

the matrix elements of each row is a permu-

tation of those of the ﬁrst row. Its eigenvalue

is just the sum of the matrix elements of

any row.

(b) Eigenvectors of a Hermitian matrix can be

made othogonal to one another.

the matrix.

Example 1: The vector v

= (1, 1, 1) is an eigen-

vector of the 3 × 3 matrix ((0,1,1),(1,0,1),(1,1,0)),

with eigenvalue E

= 2. The three vectors v

(2, −1, −1), v

= (−1, 2, −1), v

= (−1, −1, 2) are

all orthogonal to v

, but not linearly independent

of one another. Each has the eigenvalue -1. The

orthogonal eigenvectors can be taken to be v

, v

and v

= (v

− v

)/3 = (0, 1, −1).

Example 2: The vectors v

, v

and v

are also

the orthogonal eigenvectors of the Hermitian ma-

trix ((1,1,1),(1,1,1),(1,1,1)), with the eigenvalues 3,

0, and 0, respectively. This is a special case of the

general result that two matrices M

and M

have

common eigenvectors if

= M

+ aI, (252)

where I is the identity matrix. Since the parame-

ter a can be complex, we see that non-Hermitian

matrices can also have orthogonal eigenvectors.

Example 3: The arbitrarily large N × N matrix

whose matrix elements are all 1’s has one “collec-

tive” eigenvector v

= (1, 1, ..., 1) with eigenvalue

N. The remaining eigenvalues are all 0. The or-

thogonal eigenvectors can be chosen to be v

(N −1, −1, −1, ..., −1), v

= (0, N −2, −1, ..., −1),

..., v

= (0, 0, ..., 0, 1, −1).

BCS theory of superconductivity: can be un-

derstood conceptually by using a large dimensional

N × N matrix Hamiltonian with the same small

negative matrix element −ǫ everywhere. All eigen-

states have energy 0, except the collective eigen-

state v

= (1, 1, ..., 1) that has energy ∆ = −ǫN .

The energy gap ∆ that separates the ground state

from the excited states makes it hard for the sys-

tem to be excited. If the ground state contains

current-carrying electrons, then at suﬃciently low

temperatures (below a certain critical temperature

), these electrons cannot lose energy by inelastic

collisions with the crystal lattice in the conductor.

When this happens, both dissipation and resistiv-

ity vanish, and the medium becomes superconduct-

ing. (BCS = Bardeen, Cooper and Schrieﬀer who

discovered this fundamental theory of superconduc-

tivity.)

10.2 Stark eﬀect in an external electric ﬁeld:

A charge q moved a distance z against a constant

external electric ﬁeld E = Ee

has the (dipole) in-

teraction energy

′

= −q

E.dr = −qEz. (253)

The resulting perturbing Hamiltonian is

′

= −E

, where

D = q

r (254)

is called the electric dipole operator of the charge.

This operator is odd in parity, changing sign when

z → −z. Hence all ﬁrst-order energies vanish:

(1)

= hn|

′

|ni = 0. (255)

Since z = r cos θ is independent of the azimuthal

angle φ,

has magnetic quantum number m = 0.

Hence

′

has nonzero matrix elements only be-

tween two opposite-parity states i, j with the same

magnetic quantum number: m

= m

Example: In the n = 2 shell of the hydrogen atom,

there are four states: nℓm = 200; 210, 21 ± 1.

′

connects only the two opposite-parity m = 0 states.

For these two states alone, it takes the form of a

2 × 2 matrix b((01), (1, 0)) whose eigenstates are

1,2

= ∓b : ψ

1,2

√

(φ

200

± φ

210

) . (256)

Their energies E

1,2

thus move away from the undis-

turbed energy of the two states with m = ±1 by an

amount proportional to b, which is proportional to

E. [The state |ℓmi has parity (−1)

ℓ

10.3 Aharonov-Bohm eﬀect:

Gauge transformation: In classical electromag-

netism (EM) the electric and magnetic ﬁelds are

uniquely deﬁned, but the EM potential (the scalar

potential ϕ and vector potential A) are not unique.

