System of Identical particle

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Contents
12 A System of Identical Particles 565
12.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
12.1.1 A system of two identical particles . . . . . . . . . . . . . . . . . . . 565
12.1.2 Time-dependence of the symmetry and antisymmetry property . . . . 567
12.1.3 A system of more than two identical particles . . . . . . . . . . . . . 568
12.1.4 Law of nature concerning the symmetry or antisymmetry . . . . . . . 569
12.1.5 The structure of the many-particle state vector . . . . . . . . . . . . . 570
12.2 Non-Interacting Identical Particles . . . . . . . . . . . . . . . . . . . . . . . 573
12.2.1 Two-particle system . . . . . . . . . . . . . . . . . . . . . . . . . . 573
12.2.2 N -particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
12.2.3 Pauli exclusion principle . . . . . . . . . . . . . . . . . . . . . . . . 575
12.3 Interacting Two-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . 576
12.3.1 The Z − 2 ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
12.3.2 Covalent double bond . . . . . . . . . . . . . . . . . . . . . . . . . 581
12.3.3 Heisenberg’s exchange Hamiltonian . . . . . . . . . . . . . . . . . . 585
12.4 The Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
12.4.1 The periodic table . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
12.4.2 The simple electronic configurations . . . . . . . . . . . . . . . . . . 587
12.4.3 The multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
12.5 The Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
12.5.1 The central field approximation . . . . . . . . . . . . . . . . . . . . 594
12.5.2 The spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . 596
12.6 The Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
12.6.1 White dwarf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
12.7 Scattering between two identical particles . . . . . . . . . . . . . . . . . . . 601
12.7.1 Scattering between two particles without spin . . . . . . . . . . . . . 601
12.7.2 Differential cross section in the laboratory frame . . . . . . . . . . . 603
12.7.3 Scattering between two particles with spin 1/2 . . . . . . . . . . . . . 605
12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
12.9 Source Material and Further Reading . . . . . . . . . . . . . . . . . . . . . . 613
List of Figures
12.1 Direct and exchange interaction between two particles. The solid lines with
arrows represent the states of the two particles and the dashed line the inter-
action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
12.2 Scattering between two identical particles. . . . . . . . . . . . . . . . . . . 602
12.3 (a) Scattering of two particles in the center of mass frame. (b) In the rest
frame of particle 2. (c) Transformation between two frames of the particle 1
velocity before the collision. (d) Transformation of velocity after the colli-
sion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
iv
Chapter 12
A System of Identical Particles
One Fish, Two Fish
Red Fish, Blue Fish
— by Dr. Seuss.
12.1 Bosons and Fermions
We have so far been working with a system of one particle or of two different particles or
with an ensemble of noninteracting systems. We now study the quantum mechanics of a
system of two or more identical particles. The distinction between an ensemble of identical
noninteracting particles and a quantum system of identical noninteracting particles is that
in an ensemble each particle can in principle be labeled so that at no time we lose track
of the identity of that particle whereas in a quantum system when the wave functions of two
particles overlap we will no longer be able to identify each with the label previously assigned.
Thus, the state of an ensemble may be specified by the statistical description of the density
matrix but the state of a system must be described by the wave function of the many particles.
We shall see how the inability to separate the identical particles restricts the choice of the
many-particle wave function.
12.1.1 A system of two identical particles
Consider two particles which are identical in their intrinsic properties, such as the mass,
charge and spin. Now they may be in some dynamical state. If some one, unbeknownst to
us, switches the two particles’ roles, then there is no way for us to find out by any physical
565
566 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
measurement. This is true both in classical and quantum physics. However, classically it is
possible to keep track of each particle by initially labelling the particles and following their
motion. Each particle retains its identity. In quantum mechanics, when the wave functions
of the two particles overlap, we can no longer be certain of their separate identities. In the
region of the wave function overlap, the uncertainty in their positions means tracking of each
particle becomes impossible.
Let r denote the complete set of dynamical variables of a particle, such as the eigenvalues
of position, r, and the eigenvalues of the spin component S · nˆ in a direction nˆ specified, for
convenience, for all the particles. Since the eigenstates of S2 of all the particles assume the
same eigenvalue, this eigenvalue is understood. The coordinates of particle j are denoted by
rj . The wave function Ψ(r1, r2, t) describes a state of the system of the two identical particles.
Consider an observable A of the system. It may be the position of one particle or the total
energy of two particles, etc. The eigenfunctions and eigenvalues of A,
Aψn(r1, r2) = αnψn(r1, r2), (12.1.1)
give the only possible outcomes of a measurement of A. Now if in the eigenstate we have
switched the roles of the two particles, i.e., replaced the wave function ψn(r1, r2) by ψn(r2, r1)
then a measurement of A should yield the same outcome. In other words,
Aψn(r2, r1) = αnψn(r2, r1). (12.1.2)
For a non-degenerate eigenvalue, this means
ψn(r1, r2) = ±ψn(r2, r1). (12.1.3)
Since the above has to hold for all the observables, it means that at all
Ψ(r2, r1, t) = ±Ψ(r1, r2, t). (12.1.4)
A wave function which remains unchanged under the permutation of the particles is called a
symmetric wave function. A wave function which changes sign under the permutation of the
particles is said to be antisymmetric.
12.1. BOSONS AND FERMIONS 567
12.1.2 Time-dependence of the symmetry and antisymmetry property
The symmetry or antisymmetry property of a wave function of two identical particles is a
constant of motion. The total Hamiltonian of the two particles is
H = H(1, 2), (12.1.5)
where 1 stands for the relevant physical observables of particle 1, such as the momentum,
position and spin or even isospin. Since
H(1, 2) = H(2, 1), (12.1.6)
if ψ(r1, r2) is an energy eigenstate with energy E, then ψ(r2, r1) is also an energy eigenstate
with the same energy E. If E is non-degenerate, then the two functions representing the
same eigenstate are related by Eq. (11.1.4). If E is degenerate, ψ(r1, r2) and ψ(r2, r1) may
be distinct states and their sum and difference will make a symmetric and an antisymmetric
state.
To put the above procedure more formally, we may regard the permutation as a symmetry
operator,
Pψ(1, 2) = ψ(2, 1). (12.1.7)
Since P 2 = 1, P has eigenvalues ±1 associated with the wave functions of Eq. (12.1.3). The
commutation of P with H leads to the common eigenstates of H and P given by symmetric
and antisymmetric states in Eq. (12.1.3).
To satisfy the time-dependent Schro¨dinger equation, a general wave function is of theform
Ψ(r1, r2, t) =
∑
n
anψn(r1, r2)e
−iEnt/h¯, (12.1.8)
in terms of the energy eigenstates. The coefficients an are independent of time. If the initial
state of the system is symmetric, then an is non-zero only if the energy state n is symmetric.
It follows that the closed system will be in a symmetric state at all subsequent times. For the
same reason, an antisymmetric state will remain antisymmetric forever, unless it is perturbed
by an external agent.
568 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
12.1.3 A system of more than two identical particles
For a system of two identical particles, there are only two possible combinations of a wave
function ψ(1, 2), i.e. symmetric and antisymmetric
ψ(1, 2)± ψ(2, 1). (12.1.9)
For a system of three particles, the symmetrized combination of a wave function ψ(123)
is (unnormalized)
Sψ(123) = ψ(123) + ψ(312) + ψ(231) + ψ(132) + ψ(213) + ψ(321). (12.1.10)
The antisymmetrized combination is
Aψ(123) = ψ(123) + ψ(312) + ψ(231)− ψ(132)− ψ(213)− ψ(321). (12.1.11)
The sign is positive if the permutation is even and the sign is negative if the permutation is
odd. A permutation is even (or odd) if it can be carried out by an even (or odd) number of
exchanges of pairs of numbers.
For three particles, there are, of course, four other combinations of wave functions.
For N particles, there are N ! ways of permuting the coordinates of the particles. Denote a
permutation of the integers (1,2,3,. . . N) by P (12..N). Then the symmetrized wave function
is
Ψ+ = Sψ(12 . . . N) =
∑
P
ψ(P{12 . . . N}). (12.1.12)
The antisymmetrized wave function is
Ψ− = Aψ(12 . . . N) =
∑
P
epψ(P{12 . . . N}), (12.1.13)
where ep = 1 or − 1 if P is an even or odd permutation.
If we interchange only the coordinates of two particles, the symmetric wave function
remains unchanged and the antisymmetric wave function changes sign. There are still N !−2
other possible combinations of ψ(12..N).
12.1. BOSONS AND FERMIONS 569
12.1.4 Law of nature concerning the symmetry or antisymmetry
For a system of N identical particles, only the symmetric or the antisymmetric wave function
satisfies the criterion that a measurement of an observable is unaffected by the interchange of
the identities of two particles. Indeed, only these two types of states are observed in nature.
Observation further reveals a law which cannot be deduced from the non-relativistic quantum
theory, namely, that a given type of particles only exhibits either symmetric or antisymmetric
wave functions. It has never been found that one kind of particles can exist in a symmetric
state under one set of initial conditions and exist in an antisymmetric state under another set
of circumstances.
Furthermore, the symmetric states are always associated with particles of integral spins
and the antisymmetric states are only found among particles of half-integral spins.