They can change by a gauge transformation

A → A

′

= A + ∇Λ,

ϕ → ϕ

′

= ϕ −

∂Λ

∂t

, (257)

where Λ is any scalar ﬁeld, without changing the

EM ﬁelds

B = ∇ × A,

E = −∇ϕ −

∂A

∂t

. (258)

Relativistic electro dynamics: The basic dy-

namical variable in classical mechanics is the 4-

momentum (p, iE/c). To add EM, one needs an

EM 4-vector. There is a unique qualifying candi-

date: the EM 4-potential (A, iϕ/c). Hence electro-

dynamics for a particle of mass m and charge q can

be built up from the sum (p − qA, i(E − qϕ)/c).

This combination transforms like a 4-vector un-

der Lorentz transformations, thus guaranteeing the

correct result in diﬀerent inertial frames. As a re-

sult, the EM 4-potential plays a more fundamental

role in quantum mechanics than the EM ﬁelds.

The energy-momentum relation for relativistic elec-

trodynamics is then

(E − qϕ)

= (p − qA)

+ m

. (259)

One can ﬁnd from this the nonrelativistic kinetic

energy T

T ≡ E −mc

≈ T

= qϕ +

(p − qA)

. (260)

Quantum electrodynamics: We are now in a

position to quantize the mechanical terms in NR

electrodynamics, leaving the EM potential unquan-

tized for simplicity:

H =

(

p − qA)

V + qϕ. (261)

Aharonov-Bohm phase: The EM interaction re-

sides primarily in the phase (or gauge) of quantum

wave functions. For the special case where ϕ = 0,

this important result can be demonstrated readily

by showing that the wave function ψ in the pres-

ence of the vector potential A diﬀers from the wave

function ψ

′

without A by an A-dependent phase g:

ψ(r) = e

′

(r). (262)

If true, the phase factor e

serves the purpose of

removing the A dependence from the Schr¨odinger

equation when moved to the left of a “gauged” fac-

tor:



∇ − qA



′

(r) = e



∇



′

(r). (263)

Since



∇



′

(r) = e



~(∇g) +

∇



′

(r), (264)

g has to satisfy the diﬀerential equation

~∇g = qA, or ~dg = qA · dr. (265)

Hence

g =

A(r

′

) · dr

′

, (266)

where the line integral is in general path-dependent.

The integral is then said to be non-integrable. This

non-integrability can be made explicit by integrat-

ing around a closed circuit c once and simplifying

the result with the help of Stokes’s theorem:

g =

A(r

′

) · dr

′

(∇ × A) · dS =

Φ, (267)

where the total magnetic ﬂux Φ across the surface

S enclosed by the circuit c depends on the magnetic

ﬁeld ∇ ×A on and inside c. Such a dependence on

properties of the system outside the path c shows

why quantum mechanics in particular, and waves

more generally, describe nonlocal phenomena that

are spread out in space.

Magnetic ﬂux quantization: Suppose the path

c is located in a region free of any EM ﬁeld, then

the wave function must return to its original value

after one complete circuit. Hence its Aharonov-

Bohm phase can only be an integral multiple of 2π:

g = 2πn =

qΦ

, or Φ = nΦ

. (268)

As a result, the magnetic ﬂux enclosed by c can

only exist in integral multiples of a quantized unit

ﬂux Φ

= h/q. Experimental measurement deter-

mines that q = −2e in superconductors, thus show-

ing that the super current is carried by pairs of

electrons, now called Cooper pairs.

Gauge interactions: Other fundamental inter-

actions also work through the phase of quantum

wave functions. Hence they are called gauge inter-

actions. Indeed, the information contained in the

phase of a wave function is usually more important

than that contained in its amplitude. A picture of

Snow White can be made to appear in our eye as

a coherent wave. One can take the phase informa-

tion from Snow White’s wave function and use it

with the amplitude part of the wave function from

a picture of Happy, one of the seven dwarfs. A

composite wave like this has been constructed the-

oretically. Believe it or not, one sees Snow White

in the resulting hybrid picture, and not the dwarf.