Thus, all the particles in nature fall into one of two classes:
(1) Bosons. A particle with an integral spin is a Boson. A system of identical bosons always
occurs in symmetric wave functions. They obey Bose-Einstein statistics. The particles of
the quantized electromagnetic field are bosons. They are elementary particles with integral
intrinsic spin such as the photon, W±, Z0, and the gluons, with spin one and the gravitons
(not yet observed) with spin 2. Other bosons are composite particles with total spin integral
such as the α-particle, i.e., He4 nucleus made up of two neutrons and two protons which are
fermions.
(2) Fermions. A particle with a half-integral spin is a Fermion. A system of fermions always
occurs in an antisymmetric state. The fermions obey Fermi-Dirac statistics. The electron is
an elementary particle which is a spin 1/2 fermion. So are the quarks. A nucleon (proton or
neutron), with spin 1
2
, is a composite fermion made up of three quarks. He3 is a composite
particle made up of two protons and one neutron and therefore with a half-integral spin, and
is an example of a fermion.
When a composite particle is treated as a boson or a fermion, it is only a good approx-
imation for the phenomena in which the internal structure of the composite particle is not
affected. For example, liquid He4 (or liquid He3) is a boson (or fermion) system where the
helium atoms behave as particles. In the neutron capture by He3 to form He4, it is the fermions
570 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
of the nuclear particles which count.
Pauli gave an argument based on the relativistic quantum field theory for the reason relat-
ing the integral and half-integral spins to symmetry and antisymmetry.
The above considerations of spin and statistics apply to particles moving in three dimen-
sions. For particles confined to move in two dimensions, F. Wilczek suggested that the orbital
angular momentum of a particle of charge q orbiting around a tube of magnetic flux αΦ0 in
units of quantum of flux Φ0 = h/e is quantized in units of � − αq/e, where � is an integer.
Thus, a particle restricted to move in a plane and associated with a charge e and a flux tube
αΦ0 may have spin �− α. When α = 0, an integer value of � makes the particle a boson and
a half-integer value of � makes it a fermion. A particle with a fractional α between zero and
1/2 may obey “fractional statistics”, leading Wilczek to term such a particle an anyon.
12.1.5 The structure of the many-particle state vector
In the above treatment of the permutation symmetry of the state of identical particles, we have
permuted the coordinates of the particles. In analogy with the abstraction of the momentum
from a differential operator on the wave function of position coordinates to a momentum
operator acting on the state vector:
h¯
i
∂
∂x
〈r|ψ〉 = 〈r|Px|ψ〉, (12.1.14)
we wish to translate the permutation operator on the coordinates to the permutation symmetry
operator on a many-particle state vector. Before we do that, we need to examine the state
vector in the Hilbert space of many particles.
We can set up a basis set of states by using a complete set of orthonormal single-particle
states |u〉, u being an index. For example, |u〉1|u′〉2 is a two-particle basis state. The subscripts
denote the single-particle Hilbert space of each particle. Then a general two-particle state may
be regarded as the sum
|Ψ〉 =
∑
u,u′
|u〉1|u′〉2 au,u′ , (12.1.15)
where au,u′ = 〈2u′|〈1u|Ψ〉. Thus, the two-particle wave function is Ψ(r1, r2) = 〈2r2|〈1r1|Ψ〉.
Extension to the Hilbert space of N particles is straightforward.
12.1. BOSONS AND FERMIONS 571
The exchange of the roles of the identical particles in Sec. 12.1 will now be put in a more
formal framework using the permutation operators acting on the many-particle state vector.
Since any permutation may be made up of a sequence of pair exchanges, we shall explore the
properties of the exchange operator here. Those of the permutation operators follow easily
and will be given in Problem 1. An exchange operator Pij exchanges the roles of particle i
and particle j and is defined by its action on the N-particle basis state,
Pij|u1〉1|u2〉2 . . . |ui〉i . . . , |uj〉j, . . . , |uN〉N =
|u1〉1|u2〉2 . . . |uj〉i . . . , |ui〉j, . . . , |uN〉N , (12.1.16)
its action on the general state vector is
Pij|Ψ〉 =
∑
uj
[Pij|u1〉1 . . . |uN〉N ] au1,...uN , (12.1.17)
where the square brackets signify the action of the operator on the basis state only. The orig-
inal action of the exchange operator Pij on the N-particle wave function may be recovered:
PijΨ(1, 2, . . . , i, . . . , j, . . . , N) = Ψ(1, 2, . . . , j, . . . , i, . . . , N). (12.1.18)
The exchange operator has the following properties:
(a) Show that Pij is Hermitian and unitary.
Proof — By the definition of the Hermitianconjugate, for any wave function
〈Φ(1, 2, . . . , i, . . . , j, . . . , N)|P †ij|Ψ(1, 2, . . . , i, . . . , j, . . . , N)〉
= 〈PijΦ(1, 2, . . . , i, . . . , j, . . . , N)|Ψ(1, 2, . . . , i, . . . , j, . . . , N)〉
= 〈Φ(1, 2, . . . , j, . . . , i, . . . , N)|Ψ(1, 2, . . . , i, . . . , j, . . . , N)〉
= 〈Φ(1, 2, . . . , i, . . . , j, . . . , N)|Ψ(1, 2, . . . , j, . . . , i, . . . , N)〉
= 〈Φ(1, 2, . . . , i, . . . , j, . . . , N)|Pij|Ψ(1, 2, . . . , i, . . . , j, . . . , N)〉. (12.1.19)
In other word, the result is the same whether Pij acts on the left or right. Thus,
P †ij = Pij, (12.1.20)
572 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
or Pij is Hermitian. Since operating Pij twice on a wave function has no net effect on
the wave function,
PijPij = 1, (12.1.21)
or
P−1ij = Pij = P
†
ij, (12.1.22)
i.e. Pij is unitary.
(b) The eigenfunctions of Pij are symmetric and antisymmetric in i and j with the associ-
ated eigenvalues ±1.
Proof — Let
Pij|Ψ〉 = α|Ψ〉, (12.1.23)
where α is a number. By Eq. (12.1.21),
α2 = 1, (12.1.24)
and
α = ±1. (12.1.25)
Define the operator Sij by
Sij =
1
2
(1 + Pij). (12.1.26)
Then for any state |Ψ〉,
Pij(Sij|Ψ〉) = Sij|Ψ〉, (12.1.27)
showing that Sij|Ψ〉 is an eigenfunction of Pij for the eigenvalue +1. The operator Sij
is called the symmetrizing operator.
Similarly, we can define an antisymmetrizing operator by
Aij =
1
2
(1− Pij), (12.1.28)
such that Aij|Ψ〉 is an eigenfunction of Pij for the eigenvalue −1.
12.2. NON-INTERACTING IDENTICAL PARTICLES 573
(c) For the total Hamiltonian H or any other total property of the system of N particles, the
eigenstates may be classified as symmetric or antisymmetric in the exchange of (ij).
Proof — Since H or any other total property is symmetric in the exchange of i and j,
[Pij, H] = 0. (12.1.29)
So the two operators can have simultaneous eigenstates, i.e. the energy eigenstates can
be classified as symmetric or antisymmetric with respect to (ij).
12.2 Non-Interacting Identical Particles
12.2.1 Two-particle system
Take the example of the helium atom which consists of two electrons and a nucleus. Treat the
massive nucleus as a fixed point of attraction. The two electrons form a Fermi system with
the Hamiltonian
H = K1 + K2 + V (r1, r2), (12.2.1)
where Ki is the kinetic energy of the ith particle and V (r1, r2) is the potential energy of the
two particles and is composed of
V (r1, r2) = v(r1) + v(r2) + u(r1 − r2), (12.2.2)
two Coulomb terms due to the attraction of the nucleus and the mutual repulsion of the elec-
trons. Thus, the Hamiltonian of two particles is made up of two single-particle Hamiltonians
and one interaction term. Although it is possible to obtain numerically accurate solutions
of the energy eigenstates for systems containing a few particles, they shed little light on the
structure of the many-particle state in general. As a first approximation, it is useful to treat
all the particles as moving under the influence of a common potential without mutual interac-
tion. This yields an approximate structure for the many-particle state. The common potential
approximation can be refined to include the influence of the other particles (see Chapter 13).
In a non-interacting system, there is no mutual interaction term and the total Hamiltonian
is a sum of single-particle Hamiltonians
H(1, 2) = h(1) + h(2). (12.2.3)
574 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
The energy eigenstates of the two particles can be built from the energy eigenstates of the
single particle Hamiltonian
h(1)ψα(r1) = Eαψα(r1). (12.2.4)
The wave function
ψ(r1, r2) = ψα(r1)ψβ(r2) (12.2.5)
is clearly an eigenstate of the total Hamiltonian H(1, 2) with energy Eα +Eβ . It is, however,
not symmetrized or antisymmetrized. The actual wave function of the system of two particles
is either
ψ+ = Sψ(r1, r2) =
[
1√
2
δαβ + (1− δαβ)
]
1√
2
[ψα(r1)ψβ(r2) + ψα(r2)ψβ(r1)] (12.2.6)
if the particles are bosons, or
ψ− = Aψ(r1, r2) = 2−1/2[ψα(r1)ψβ(r2)− ψα(r2)ψβ(r1)]
= 2−1/2
∣∣∣∣∣∣
ψα(r1) ψα(r2)
ψβ(r1) ψβ(r2)
∣∣∣∣∣∣
, (12.2.7)
if the particles are fermions.