10.4 Magnetic resonance:

Rabi measured the magnetic moments of atomic

states by ﬁnding the Zeeman splitting of the en-

ergies of diﬀerent magnetic substates by resonance

matching. Atomic magnetic moments are tradition-

ally expressed as gµ

in units of the Bohr magneton

= e~/2m, the theoretically value g being called

the Land`e g factor. The Hamiltonian involved is

H =

′

, (269)

where

contains the usual

NR Hamiltonian

, the relativistic spin-orbit term

, and the Zeeman Hamiltonian

= −

M · B

gµ

= ω

(270)

that gives the splitting between the magnetic sub-

states of the same j. Finally a small oscillatory

magnetic ﬁeld is applied in the x-direction to give

a perturbing Hamiltonian

′

= ω

cos(ωt), where ~ω

= gµ

. (271)

This perturbation causes transitions between the

magnetic substates. Any quantum state

|Ψ(t)i =

(t)|Φ

(t)i

(t)e

−iω

|φ

i, (272)

where ω

= E

(0)

/~, can be expressed in term of the

unperturbed states |Φ

(t)i. The time-dependent

expansion coeﬃcient c

(t) then satisﬁes the TD-

SchEq

i~ ˙c

(t) =

′

iω

(t), (273)

where ω

= ω

− ω

Example: The problem can be solved approxi-

mately for c

(t) for the atomic n[ℓ]

= 2p

1/2

con-

ﬁguration that has only two magnetic substates φ

of energies E

(0)

= ±~ω

/2. For example, if the sys-

tem is initially in the upper state (c

= 1, c

−

= 0),

then the probability of ﬁnding the system in the

upper and lower states at a later time t are respec-

tively:

= |c

= cos

(ω

t/2) + sin

χ sin

(ω

t/2),

−+

= |c

−

= cos

χ sin

(ω

t/2), where (274)

(∆ω)

+ (ω

/2)

, ∆ω = ω

− ω,

sin χ =

∆ω

, cos χ =

. (275)

Magnetic resonance: As the applied frequency

ω passes the Zeeman frequency ω

, the minimum

population of the φ

state falls sharply to 0 at ω

and then rises sharply back up to ≈ 1 again, thus

giving a clear signal for the measurement of ω

The third frequency ω

controls the period τ of

oscillation of the population between the two mag-

netic substates. At full resonance (ω = ω

), one

ﬁnds τ = 4π/ω

XI. Scattering theory

Information about the dynamical properties of micro-

scopic systems like atoms, nuclei and particles can be

obtained by scattering beams of projectiles from targets

containing them, as ﬁrst demonstrated by Rutherford,

Geiger and Marsden who elucidated atomic structure by

scattering α particles from a gold foil.

11.1 The scattering cross sections:

The diﬀerential cross section

dσ

dΩ

(276)

is the angular distribution dR/dΩ of the reaction

rate R per unit luminosity L. L itself is the product

, where I is the ﬂu x or current density of the

incident beam (number of incoming particles per

second per cross sectional area of the beam, or

for a beam of uniform cross section a) and N

the number of target particles illuminated by this

beam. L depends on both beam and target. dΩ

is the diﬀerential solid angle in a suitable inertial

frame.

Example: Two beams of the same cross sectional

area A collide head-on where their paths cross each

other. Each beam is made up of bunches of N

(i =

1, 2) particles per bunch. The target can be taken

to be one bunch of N

particles in beam 2. Then

the eﬀective incident ﬂux is I = fN

/A, where f is

the frequency of collision between bunches.

Rutherford cross section: For the scattering of

α-particles of charge Z

= 2 from an atomic nu-

cleus of charge Z

, Rutherford found from classical

mechanics that in the CM frame

dσ

dΩ





sin

(θ/2)

where d =

, (277)

is the distance of closest approach (or turning

point) for head-on collision at the NR kinetic en-

ergy T

in the CM frame. Note that for backscat-

tering θ = π, the Rutherford diﬀerential cross sec-

tion d

/16 gives a direct measurement of the posi-

tion d of the classical turning point.

Total cross section: is proportional to the total

reaction rate:

σ =

dσ =

dσ

dΩ

dΩ =

. (278)

11.2 Quantum theory of scattering in a nutshell:

Partial-wave expansio n: of a plane wave into

spherical waves of good angular momentum ℓ

around the origin of coordinates has the Rayleigh

form:

(r) = e

ik·r

∞

ℓ=0

ℓ

(2ℓ + 1)P

ℓ

(cos θ)j

ℓ

(kr),

∼

r→∞

∞

ℓ=0

ℓ

(2ℓ + 1)P

ℓ

(cos θ)



−i(kr−ℓπ/2)

−

i(kr−ℓπ/2)



(279)

This is the time-independent wave function of a sta-

tionary scattering state. The time-dependent wave

function carries an additional time factor e

−iωt

where ω = ~k

/2m. The additional time factor

shows that the wave function e

−ikr

/r describes an

ingoing spherical wave collapsing towards the ori-

gin, while the wave function e

ikr

/r describes an

outgoing spherical wave expanding out from the ori-

gin.

Scattered wave: When a target particle is placed

at the origin of coordinates, the radial part j

ℓ

of the

wave function (279) will be replaced by the func-

tion R

ℓ

(r) from the SchEq with a potential in it. At

large distances r, the ingoing waves are still collaps-

ing towards the origin, and therefore do not know if

a potential is present there. In elastic scatterings,

an outgoing spherical wave has the same normal-

ization as before, and can be changed at most by a

phase, here taken to be 2δ

ℓ

, that resides in a phase

factor S

ℓ

= e

2iδ

ℓ

called an S-matrix element:

ψ(r) ∼

r→∞

∞

ℓ=0

ℓ

(2ℓ + 1)P

ℓ

(cos θ)



−i(kr−ℓπ/2)

− S

ℓ

i(kr−ℓπ/2)



= ψ

(r) + ψ

(r) where (280)

(r) ∼

r→∞

∞

ℓ=0

ℓ

(2ℓ + 1)P

ℓ

(cos θ)



ℓ

− 1

2ik



i(kr−ℓπ/2)

= f(θ)

ikr

. (281)

is the scattered wave function, the additional wave

function generated by the potential. Containing

only outgoing spherical waves, it can be written

asymptotically (r → ∞) in terms of an angle-

dependent quantity called the scattering amplitude

f(θ) =

∞

ℓ=0

(2ℓ + 1)f

ℓ

(cos θ),

where f

ℓ

− 1

2ik

iδ

ℓ

sin δ

ℓ

(282)

if the scattering is elastic, meaning that the phase

shift δ

ℓ

is real.

Flux: or probability current density is

j =

2im

(ψ

∗

∇ψ − ψ∇ψ

∗

) ≈ j

+ j

, (283)

where j

= ~k/m is the ﬂux of the plane wave, and

|f(θ)|

(284)

is the ﬂux of the scattered wave at large distances.

Diﬀerential cross se ction: A detector of area

dA = r

dΩ then measures the diﬀerential cross sec-

tion

dσ

dΩ

~k/m

· e

)

dΩ

= |f(θ)|

. (285)

Total cross section: is the result integrated over

the solid angle dΩ:

tot



dσ

dΩ



dΩ =

∗

(θ)f (θ)dΩ

ℓ

tot,ℓ

, (286)

where the two sums over ℓ, one from each f, has

been simpliﬁed to only one sum by using the or-

thogonality relation for Legendre polynomilas

−1

ℓ

(cos θ)P

ℓ

′

(cos θ)dΩ =

4π

2ℓ + 1

ℓℓ

′

, and (287)

tot,ℓ

4π

(2ℓ + 1) sin

ℓ

. (288)

Optical theorem: If δ

ℓ

is real, one ﬁnds that

Imf(θ = 0) =

ℓ

(2ℓ + 1)Imf

ℓ

(2ℓ + 1)

sin

ℓ

4π

tot

. (289)

This theorem holds even when the phase shift δ

ℓ

+ iβ

ℓ

becomes complex.

11.3 Breit-Wigner resonance:

The partial-wave total cross section σ

ℓ

ℓ, max

sin

ℓ

reaches a maximum value of σ

ℓ, max

(4π/k

)(2ℓ + 1) whenever δ

ℓ

(k) = π/2. This maxi-

mum value is called the unitarity limit.