Let us examine the probability density of one particle at r1 and one particle at r2 without
specifying which particle is at one place or the other. It is given for α �= β by
|ψ±|2 = 1
2
{[|ψα(r1)|2|ψβ(r2)|2 + |ψα(r2)|2|ψβ(r1)|2
]
± [ψ∗α(r1)ψβ(r1)ψ∗β(r2)ψα(r2) + complex conjugate
]}
. (12.2.8)
The terms in the second pair of braces are the overlap of the wave functions in state α and
state β for each particle or, equivalently, the overlap of the wave function of particle 1 in state
α and the wave function of particle 2 in state β and vice versa. In the classical limit, the wave
functions for the single particle states α and β are well separated spatially so that the particle
following the trajectory α is distinct from the particle following the trajectory β. Then the
overlap terms are negligible and the probability density is given by the terms in the first pair
of braces in Eq. (11.2.8), as would be expected. Note that the classical limit is independent
of the boson or fermion origin of the particles.
12.2. NON-INTERACTING IDENTICAL PARTICLES 575
12.2.2 N -particle system
It is straightforward to generalize the foregoing to a system of N particles. If the particles
are non-interacting, the total Hamiltonian is a sum of single-particle Hamiltonians h(i), i =
1, 2, ...N . A product of the single-particle wave functions
ψ(1, 2, . . . N) =
N∏
j=1
ψαj(rj) (12.2.9)
is an eigenstate of the total Hamiltonian with energy
E =
N∑
j=1
Eαj . (12.2.10)
The wave function (11.2.9) has to be antisymmetrized for fermions and has to be symmetrized
if the particles are bosons. The normalized antisymmetric wave function can be written as a
determinant
ψ− =
1√
N !
∑
P
ePPψ(1, 2, . . . N) =
1√
N !
∣∣∣∣∣∣∣∣∣∣∣∣
ψα1(r1) ψα1(r2) . . . ψα1(rN)
ψα2(r1) ψα2(r2) . . . ψα2(rN)
. . . . . . . . .
ψαN (r1) ψαN (r2) . . . ψαN (rN)
∣∣∣∣∣∣∣∣∣∣∣∣
,(12.2.11)
where eP is the parity of the permutation (depending on the even or odd number of exchanges)
and the determinant is known as the Slater determinant. The normalized symmetric wave
function is given by
ψ+ =
√∏
k nk!
N !
1∏
k nk!
∑
P
Pψ(1, 2, . . . N) =
1√
N !
∏
k nk!
∑
P
Pψ(1, 2, . . . N),(12.2.12)
where nk is the number of particles in the same state αk. We need to divide the symmetrized
sum by nk! for each k to find the sum of distinctive terms, numbering is N !/(
∏
k nk!), which
explain the normalization factor in the second expression.
12.2.3 Pauli exclusion principle
For a many-fermion system, the wave function is antisymmetric with respect to interchange of
a pair of particles. If two fermions have exactly the same dynamical state, then an interchange
576 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
of the coordinates of the two particles also does not change the wave function which must be
zero. Therefore, in a many-fermion system, no two particles can be in exactly the same state.
This statement, in particular, applies to the electrons, as was first enunciated by Pauli. Two
electrons are in the same state only if both their orbitals and spin states are identical.
In a non-interacting system, the exclusion principle is transparent, since the Slater deter-
minant will vanish if two of the single-particle wave functions are the same.
12.3 Interacting Two-Electron Systems
While the independent particle model has yielded a number of important results, there are
properties of the system for which the inter-particle interaction plays a key role. The two-
electron system is the simplest interacting system for which we should beable to find some
key features of interaction. The example serves to introduce concepts for more complicated
many-particle systems. It also has direct applications to the helium atom, the covalent bond
in the hydrogen molecule and other diatomic molecules, and the root cause for magnetism.
Below, we choose three examples with similar interaction effects but contrasting results.
12.3.1 The Z − 2 ion
Consider the system of two electrons and a nucleus of atomic number Z. We shall consider
the nucleus to be sufficiently massive to be treated as the fixed center in the dynamics of the
two electrons. We express the energy in the atomic units (mee′4/h¯
2 = 2 Ry), the distance
in units of a0/Z (a0 = h¯
2/mee
′2), and the angular momentum in units of h¯. Note that the
electron mass me is used instead of the reduced mass. The Hamiltonian is composed of three
terms,
HT = H0 + U + Hso, (12.3.1)
where
H0 = Z
2
2∑
j=1
(
−1
2
∇2j −
1
rj
)
, (12.3.2)
12.3. INTERACTING TWO-ELECTRON SYSTEMS 577
is the noninteracting part,
U =
Z
|r1 − r2| , (12.3.3)
is the Coulomb interaction between the two electrons,
Hso = α
2Z4
2∑
j=1
1
2r3j
Lj · Sj, (12.3.4)
is the spin-orbit interaction, α being the fine structure constant. The utility of the choice of
units is now apparent. The explicit factors of Z will give us important qualitative guidelines
in constructing the ground state configurations of the elements in the periodic table. It ap-
pears that we should be able to use the hydrogenic solution as the basis to construct single
Slater determinants as the zeroth order approximation for the two-electron state and treat the
electron-electron interaction and the spin-orbit interaction as corrections. This is true pro-
vided that Z is large. The two corrections are comparable when Z ∼ 30. Below that the
interaction term should be treated as the first (more important) correction. This is the most
common route. For larger Z, the spin-orbit correction should come first.
The spin structure of the two-electron states
If the spin-orbit term is neglected, the Hamiltonian H = H0 + U is independent of the
spin coordinates. The spin structure of the two-electron states derives then from the fermion
nature, or equivalently, the exclusion principle. Since H commutes with the spins of the
electrons, it commutes with the total spin of the two electrons,
S = S1 + S2. (12.3.5)
Hence, the energy eigenstates may be written as the eigenstates of S2 and Sz:
Ψ(r1, r2) = ψS(r1, r2)χS,MS . (12.3.6)
The spin part χS,MS includes the possibilities of the S = 1 triplets and the S = 0 singlet.
(We hope that the use of S for the quantum number of the total spin would not be confused
with the operator.) Since the triplets are symmetric under the exchange of the two electrons,
578 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
the corresponding spatial part ψ1(r1, r2) must be antisymmetric in order that the whole wave
function is fermionic. Since the singlet is antisymmetric under exchange, its spatial part
ψ0(r1, r2) must be symmetric. If only the H0 part is kept, the spatial part would follow
from a single Slater determinant approximation. In this approximation, the energy of the two
electrons would have in general a certain amount of degeneracy. The exception is when both
electrons are in the same spatial orbital. In that case, there are no triplets by the antisymmetry.
The two electrons are in a singlet state. Thus, we expect the ground state to derive from the
configuration (1s)2 and to be in the singlet, which, in the spectroscopic notation, is 1S0. When
two different orbitals are occupied, Eq. (12.2.8) shows that the two electrons tend to avoid
each other in the triplet and spatially antisymmetric state and to overlap in the singlet and
spatially symmetric state. Therefore, the electron-electron interaction energy is lower in the
spin triplet states than in the single state. This energy difference is caused not by any explicit
spin-dependent forces but by the fermion nature. This conclusion of the triplet being lower
in energy than the singlet is a special example of Hund’s rule to be considered later.
Perturbation treatment of the multiplets
The zeroth order wave function, i.e., the eigenstates of H0, is a Slater determinant with two
orbitals, j and k, where k is short for the quantum numbers nk, �k,m�k, sk,msk. If the |j, k〉
state is nondegenerate, then 〈j, k|U |j, k〉 gives the first order correction in energy. If there
is a set of degenerate states with common nj, nk, then the degenerate perturbation theory
dictates the diagonalization of the matrix with elements 〈j, k|U |j′, k′〉 to yield the splitting of
the degenerate set of states.
The diagonalization may be simplified by a symmetry consideration. The symmetry of
the spin rotation leads to the singlet and triplet spin structure considered above. Without
the spin-orbit interaction, the Hamiltonian H also is invariant under spatial rotation and,
therefore, commutes with the total orbital angular momentum
L = L1 + L2. (12.3.7)
The spatial part of the wave function may be taken as the eigenstates of L2 and Lz. In other
words, the state |j, k〉 = |njnk�j�km�jm�kmsjmsk〉 may be transformed to the eigenstates of
12.3. INTERACTING TWO-ELECTRON SYSTEMS 579
L2 and Lz and S2 and Sz, denoted by |njnk�j�kLMLSMS〉. In the spectroscopic notation,
this set of degenerate states is labeled 2S+1L. We note the separation of the spatial and spin
part as |njnk�j�kLML〉|SMS〉, where the spin part was given above. For the spatial part,
instead of proceeding generally, we shall examine only the ground state configuration (1s)2
and the first excited states from the two degenerate configurations 1s2s and 1s2p.
The ground state term is 1S. It is a singlet with the symmetric spatial wave function
χ0(r1, r2) = ψ100(r1)ψ100(r2) in terms of the 1s hydrogenic wave function. The unperturbed
ground state energy is
E(0)(1S) = −Z2 a.u. (12.3.8)
The interaction matrix element gives the first order correction in energy,
E(1)(1S) =
∫
d3r1
∫
d3r2ψ
∗
100(r1)ψ
∗
100(r2)
Z
|r1 − r2|ψ100(r1)ψ100(r2) =
5Z
8
a.u.(12.3.9)
The Coulomb integral may be evaluated in two ways: (1) by expressing the Coulomb inter-
action in terms of its Fourier integral thus transforming the Coulomb integral into an integral
of the Fourier components,
∫
d3r1
∫
d3r2f(r1)
1
|r1 − r2|g(r2) =
∫
d3q
(2π)3
f ∗(q)
4π
q2
g(q), (12.3.10)
where g(q) denotes the Fourier transform of g(r); or (2) in terms of the power series of
Legendre polynomials,
1
|r1 − r2| =
∞∑
�=0
r�<
r�+1>
P�(cos(rˆ1 · rˆ2)), (12.3.11)
where r< = min(r1, r2), r> = max(r1, r2).