If δ

ℓ

(E) rises through δ

ℓ

(E = E

) = π/2 siﬃciently

rapidly, a Taylor expansion about E = E

needs

to be taken only to the terms

ℓ

≈

+ (E −E

)

dδ

ℓ



. (290)

Then tan δ

ℓ

≈ −

Γ/2

E − E

, where

(dδ

ℓ

/dE)

> 0. (291)

Resonance: In the neighborhood of a resonance,

the partial-wave total cross section has the simple

E-dependence:

ℓ

= σ

ℓ, max

tan

ℓ

1 + tan

ℓ

= σ

ℓ, max

(Γ/2)

(E − E

)

+ (Γ/2)

. (292)

This function has a sharp maximum at E = E

and

falls rapidly to half its maximal value at E −E

±Γ/2. Hence Γ is called the resonance width (or

full width at half maximum, FWHM).

Decaying state: The partial-wave scattering am-

plitude near resonance has the simple form

ℓ

iδ

ℓ

sin δ

ℓ

cot δ

ℓ

− i

−Γ/2

(E − E

) + iΓ/2

. (293)

At the complex energy E = E

− iΓ/2, the time-

dependent probability density decays exponentially

|Ψ(r, t)|

= |ψ(r)e

−i(E

−iΓ/2)t/~

= |ψ|

−Γt/~

. (294)

11.4 Yukawa’s theory of interactions:

Yukawa showed in 1935 that both the Coulomb

interaction mediated by the exchange of massless

photons between charges, and the strong or nuclear

interaction mediated by the exchange of massive

bosons called mesons, can be derived from quan-

tum mechanics.

Klein-Gordon equation: The Einstein energy-

momentum relation E

= p

can be quan-

tized into the Klein-Gordon equation in free space:

Φ(r, t) =



+ m



Φ(r, t), (295)

where

H = i~∂/∂t and

p = (~/i)∇. The result-

ing wave functions Φ oscillates in time as e

−iEt/~

These free-space solutions are said to be on the

energy shell, because the E-p relation is satisﬁed

on a spherical shell in p-space.

Static solutions of the KG equation: The

KG equation does not have time-independent so-

lutions in free space, because these solutions have

E = 0 and therefore cannot satisfy Einstein’s

E-p relation. Yukawa showed that there are

static (hence energy-nonconserving) solutions near

a point charge q located at the origin that satisfy

the inhomogenious DE:



+ m



φ(r) = 4π(~c)

qδ(r), (296)

For m = 0, the Yukawa equation simpliﬁes to the

Poisson equation

∇

φ(r) = −4πqδ(r). (297)

One can show with the help of Gauss’s theorem

(or Gauss’s law in electrostatics) that the resulting

static wave function of a massless photon around a

point charge that vanishes at r = ∞ is

φ(r) =

. (298)

This is just the Coulomb potential around the point

charge q (in Gaussian units, or q = q

√

4πǫ

The photon in question cannot propagate freely. It

can only exist momentarily around its point source

(the point charge), and is then said to be a virtual

photon. In Feynman’s diagrammatic language, the

Coulomb interaction arises from the emission of a

virtual photon by a source (the q here) and its ab-

sorption by a test charge at distance r from it, or

vice versa.

The virtual particle in Yukawa’s theory of inter-

actions is a boson, otherwise it cannot be emitted

because of the conservation of fermion number. A

virtual boson that is massive satisﬁes the Yukawa

equation

(∇

− µ

)φ(r) = −4πgδ(r), (299)

where the unit of charge is now denoted g, and µ =

mc/~ is the inverse reduced Compton wavelength.

The solution of this DE that vanishes at r = ∞ can

be shown to be the Yukawa potential

φ(r) =

−µr

. (300)

Nuclear forces: were known in 1937 to have a

range of 1/µ ≈ 1.4 fm. (fm = 10

−15

m.) This

fact allowed Yukawa to predict that these forces

are caused by the exchange of bosons of mass

m = µ~/c ≈ 140 MeV/c

. Such strongly inter-

acting bosons, now called π mesons or pions, were

discovered in 1947.

11.5 Scattering at low energies:

Impact parameter: In classical mechanics, a par-

ticle impacting at a transverse distance b (the im-

pact parameter) from the center of a square-well

potential of range R will not “see” the potential if

b > R. If the incident momentum in the center-

of-mass frame is p, the interaction vanishes when

the angular momentum ℓ = r×p exceeds the value

ℓ

max

= Max(kR/~), where Max(z) is the largest

integer in z.