The binding energy for the helium atom (Z = 2) is
−E(1S) =
(
2Z2 − 5Z
4
)
Ry = 74.83 eV, (12.3.12)
within 5% of the experimental value of 79.01 eV.
As a preliminary to the perturbation computation of configurations 1s2s and 1s2p, con-
sider the Coulomb integral of the spatial part of the state of two electrons in orbitals j and k.
j k
kj
kj
jk
Direct Exchange
r 1 r 1
r 2 r 2
580 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
Figure 12.1: Direct and exchange interaction between two particles. The solid lines with
arrows represent the states of the two particles and the dashed line the interaction.
Since the spin structure has already been taken care of, the quantum numbers in j and k are
only orbital, e.g., k stands for nk, �k,m�k. The symmetric (S = 0) and antisymmetric (S = 1)
spatial parts of the wave functions are given by,
χS =
1√
2
[
ψj(r1)ψk(r2) + (−1)Sψk(r1)ψj(r2)
]
, (12.3.13)
in terms of the hydrogenic functions ψk(r2). The Coulomb integral is then
〈jk|U |jk〉S = ujkkj + (−1)Sujkjk, (12.3.14)where ujkkj =
∫
d3r1
∫
d3r2ψ
∗
j (r1)ψ
∗
k(r2)
Z
|r1 − r2|ψk(r2)ψj(r1), (12.3.15)
ujkjk =
∫
d3r1
∫
d3r2ψ
∗
j (r1)ψ
∗
k(r2)
Z
|r1 − r2|ψj(r2)ψk(r1) (12.3.16)
Fig. 12.1 illustrates the direct term ujkkj which is the electrostatic energy between the two
electrons in states j and k, and the exchange term ujkjk of the Coulomb integral in which
the incoming pair of states have been exchanged from j, k to k, j. For use in the next chap-
ter, we adopt the convention that the indices of each integral follow the assignment in the
corresponding diagram starting from the upper left corner running clockwise.
When the two orbital states j and k are the same, there is no triplet. When j and k
represent different orbital states, the singlet and triplet states are split in energy by twice the
12.3. INTERACTING TWO-ELECTRON SYSTEMS 581
exchange integral, 2ujkjk, with the triplet being lower in energy. For He, the 1s2s multiplet
lies lower in energy than the 1s2p multiplet. The determinating factor is the larger direct
Coulomb term. From Fig. 11.5, it is barely discernible that the radial density distribution,
r[Rn�(r)]
2 has a mean distance rn� slightly larger for 2s than for 2p. Thus, the direct Coulomb
interaction between 1s and 2s is weaker than that between 1s and 2p.
12.3.2 Covalent double bond
We have studied in Chapter 5 the covalent bond formed by one electron between two protons
in H+2 . We now examine the double bond formed by two electrons in a neutral hydrogen
molecule H2. The two protons are treated as fixed at a distance R apart at positions R1 and
R2. The total energy of the system then contains the contribution from the two-electron state
and the Coulomb repulsion between the two protons. Bonding occurs when the total energy
is lower at the equilibrium distance than when R → ∞, i.e., when the hydrogen atoms
dissociate.
For more generai use, we consider a generic problem of two electrons at positions rj, j =
1, 2, under the influence of two identical ion potentials v(rj−R�) centered at two ion positions
R�, � = 1, 2, and the usual Coulomb repulsion between the two electrons. The Hamiltonian
of the system can be put in the form
H =
∑
j
hj + V + U, (12.3.17)
where the single ion Hamiltonian for each electron in isolation is
hj = − h¯
2
2m
∇2j + v(rj −Rj), (12.3.18)
the total influence of the “other” ion on each electron is given by
V = v(r1 −R2) + v(r2 −R1), (12.3.19)
and the Coulomb interaction between the two electrons is
U =
(e′)2
|r1 − r2| . (12.3.20)
582 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
Let us examine two approximate ways of obtaining the two-electron ground state based on
the ground state of the electron in the single ion,
hjφ(rj −Rj) = εφ(rj −Rj), (12.3.21)
with the normalized wave function.
(a) The molecular orbital state
First we construct the bonding orbital for one of the electrons from the single-ion or-
bitals φ(rj −−Rj) centered about one each ion much like in the H+2 ion:
ψb(r) =
1√
2
[φ(r−R1) + φ(r−R2)]. (12.3.22)
The antibonding orbital is not used since it has higher energy than the separate orbital
around a proton. By the Pauli exclusion principle, we can put two electrons in this
orbital to give a wave function which is the product ψb(r1)ψb(r2) but they must have
opposite spins, i.e., they form a spin singlet
χ0(1, 2) =
1√
2
[χ+(1)χ−(2)− χ−(1)χ+(2)]. (12.3.23)
Thus, the ground-state wave function is
Ψ(1, 2) = ψb(r1)ψb(r2)χ0(1, 2) (12.3.24)
The approximate total energy of the molecule is obtained by the variational integral of
this wave function with respect to the Hamiltonian of the system. The qualitative result
is that the bonding orbital energies of the two electrons are large enough to more than
compensate for the Coulomb energy of the two electrons and the two protons. Thus,
this molecular orbital approximation yields binding for the hydrogen molecule. The
defect of the wave function is that it allows for two electrons to be on the same atom
which overestimates the Coulomb repulsion between the two electrons. At a large
separation, R → ∞, it does not yield the expect state of one electron in each atom
separately.
12.3. INTERACTING TWO-ELECTRON SYSTEMS 583
(b) The Heitler-London state or the valence-bond state
To remedy the shortcoming of the molecular orbital approximation in the large R re-
gion, we replace the molecular orbitals with a correlated Heitler-London form ψSχS in
which the spatial part ψS is given by
ψS(r1, r2) =
1√
2
[φ(r1 −R1)φ(r2 −R2) + (−1)Sφ(r1 −R2)φ(r2 −R1)], (12.3.25)
for the singlet (S = 0) and the triplet (S = 1) in Eq. (12.3.13). This wave function
eliminates the possibility of both electrons being in the same atomic orbital regardless
of their spin configuration. The energy of each state is given by the expectation value,
ES =
〈ψS|H|ψS〉
〈ψS|ψS〉 . (12.3.26)
The norm of the state is given by
〈ψS|ψS〉 = 1 + (−1)SN, (12.3.27)
where
N =
∫
d3r φ∗(r−R1)φ(r−R2) (12.3.28)
is a measure of the overlap of the wave functions on two ion sites. The expectation
value can be simplified. Denote the part of the wave function in Eq. (12.3.25) in which
each electron resides in its own ion, φ(r1 −R1)φ(r2 −R2), by D, the direct product,
and its exchange counterpart, φ(r1 −R2)φ(r2 −R1), by X . Then,
〈ψS|H|ψS〉 = 1
2
[〈D|H|D〉+ 〈X|H|X〉+ (−1)S(〈X|H|D〉+ 〈D|H|X〉)]
= 〈D|H|D〉+ (−1)S〈X|H|D〉, (12.3.29)
where 〈X|H|X〉 = 〈D|H|D〉 is shown by exchanging the electron positions and
〈X|H|D〉 = 〈D|H|X〉 by choosing φ to be a real function. The direct term is given by
〈D|H|D〉 = 2ε + F, (12.3.30)
584 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
where the direct potential integrals are grouped in
F = 2
∫
d3r φ(r−R1)v(r−R2)φ(r−R1) (12.3.31)
+
∫
d3r1
∫
d3r2φ(r1 −R1)φ(r2 −R2)u(r1 − r2)φ(r2 −R2)φ(r1 −R1).
The cross term is given by
〈X|H|D〉 = 2N2ε + G, (12.3.32)
where the exchange integrals are grouped in
G = 2N
∫
d3r φ(r−R1)v(r−R2)φ(r−R2) (12.3.33)
+
∫
d3r1
∫
d3r2φ(r1 −R1)φ(r2 −R2)u(r1 − r2)φ(r2 −R1)φ(r1 −R2).
The expectation energy values are given by
ES = 2ε +
F + (−1)SG
1 + (−1)SN2 . (12.3.34)
The energy difference between the singlet and the triplet is
E0 − E1 = 2G−N
2F
1−N4 . (12.3.35)
At the equilibrium distance between the two hydrogen ions R = |R1 − R2| where
the total energy (including the proton-proton interaction) is a minimum, the singlet lies
lower than the triplet. The key to the binding energy of the bond is the overlap density
φ(r−R1)φ(r−R2) for either electron which sees an attraction from both nuclei in the
first term on the right of Eq. (12.3.33). It makes G negative. The Heitler-London wave
function avoids the penalty of large Coulomb repulsion energy due to two electrons on
the same atom and has the correct limiting form of each electron residing in one ion as
R →∞. However, for small separation R = |R1 −R2|, by not allowing one electron
wave function on one atom to reflect the presence of the other proton, its orbital does
not tend to that of the helium atom as R → 0. Nonetheless, the variational energy
for the Heitler-London wave function is in good agreement with experiment and is
somewhat better than the molecular orbital state.
12.4. THE ATOMIC STRUCTURE 585
12.3.3 Heisenberg’s exchange Hamiltonian
We consider the interaction effects of two electrons in two atomic sites of a molecule or
crystal lattice. The singlet and triplet states and energies can be reproduced by the effective
Hamiltonian, called the Heisenberg Hamiltonian,
Heff = E − 2JS1 · S2, (12.3.36)
where
E = 2ε +
F (1 + 1
2
N2)−G(1
2
+ N2)
1−N4 ≈ 2ε + F (12.3.37)
and the exchange constant
J =
G−N2F
1−N4 ≈ G. (12.3.38)
If each of the two electronsbelongs to an inner shell, they are not responsible for the
bonding. Therefore, it is possible that the G term is dominated by the Coulomb exchange,
the second tem on the right-hand side of Eq. (12.3.33). The exchange energy G may also be
viewed as the Coulomb self-energy of the overlap charge density, φ(r−R1)φ(r−R2). Then,
J > 0 and the triplet lies below the singlet. This applies also to the excited states 1s2s and
1s2p of helium, discussed above. The difference in the conclusion from the covalent bond
model is clearly due to the different degrees of the overlap leading to the dominance of the
single electron overlap energy in Eq. (12.3.33) in the bonding case and the dominance of the
exchange energy G in the Heisenberg exchange case. Similarly, in solids, positive J leads to
parallel spins on all the sites and hence ferromagnetism and negative J leads to antiparallel
spins between neighboring sites and hence antiferromagnetism.
12.4 The Atomic Structure
An atom is made up of a nucleus and a number of electrons. For the study of the electron
dynamics here, the nucleus is treated to an excellent approximation as a point particle. The
Pauli exclusion principle governs the electronic structure in the atom. The atoms, in turn, are
the building blocks of molecules and matter in various states, which comprise a large part of
586 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
chemistry and physics. A basic knowledge of the theory of electronic structure of the atom is
essential to the further studies in a number of branches of physics: laser and quantum optics
including the new Bose-Einstein condensation of atoms, astrophysics, and condensed matter
(especially magnetism). The atomic theory also serves as a paradigm for the study of the
nucleus and the particles in it.
12.4.1 The periodic table
Many important properties (chemical and physical) of the elements have regular trends, easily
evident by their positions in the periodic table. The structure of the periodic table can be
justified by the electronic structure of the atoms. The many-electron system in an atom has
strong Coulomb interaction between the electrons. In the first approximation, each electron
moves independently under the influence of a spherical potential, called the central field,
which includes the Coulomb potential of the nucleus and approximately the average influence
of the other electrons. Spin-orbit interaction is temporarily neglected. The next chapter will
treat the approximations to produce a central field. Here, we consider the structure of the
electronic states given the central field. Clearly the central field is no longer Coulombic.
The single-particle Hamiltonian is then diagonalized in the same way as the hydrogen atom,
except that the radial equation has to be solved numerically. The quantum numbers are exactly
the same as in the hydrogen atom, n�m�ms for each electron but the degeneracies are not the
same as in the hydrogen atom. States with the same n and different � do not have the same
energy. In fact, for a given n, the energy increases with the angular momentum �. To build a
Slater determinant for all the electrons of the atom, each single-particle state (including spin)
can only be occupied by one electron. The periodic table is then constructed as explained in
an introductory chemistry course.
For a given pair of values for (n, �), there are 2(2� + 1) states with the same energy from
the ranges of ms and m�. The order of the energy levels in increasing energies is:
1s,
2s, 2p,
3s, 3p,
12.4. THE ATOMIC STRUCTURE 587
4s, 3d, 4p,
5s, 4d, 5p,
6s, 5d, 4f, 6p,
7s, 6d, 5f, ..
There is a distinct discontinuity in energy from one row to the next but within each row the
order is not always the same for different atoms. For example, the order of 5s and 4d may be
reversed. The magic numbers, i.e. the total numbers of electrons of all the filled shells for
each row are 2, 10, 18, 36, 54, 86.
12.4.2 The simple electronic configurations
To specify the ground state configuration of the electrons in an atom, such as argon (Ar), we
take the atomic number Z and indicate the number of electrons occupying each n� shell. For
example Z = 18, argon, the configuration is: (1s)2,(2s)2,(2p)6,(3s)2,(3p)6. By convention,
s, p, d, f, . . . , are used to represent � = 0, 1, 2, 3, . . .. The spectroscopist’s notation for the
many-electron state (Section 11.9), 2s+1Lj now indicates the total spin degeneracy, the total
orbital angular momentum and the total orbital plus spin angular momentum. For Ar, it is
1S0, which indicates that for the full shell the total orbital angular momentum and the total
spin are both zero. The group VIII atoms in the periodic table (He, Ne, Ar, Kr, Xe, Rn, known
collectively as the noble atoms) with electrons filled up to the ns and np levels in each row
have the same 1S0 configuration and are chemically inert.
To show explicitly the Slater determinant, we use the example of a lower atomic number,
Z = 3 for the lithium atom. Its electronic configuration is 1s22s. The ground state 2S1/2
shows that, after the closed shell of He, Li behaves like a one-electron atom, similar to H.
Similarly, in group I elements, the alkalis, or the group IA noble metals, there is a single
electron in the outermost ns shell, which is chemically active. The X-ray and optical spectra
of the alkalis historically established the electronic structure of the atom. For example, in
sodium, with the configuration (1s)2,(2s)2,(2p)6,(3s), the X-ray ionizes the 1s electron and
the subsequent transition from 2p to the 1s hole emits a photon of energy 76.6 Ry., yielding a
line in the so-called K series spectrum. The optical transition from the excited state 3p to 3s
588 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
gives the famous double sodium “D lines”, which can be explained by the spin-orbit splitting
of the 3p level into 2P3/2 and 2P1/2 states.
We can examine more closely the separate roles played by the spatial motion and the spin
dynamics. For the single-particle state with the quantum number nsσ, we let
φns(r) = 〈r|φns〉 (12.4.1)
be the spatial part of the ns orbital and
χ±(ms) = 〈ms|χ±〉 (12.4.2)
denote the spin up or down state along the z-axis, where ms = ±1/2 denotes the eigenvalues
of Sz of the particle. Thus, in the basis set of |ms〉, the spin up and down states are
χ+ =

 1
0

 ;χ− =

 0
1

 . (12.4.3)
For the j-th particle, its position is rj and spin state is χσ(j). We also let the spin singlet
between two particles be denoted by
χs(1, 2) =
1√
2
[χ+(1)χ−(2)− χ−(1)χ+(2)]. (12.4.4)
The Slater determinant of the electron configuration 1s+, 1s−, 2sσ where σ = ± is
1√
6
∣∣∣∣∣∣∣∣∣
φ2s(r1)χσ(1) φ1s(r1)χ+(1) φ1s(r1)χ−(1)
φ2s(r2)χσ(2) φ1s(r2)χ+(2) φ1s(r2)χ−(2)
φ2s(r3)χσ(3) φ1s(r3)χ+(3) φ1s(r3)χ−(3)
∣∣∣∣∣∣∣∣∣
=
1√
3
[φ2s(r1)χσ(1)φ1s(r2)φ1s(r3)χs(2, 3)
+(1, 2, 3) → (2, 3, 1) + (1, 2, 3) → (3, 1, 2)], (12.4.5)
where the two terms are similar to the first term with the dynamical variables (1,2,3) replaced
by (2,3,1) and (3,1,2). The first term tells us that the two 1s electrons form a close shell with a
product of two spatial wave functions and a singlet of two spins and that the the third electron
is the valence electron on the 2s shell with spin pointing either up or down.
12.4. THE ATOMIC STRUCTURE 589
12.4.3 The multiplets
The Hamiltonian of Z electrons in the static potential of the nucleus in an atom may be
thought of as composed of three terms:
HA = Hcf + Hic + Hso, (12.4.6)
where Hcf is the central field approximation used above,
Hcf =
Z∑
j=1
[
p2j
2m
+ Vcf (rj)
]
, (12.4.7)
Hic is the correction due to the electron-electron interaction minus whatever interactionef-
fects which have been simulated as the average potential which has been included in the
central field approximation,
Hic =
1
2
∑
j �=k
[
e′2
|rj − rk|
]
−
Z∑
j=1
[
Vcf (rj) +
Ze′2
|rj|
]
. (12.4.8)
Thus, the Hamiltonian of the electrons in the nonrelativistic limit is
H = Hcf + Hic =
Z∑
j=1
[
p2j
2m
− Ze
′2
|rj|
]
+
1
2
∑
j �=k
[
e′2
|rj − rk|
]
. (12.4.9)
The most important of the relativistic corrections – the spin orbit term,
Hso =
Z∑
j=1
ζ(rj)Lj · Sj, (12.4.10)
where ζ(r) =
Ze′2
2m2c2r3
. (12.4.11)
The starting approximation we have used is to construct the eigenstates of the central field
approximation Hcf . In this subsection, we consider the effects of the two correction terms.
For small Z, up to about 40, say, the spin-orbit interaction Hso is a correction to the central
field smaller than the electron-electron interaction correction Hic. Therefore, the interaction
correction has to be treated first. To the zeroth approximation, the many-electron eigenstate
of Hcf is a Slater determinant made up of each electron occupying a single electron state.