The wave spreading in quantum mechanics makes

the connection between ℓ and b less sharply deﬁned,

but it remains true that the scattering phase shift

ℓ

vanishes sharply when ℓ > ℓ

max

. Consequently,

f(θ) ≈

ℓ

max

ℓ=0

(2ℓ + 1)f

ℓ

(cos θ), (301)

Hence at suﬃciently low energies, the S-wave (ℓ =

0) term dominates:

tot

≈ 4π

sin

4π

+ k

cot

. (302)

The fact that the leading Taylor term for ℓ = 0 is

independent of k comes from the properties of the

scattering wave function.

Eﬀective-range expansion:

k cot δ

= −

a(k)

= −

+ ..., (303)

where a

is called the scattering length and r

called the eﬀective range.

11.6 Phase shifts for ﬁnite-range potentials:

Radial wave function: The wave function in the

presence of a potential V is

ψ(r) =

∞

ℓ=0

ℓ

(2ℓ + 1)P

ℓ

(cos θ)

ℓ

(r)

, (304)

where the radial wave function u

ℓ

satisﬁes the ra-

dial wave equation

ℓ

2µ

(E − V )u

ℓ

−

ℓ(ℓ + 1)

ℓ

= 0. (305)

where µ is the reduced mass in the center-of-mass

system.

Attractive S- wave square-well potential: For

ℓ = 0 in an attractive square-well potential of depth

, the radial wave equation is

For r < R :

+ κ

= 0,

= sin(κr); (306)

For r > R :

out

+ k

out

= 0,

out

= sin(kr + δ

); (307)

2mE

, κ

2mV

, κ

= k

+ κ

. (308)

The phase shift δ

is determined by matching the

inside and outside logarithmic derivatives L

out

at r = R, where L = (du/dr)/u, to give

tan κR =

tan(kR + δ

), or

= tan

−1



tan κR



− kR. (309)

Levinson’s theorem: If there are m bound states

inside the potential,

ℓ

(k = 0) = mπ, (310)

except for an S-wave bound state at E = 0, which

contributes only π/2, half of the normal contribu-

tion.

11.7 Born approximation for two-particle scatter-

ings:

The Born (or ﬁrst Born) approximation for the

scattering of two particles is a ﬁrst-order TDPT

that can be obtained from the

Golden rule: A weak two-body interaction

′

V e

−iωt

†

iωt

(311)

causes the scattering from an initial two-particle

state i to a ﬁnal two-particle state f within the solid

angle d

Ω

around the ﬁnal relative momentum k

The transition rate is given, in ﬁrst-order TDPT,

by the Golden Rule:

dΩ

2π

|hf|

V |ii|

(E)

4π

(~k

)

(2π)



|hf|

V |ii|



. (312)

Here µ

, k

are the reduced mass and relative mo-

mentum, respectively, in the state j (= i, f ) of the

two-body system in a cube of side L, and the den-

sity of ﬁnal state factor is

(E)

4π



2π



. (313)

The scattering is elastic if k

= k

, and inelastic if

, k

The resulting diﬀerential scattering cross sec-

tion

dσ

dΩ

≡

inc



dΩ



(314)

is just the transition rate per unit incident ﬂux (or

current density), which is

inc

(315)

in a cube of side L. Note that J

inc

has the expected

dimension of m

−2

−1

. The ﬁnal result for the Born

approximation can be written compactly as

dσ

dΩ

(2π)





V (q)|

, (316)

where q = k

− k

, and

V (q) ≡ L

hf|

V |ii =

−iq·r

V (r)d

r (317)

is obtained by using the plane-wave wave functions

in the relative coordinate r normalized to one par-

ticle in a cube of side L:

hr|ii =

3/2

·r

, hf|ri =

3/2

−ik

·r

. (318)

Thus the cross section in the Born approximation

can be calculated directly from the Fourier trans-

form

V (q) of the interaction potential V (r).

Examples of

V (q) are those from the δ-shell and

the Yukawa potentials:

V (r) = A

δ(r − R) ⇒

V (q) = A

4πR

sin(qR);

V (r) ∝

−αr

⇒

V (q) ∝

4π

+ q

. (319)