Since the Hamiltonian of each electron is invariant under rotation with respect to position
590 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
and to spin of each electron separately, the single electron state has the quantum numbers
n�m�ms. Because the central field potential is not the Coulomb attraction, the single-electron
Hamiltonian is symmetric under O(3) but not O(4). Thus, the single particle energy in the
central field, εn�, depends on both the quantum number n and �. The Hamiltonian H in
Eq. (12.4.8) without the spin-orbit interaction is invariant under the rotation of either position
or spin of all the electrons. The total spin angular momentum S2 and Sz and the total orbital
angular momentum L2 and Lz are conserved, where
S =
Z∑
j=1
Sj, (12.4.12)
L =
Z∑
j=1
Lj. (12.4.13)
Therefore, the Z-electron state may be characterized by the quantum numbers L, S,ML,MS .
Note that the scalars L and S used as quantum numbers should not be confused with the
operators. Since the group theory now guarantees the degeneracy of each L, S term with
different ML and MS , each energy level is denoted by 2S+1L where a letter is used for L.
Thus, when the interaction correction is taken into account as a perturbation, the degenerate
states of a configuration are split into different 2S+1L states, each is known as a “term”.
As an example, consider the excited states of beryllium (Be) in the configuration (1s)2,2s,2p.
The degeneracy is 12 from two spin 1/2 electron states in 2s and 2p and the three orbital states
of 2p. The helium core of (1s)2 may be ignored in the spin and orbital angular momentum
addition since they add up to zero. The possible values of S is the sum of the spins of 2s and
2p, i.e., 0 or 1. The sum of the orbital angular momenta is L = 1. The interaction correction
can split the 12 states into terms 0P of three-fold degeneracy and 3P of nine-fold degeneracy.
In general, when the L values of two terms are different, we expect their electron density
distributions and, hence, the expectation values of the interaction energy in Hic to be differ-
ent. Therefore, the term values of different L’s are split. Although in the Be example both
terms are P , nonetheless, the difference in the spin configurations creates a difference in the
electron spatial distributions because of the Fermion nature of the electrons. the The singlet
spin state is antisymmetric making the spatial part symmetric with larger overlaps between
12.4. THE ATOMIC STRUCTURE 591
the two electron wave functions and, thus, higher repulsive energy. The triplet spin states are
symmetric, causing the spatial part to be antisymmetric, keeping the electrons apart and their
repulsion lower. Hence, we expect 3P to have lower energy than 0P . In Section 12.3, we
have examined the interaction of two electrons in more depth.
To correct for the spin-orbit interaction, we use the sum of the total angular momenta,
L + S = J. The resultant states can be designated by the spectroscopist’s term, 2S+1LJ in
terms of the quantum numbers S, L, J,MJ of the sum of the spin angular momenta of all the
electrons, S2, their total orbital angular momentum L2, and the total angular momentum J2
and Jz. This classification method is known as the Russel-Saunders or L-S coupling scheme.
Thus, the origin of the sodium double yellow lines was explained above. Problem 7 gives an
exercise in this reasoning for the multiplets of an excited state in the calcium atom.
There is an empirical rule, called Hund’s rule, which determines the ground state among
the different terms. Since the quantum numbers S, L, and J come from the Slater determi-
nant, they satisfy the Pauli exclusion principle. The ground state is governed by the criteria
in order: (1) the highest possible S value, (2) for the highest possible S value, the highest
possible L value, and (3) for the given S and L, the total angular momentum J be given
by the smallest possible value J = |L − S| if the shell is less than half full and the largest
possible value J = L + S if the shell is more than half full. The physical reasons follow the
two-electron example above. When the electron spins in the shell are all parallel, they have to
avoid one another spatially by the exclusion principle. When the total orbital angular momen-
tum is largest, the electrons can all keep farther away from the nucleus and have more room
to avoid one another. When the shell is less than or equal to half full, the spin orbit energy
is minimized by the smallest J . When the shell is completely full, there is zero spin-orbit
energy. Thus, for a shell which is more than half full, the spin-orbit energy is determined by
the total angular momentum of the holes in the shell. Hund’s rule works well for most of the
ground states but is not reliable for higher energy states (see the first excited states of helium
above).
Consider the example of vanadium atom (Z = 23). The configuration is (1s)2, (2s)2,
(2p)6, (3s)2, (3p)6, (4s)2, (3d)3. If we ignore the argon core and the 4s shell, we are left with
592 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
the three d electrons, (3d)3. The large value of S is 3/2 and the largest L is 6. The ground
state term is 4I . Its magnetic moment is 3µB. However, be mindful that the atomic ground
state may not be an accurate indicator of the magnetic properties of its compounds, such as
V2O3,V2O5 or VO, which depend on the valence of the vanadium ion (which could be mixed)
and its electron configuration.
For atoms with large Z, the spin-orbit interaction of an individual electron may be stronger
than the electron-electron interaction correction to the central field, in which case, the single
electron state should be classified first by its own sum of the orbital and spin angular mo-
menta. The method is known as the j-j coupling scheme. The single particle states have the
quantum numbers n�jmj instead of n�m�ms used above. The many-electron state is then
classified by the total angular momentum J2 and Jz. The total S, L are no longer good quan-
tum numbers. Each term is labeled by (j1, j2, . . .)J , where the j’s are the quantum numbers
for the individual orbitals and J is the quantum number of the total angular momentum of all
the electrons. For example, lead (Pb), Z = 82, has four active electrons, (6s)2(6p)2, outside
a closed shell. The two 6s orbitals may be considered part of the closed shell. The 6p orbital
has j = 1/2, 3/2. By the j-j coupling scheme, the terms are, in order of increasing energy,
in three clusters of (1) (1/2, 1/2)0; (2) (3/2, 1/2)1 and (3/2, 1/2)2; and (3) (3/2,3/2)2 and
(3/2, 3/2)0. The term (3/2, 3/2)1 is excluded since it would be symmetric under exchange.
The reason that the set (3) has the highest energy is the strong Coulomb repulsion for the two
electrons with the same j states and, therefore, the same spatial distribution. Compare the
sequence of energy states of Pb with the much lighter element with the same outer electron
configuration, carbon (C), Z = 6 with the configuration of (2s)2(2p)2. In the L-S coupling
scheme, the levels of C are, in increasing energy, in three clusters of (1) 3P0, 3P1, 3P2; (2)
1D2; and (3) 1S0. Note that, although the J value is in the same order of 0, 1, 2, 2, 0 as
in Pb, the first three energy levels of C are close together (split by the very weak spin-orbit
interaction) and well separated from the two higher levels.
For the rare earth series, Z = 57 — 71, the interaction correction and the spin-orbit are
comparable. The situation is intermediate between the L-S coupling limit and the j-j coupling
limit. In this regime, it is convenient to start with the L-S coupling as the basis to find the
12.5. THE NUCLEAR SHELL MODEL 593
spin-orbit corrections.
12.5 The Nuclear Shell Model
A nucleus of mass number A and charge Ze is composed of Z protons and N = A − Z
neutrons, and is denoted by AZX or
A
ZXN , where X is the chemical symbol for the element. An
old example of ours is 5626Fe or
56
26Fe30. The proton and the neutron are the two members of the
family of particles known collectively as the nucleons. The interaction between two nucleons
has been determined by low-energy scattering experiments of nuclei by nucleons. It is more
complicated than the Coulomb interaction between a proton and an electron or between two
electrons. At distances of the order of 1 fm (10−15 m), the interaction is attractive and much
stronger than the Coulomb repulsive between two protons at these distances. (In Chapter 1,
there is a problem applying the uncertainty principle to show that confinement to a distance of
1 fm cannot be due to the electrostatic attraction.) The strong interaction is nearly independent
of the nucleon charges, which makes it convenient to lump protons and neutrons together as
nucleons. The attraction is short-ranged, i.e., it decreases to zero rapidly when the distance
between two nucleons exceeds about 1 fm. As two nucleons come closer than the 1 fm range,
they repel each other before attraction. Besides a component of the interaction which can be
described as dependent on the distance between two nucleons, it has a complex functional
form, with its dependence on the spins of the nucleons and with its noncentral dependence on
the positions or the tensorial nature, which belies the complicated microscopic origin. The
interaction between two nucleons is regarded as phenomenological since it has deeper roots
in terms of exchange of other particles.
The structure of the nucleons in a nucleus, if regarded as a many-body problem of nucle-
ons interacting with the tensor and spin-dependent forces, is a complicated if not intractable
computational problem, particularly for the heavier nuclei. A simple model which yields a
good picture of the nucleon structure in a nucleus is analogous to the independent-electron
model for the electronic structure in an atom. In the atom case, electrons are filled in succes-
sive shells. By analogy, the nuclear model is known as the shell model.
In the nuclear shell model, the nucleons are regarded as two types (protons and neutrons)
594 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
of independent fermions moving in a common potential. As in the case of the electrons in an
atom, as the nucleons fill the one-particle states, each of which is in an orbital state, n�m�,
with a four-fold degeneracy, two from the spin and two from the isospin, or charged-state,
degeneracy). Across a large energy separation between two levels there is an abrupt change in
the properties, such as the nuclear radius, the two-nucleon separation energy, neutron capture
cross section, etc. The observed magic numbers for the proton number Z or the neutron
number N = A− Z are 2, 8, 20, 28, 50, 82, 126.
12.5.1 The central field approximation
To explain the magic numbers, the first thing to do is to see whether a central potential can
produce the right order of orbital energies. The simplest potential which approximates the
strong confining potential for each of the nucleon is the three dimensional harmonic potential.
The energy levels are given by (n+ 1
2
)h¯ω, where n is an integer starting from one and ω is the
frequency. The three dimensional harmonic oscillator has SU(3) symmetry, which is higher
than the rotational group of O(3) or SU(2). The degeneracy of the nth level is n(n + 1),
counting the double degeneracy of the spin states. The quantum numbers of a level is given
by (n, �), where � is the angular momentum quantum number, and n is an integer starting
from � + 1 labeling the states of a given � in the order of increasing energy (in the same way
as the principal quantum number in the Coulomb potential case). Analogous to the atomic
case, the order of the energy levels in increasing energies by steps of h¯ω line by line is:
1s,
2p,
3d, 2s,
4f, 3p,
5g, 4d, 3s,
6h, 5f, 4p,
7i, 6g, 5d, 4s,
. . .
Unlike the atomic potentials and the potentials considered below, the states in each row here
12.5. THE NUCLEAR SHELL MODEL 595
have the same energy. The magic numbers from this table are 2, 8, 20, 40, 70, 112, 168
. . . . The first three numbers are in agreement with the ones deduced from experiment but the
higher levels appear to have higher degeneracy than experiment. For example, the next group
after the third shell should have 8 states and not 20.
A different potential breaking the SU(3) symmetry will produce a different set of magic
numbers. The infinite-well potential (with confinement inside a sphere of radius R) gives the
energy levels in the order:
1s,
2p,
3d, 2s,
4f,
5g,3p,
6h, 4d, 3s,
7i, 5f, 4p,
8j, 6g,
. . .
The magic numbers are then 2, 8, 20, 34, 58, 92, 138. They are about as good as the harmonic
potential in comparison with experiment.
A more realistic potential would be a square-well potential (−V0 insider a sphere of radius
R and zero outside). A further improvement is provided by smoothing the sharp step with a
Fermi function:
V (r) = − V0
1 + e(r−R)/a
. (12.5.1)
The step is rounded off in a thickness of the order of a about the radius R. With the phe-
nomenological formulas, R = 1.25A1/3 fm and a = 0.524 fm, and V0 ∼ 50 MeV, the energy
levels appear in the order:
1s,
2p,
3d, 2s,
4f, 3p,
596 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
5g,
6h, 4d, 3s,
5f, 4p,
7i, 6g, 5d, 4s,
. . .
The magic numbers are 2, 8, 20, 40, 58, 92, 112, . . . .
12.5.2 The spin-orbit interaction
Once we have a physically reasonable central potential, clearly there is no point in playing the
game of finding a central potential which would gives the right magic number if the price is
a contorted potential which makes no physical sense. What made the shell model successful
was the introduction of the spin-orbit interaction by Mayer, Haxel, Suess, and Jensen:
Hso = −λL · S, (12.5.2)
where λ is positive. For the spin-orbit interaction for a nucleon to be strong enough to alter
the energy levels, it cannot be of electromagnetic origin as is the case in the atoms. The
energy splitting increases with �. The energy levels appear in the order:
1s1/2,
2p3/2, 2p1/2,
3d5/2, 2s1/2, 3d3/2,
4f7/2,
3p3/2, 4f5/2, 3p1/2, 5g9/2,
5g7/2, 4d5/2, 4d3/2, 3s1/2, 6h11/2,
6h9/2, 5f7/2, 5f5/2, 4p3/2, 4p1/2,, 7i13/2,
6g9/2, 5d5/2, 7i11/2, 6g7/2, 4s1/2, 5d3/2, 8j15/2,
. ..
The magic numbers are now 2, 8, 20, 28, 50, 82, 126, 184. The splitting of the 2p and 3d
levels do not change the groupings of the levels. The splitting of the 4f level divides the third
shell into a new third shell of 4f7/2 only and a fourth shell containing 4f5/2, which decreases
12.6. THE FERMI GAS 597
the third magic number from 40 to 28. The isolated 1g level is split into 5g9/2 which joins the
lower shell and 5g9/2 which joins the upper shell. A similar event occurs with the 6h level.
Because of the strong spin-orbit coupling, the j-j coupling model seems to be a good
approximation to account for the nuclear ground states. If a nucleus has an even number of
protons and an even number of neutrons, then the total J for each species of nucleons is zero.
If the mass number A is odd, then the total J must be the same as the j of the nucleon at the
highest energy level. The ordering of the more closely spaced levels above may be changed
by the pairing energy of two nucleons.
12.6 The Fermi Gas
The Fermi gas is a system containing a large number of identical fermions all moving in a
constant external potential. Such a system is a good model for a number of physical systems.
It serves as the Sommerfeld model for a simple metal in which the conduction electrons are
treated as the fermions in a constant potential provided by the uniform background of the
positive ions. In the nuclear binding, there is a term proportional to the mass number A.
Thus, the binding energy per nucleon can be studied as in the system of a large number of
nucleons in a constant potential, known as the nuclear matter. In astrophysics, the white
dwarf plays an important role in the evolution of stars. The radius of the white dwarf is
an equilibrium property achieved by the balancing of the Pauli repulsion of the electron gas
against the gravitation attraction of the nuclear mass. On the other hand, at the higher density
of the neutron star the Pauli exclusion effect of the electron gas is too weak to resist the
gravitational collapse. The interior of the neutron star is modeled by a solid of neutrons.
We shall consider the simple case where the interaction between two fermions is ne-
glected. It is a necessary first step in treating the particle interaction in the next chapter.
The only consequence is then the Pauli exclusion principle. Consider a large number N of
fermions in a cube of side L. Since we are interested in the average properties per fermion
or per unit volume, surface effects will be neglected. Then, the boundary conditions for the
wave functions are unimportant. (See problems 13 and 14.) Instead of the wave function
598 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
vanishing at the boundaries of the cube, we impose the periodic boundary conditions:
ψ(x, y, z) = ψ(x + L, y, z) = ψ(x, y + L, z) = ψ(x, y, z + L). (12.6.1)
In a constant potential, the energy eigenfunction of each fermion is given by:
ψk,σ(r) = L
−3/2eik·rχσ, (12.6.2)
where χ± denotes the spin-up and down states, and k is the wave vector giving the energy
Ek,σ =
h¯2k2
2m
. (12.6.3)
The boundary conditions give
eikxL = 1, eikyL = 1, eikzL = 1, (12.6.4)
which restrict the wave vectors to
kx =
2π
L
nx, ky =
2π
L
ny, kz =
2π
L
nz, (12.6.5)
where (nx, ny, nz) are a trio of integers.
The ground state of the system of N fermions is made up of a Slater determinant of N
lowest single fermion energy states which are occupied by the fermions. To keep track of the
occupied states, we plot each permissible k as a point in a three-dimensional space, which
is known as the wave-vector space or the reciprocal space in contrast to the real or position
space of the particles. The highest occupied energy is known as the Fermi energy, EF . The
magnitude of the wave-vector at the Fermi energy is known as the Fermi radius kF , related to
the Fermi energy by
EF =
h¯2k2F
2m
. (12.6.6)
Thus, the occupied states are those with wave vectors inside the sphere of radius kF in the
reciprocal space. The spherical surface dividing the occupied and unoccupied states is known
as the Fermi surface.
To study systematically the ground-state properties of the system, we introduce the occu-
pation function for state k:
fk = 1 (12.6.7)
12.6. THE FERMI GAS 599
if the state is occupied, i.e. k ≤ kF or Ek ≤ EF ;
fk = 0 (12.6.8)
if the state is unoccupied, i.e. k > kF or Ek > EF .
The number of occupied states is twice the number of k points inside the Fermi surface
since there are two spin states for each wave vector. Thus,
N = 2
∑
n
fk = 2
(
L
2π
)3 ∫
d3k fk = 2
(
L
2π
)3
4π
3
k3F = L
3 k
3
F
3π2
. (12.6.9)
This formula is correct for large L and N . We have neglected the minor zigzags around the
Fermi surface of the occupied wave vectors. The density of fermions is
n =
N
L3
=
k3F
3π2
. (12.6.10)
In terms of the Fermi energy, the density is
n =
1
3π2
(
2m
h¯2
)3/2
E
3/2
F . (12.6.11)
From the above formula, we see that the number of states per unit volume with energy
less than E is given by
n(E) =
1
3π2
(
2m
h¯2
)3/2
E3/2. (12.6.12)
The density of states which is defined as the number of states per unit energy range per unit
volume is
g(E) =
dn(E)
dE
=
1
2π2
(
2m
h¯2
)3/2
E1/2, (12.6.13)
proportional to the square root of the energy E.
The ground-state energy of the system is given by
ET = 2
∑
n
fkEk. (12.6.14)
To evaluate the sum, we may proceed to integrate as in the case of the total number of
fermions N or we may make use of the density of states,
ET = L
3
∫ EF
0
dE g(E)E =
3
5
NEF . (12.6.15)
600 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
The average energy per particle is 3/5 of the Fermi energy.
The corresponding properties at finite temperatures may be obtained by replacing the
occupation function by the Fermi distribution function.
12.6.1 White dwarf
A white dwarf is a cold star made up of roughly equal numbers of protons, neutrons and
electrons. The kinetic energy of the electrons is much large than that of the protons and of
the neutrons. The star is prevented from the collapse due to gravitational attraction by the
pressure generated from the electron kinetic energy. Assuming that the electron density in a
white dwarf of radius R is constant, we wish to deduce the equilibrium radius of the star of
mass M .
The potential energy of the spherical star due to gravitational attraction is
V = − 3
5
GM2
R
. (12.6.16)
The number of protons and neutrons and electrons are
Np = Nn = Ne. (12.6.17)
The mass of the star is thus
M = Ne(mp + mn + me) � 2Nemp. (12.6.18)
The electron density is
n =
3Ne
4πR3
=
3M
8πmpR3
, (12.6.19)
and the Fermi energy is
EF =
h¯2
2me
(3π2n)2/3 =
h¯2
2me
(
9πM
8R3mp
)2/3
. (12.6.20)
The total kinetic energy of the electron gas is then
K =
3
5
NeEF =
3(9π)2/3h¯2M5/3
80mem
5/3
p R2
. (12.6.21)
12.7. SCATTERING BETWEEN TWO IDENTICAL PARTICLES 601
If we approximate the total energy of the star by the sum of the electron kinetic energy
and the gravitational energy and minimize it with respect to R, we obtain the equilibrium
radius
R =
(9π)2/3h¯2
8Gmem
5/3
p M1/3
. (12.6.22)
Note that the radius of the white dwarf is inversely proportional to the cube root of its mass.
With the mass of the star of the order of the solar mass (1030 Kg), the density is 109 Kg/m3
or a million times that of water.
12.7 Scattering between two identical particles
The question is how the identity of two particles changes the scattering probability in a col-
lision experiment. By classical reasoning, we would expect thedifferential cross section for
the spherical potential scattering to be
dσ
dΩ
= |f(θ)|2 + |f(π − θ)|2, (12.7.1)
where f(θ) is the scattering amplitude at the deflection through an angle θ. Figure 12.2 shows
how the indistinguishability of the two particles adds an additional term. Not surprisingly,
the permutations symmetry of the wave functions of the two particles leads to an interference
term due to the overlap on close approach. In the following, we give a simple treatment to
illustrate the differences to the classical result for bosons and fermions.
12.7.1 Scattering between two particles without spin
Consider first two particles without spin. An example is the alpha particle (the 4He ion made
up of 2 protons and 2 neutrons). In the scattering of two identical particles moving towards
each other, the center of mass is at rest. The laboratory frame of reference is the center of
mass frame. If the incoming particle 1 wave function is given by eik·r1 , the second particle
coming in the opposite direction is e−ik·r2 . The incoming symmetrized wave function for the
two bosons is
φk(r1, r2) = e
ik·r + e−ik·r, (12.7.2)
π−θθ
k −k
602 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
Figure 12.2: Scattering between two identical particles.
where the relative position is r = r1 − r2. In the case of the potential scattering of a single
particle, the incoming wave function was chosen to have unit amplitude. The center of mass
wave function is not shown. We have not normalized the two-particle wave function because
we choose to let the total number of particles be counted in each incoming beam to be one per
unit area. This choice is convenient for later use of determining the partial scattering cross
section and would not influence the final result.
From Chapter 7, we know that the scattering wave function of a particle by a central
potential is asymptotically
ψk(r) ∼ eik·r + e
ikr
r
f(θ), (12.7.3)
θ being the angle between k and r. The symmetrized version for two particles is
ψk(r1, r2) ∼ eik·r + e−ik·r + e
ikr
r
[f(θ) + f(π − θ)]. (12.7.4)
The differential cross section is given by
dσ
dΩ
= |f(θ) + f(π − θ)|2, (12.7.5)
The flux of scattered particles captured at the detector in the r direction may come from either
incoming beam. Figure 12.2 shows that from the k beam the particle suffers a deflection of
θ while from the k beam the particle suffers a deflection of π − θ. The expression has
not been divided by 2 because we have adopted the standard convention of counting the
scattered particles per unit flux of one incoming beam. The reason for the convention might
12.7. SCATTERING BETWEEN TWO IDENTICAL PARTICLES 603
be surmised to be historical: experiments were first done by a beam of particles moving
relative to the laboratory frame against a stationary target which contained particles identical
to those in the incoming beam. Only in recent decades were accelerators designed to scatter
two beams (electrons or protons) moving in opposite directions. Thus, Eq. (12.7.5) gives the
differential cross section in the center of mass frame of the two beams.
12.7.2 Differential cross section in the laboratory frame
We wish to transform the differential cross section of two bosons obtained in the last subsec-
tion from the center of mass frame to the frame of reference in which one beam is replaced
by a stationary target of identical particles, now referred to as the laboratory frame. We first
digress to the transformation of the scattering between two particles of mass m1 and m2
between these two frames.
Figure 12.3 (a) shows the scattering between two particles in their center of mass frame.
Then the momenta of the two particles are equal and opposite both before and after the colli-
sion. The angle between the momentum p′ of particle 1 after the collision and its momentum
p before is shown as θ. For elastic scattering, the magnitudes are equal, p = p′. Figure
12.3 (b) shows the scattering of particle 1 with momentum pL colliding with particle 2 at
rest. The momentum of particle 1 after collision is p′L with the angle of deflection θL in the
laboratory frame. The center of mass momentum is also pL and, thus, the relative velocity of
the laboratory frame to the center of mass frame is pL/M , where M = m1 + m2. Thus, the
transformation of the particle 1 velocity before collision between the two frames is depicted
by Figure 12.3 (c),
pL
m1
=
pL
M
+
p
m1
. (12.7.6)
Figure 12.3 (d) depicts the particle 1 velocity transformation after the collision,
p′L sin θL = p sin θ, (12.7.7)
p′L
m1
cos θL =
pL
M
+
p
m1
cos θ. (12.7.8)
Elimination of the momenta in the last three equations leads to the relation between the de-
1 2
p
p'
-p
-p'
1
2
p
p'
(a) (b)
θ θLL
L
(d)
pL/M
p'L/m1
θL θ
pL/M p/m1
pL/m1
(c)
604 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
Figure 12.3: (a) Scattering of two particles in the center of mass frame. (b) In the rest frame
of particle 2. (c) Transformation between two frames of the particle 1 velocity before the
collision. (d) Transformation of velocity after the collision.
12.7. SCATTERING BETWEEN TWO IDENTICAL PARTICLES 605
flection angles,
tan θL =
sin θ
cos θ + γ
, (12.7.9)
where γ = m1/m2.
Equating the numbers of particles scattering into the same solid angle cone leads to
[
dσ
dΩ
]
L
sin θLdθL =
dσ
dΩ
sin θdθ. (12.7.10)
Hence, the scattering cross section in the laboratory frame is related to that in the center of
mass frame by
[
dσ
dΩ
]
L
=
1 + γ2 + 2γ cos θ
|1 + γ cos θ|
dσ
dΩ
. (12.7.11)
For the scattering of one boson against another as a target in the laboratory frame, γ = 1
and θL = θ/2. The differential cross section is from Eq. (12.7.5),
dσ
dΩL
= 4 cos θL|f(2θL) + f(π − 2θL)|2. (12.7.12)
12.7.3 Scattering between two particles with spin 1/2
For two fermions with spin 1/2 interacting via a potential independent of the spin variables,
the incoming state in the center of mass reference frame of one fermion in the plane wave
state k and spin state σ1 and the second in the plane wave state −k and spin state σ2 has the
antisymmetrized wave function,
eik·r1χσ1(m1)e
−ik·r2χσ2(m2)− e−ik·r1χσ2(m1)eik·r2χσ1(m2), (12.7.13)
where χσ1(m1) is the spinor 〈m1|σ1〉. The scattered wave function of the two fermions by a
spherical potential is
ψkσ1σ2(r1,m1; r2,m2) = e
ik·rχσ1(m1)χσ2(m2)− e−ik·rχσ2(m1)χσ1(m2) (12.7.14)
+ [χσ1(m1)χσ2(m2)f(θ)− χσ2(m1)χσ1(m2)f(π − θ)]
eikr
r
,
where r = r1 − r2.
606 CHAPTER 12. A SYSTEM OF IDENTICAL PARTICLES
Thus, the differential cross section is given by
dσ
dΩ
= [f ∗(θ)〈σ1σ2| − f ∗(π − θ)〈σ2σ1|][|σ1σ2〉f(θ)− |σ2σ1〉f(π − θ)], (12.7.15)
where |σ1σ2〉 denotes the spin state of the first fermion one in σ1 and the second one ins σ2.
Note that in the bra state the order of the symbols does not change. Hence,
〈σ1σ2|σ1σ2〉 = 〈σ1|σ1〉〈σ2|σ2〉 = 1, (12.7.16)
〈σ1σ2|σ2σ1〉 = 〈σ1|σ2〉〈σ2|σ1〉 = 1
2
(1 + cosϑ), (12.7.17)
where ϑ is the angle between the two spin vectors of state σ1 and σ2, not to be confused with
the angle of deflection θ. The last expression is evaluated by taking the σ2 state to be along
the z axis so that the matrix representation of the σ1 and σ2 states are respectively,

 cos
ϑ
2
sin ϑ
2

 ,

 1
0

 . (12.7.18)
Hence,
〈σ1|σ2〉 = cos ϑ
2
. (12.7.19)
The differential cross section is given by
dσ
dΩ
= |f(θ)|2 + |f(π − θ)|2 − 1
2
(1 + cosϑ)[f ∗(θ)f(π − θ) + f ∗(π − θ)f(θ)]
=
1
4
(1− cosϑ)|f+(θ)|2 + 1
4
(3 + cosϑ)|f−(θ)|2, (12.7.20)
where the scattering cross sections for the symmetric and antisymmetric spatial states are
f±(θ) = f(θ)± f(π − θ). (12.7.21)