Chapter 3: Identical ParticlesY. D. Chong (2018) PH4401: Quantum Mechanics III Chapter 3: Identical Particles These our actors, As I foretold you, were all spirits and Are melted into

Y. D. Chong (2018) PH4401: Quantum Mechanics III

Chapter 3: Identical Particles

These our actors,

As I foretold you, were all spirits and

Are melted into air, into thin air:

And, like the baseless fabric of this vision,

The cloud-capp’d towers, the gorgeous palaces,

The solemn temples, the great globe itself,

Yea, all which it inherit, shall dissolve

And, like this insubstantial pageant faded,

Leave not a rack behind.

William Shakespeare, The Tempest

I. PARTICLE EXCHANGE SYMMETRY

In the previous chapter, we discussed how the principles of quantum mechanics apply tosystems of multiple particles. That discussion omitted an important feature of multi-particlesystems, namely the fact that particles of the same type are fundamentally indistinguishablefrom each other. As it turns out, indistinguishability imposes a strong constraint on theform of the multi-particle quantum states, and looking into this will ultimately lead us to afundamental re-interpretation of what “particles” are.

Suppose we have two particles of the same type, e.g. two electrons. It is a fact of Naturethat all electrons have identical physical properties: the same mass, same charge, same totalspin, etc. As a consequence, the single-particle Hilbert spaces of the two electrons must bemathematically identical. Let us denote this space by H (1). For a two-electron system, theHilbert space is a tensor product of two single-electron Hilbert spaces, denoted by

H (2) = H (1) ⊗H (1). (1)

Moreover, any Hamiltonian must affect the two electrons in a symmetrical way. An exampleof such a Hamiltonian is

H =1

2me

(|p1|2 + |p2|2

)+

e2

4πε0|r1 − r2|, (2)

consisting of the non-relativistic kinetic energies and the Coulomb potential energy. Oper-ators p1 and r1 act on electron 1, while p2 and r2 act on electron 2.

Evidently, this Hamiltonian is invariant under an interchange of the operators acting on thetwo electrons (i.e., p1 ↔ p2 and r1 ↔ r2). This can be regarded as a kind of symmetry, calledexchange symmetry. As we know, symmetries of quantum systems can be represented byunitary operators that commute with the Hamiltonian. Exchange symmetry is representedby an operator P , defined as follows: let {|µi〉} be a basis for the single-electron Hilbert

1


space H (1); then P interchanges the basis vectors for the two electrons:

P(∑

ij

ψij|µi〉|µj〉)≡∑ij

ψij|µj〉|µi〉

=∑ij

ψji|µi〉|µj〉 (interchanging i↔ j in the double sum)(3)

The exchange operator has the following properties:

1. P 2 = I , where I is the identity operator.

2. P is linear, unitary, and Hermitian (see Exercise 1).

3. The effect of P does not depend on the choice of basis (see Exercise 1).

4. P commutes with the above Hamiltonian H, or indeed any Hamiltonian where thesingle-particle operators appear in a symmetrical manner (see Exercise 2).

According to Noether’s theorem, any symmetry implies a conservation law. In the case ofexchange symmetry, P is both Hermitian and unitary, so we can take the conserved quantityto be the eigenvalue of P itself. We call this eigenvalue, p, the exchange parity. Giventhat P 2 = I, there are just two possibilities:

P |ψ〉 = p|ψ〉 ⇒ p =

{+1 (“symmetric state”), or

−1 (“antisymmetric state”).(4)

Since P commutes with H, if the system starts out in an eigenstate of P with parity p, itretains the same definite parity at subsequent times.

The concept of exchange parity generalizes to systems of more than two particles. GivenN particles, we can define a set of exchange operators Pij, where i, j ∈ {1, 2, . . . , N} and

i 6= j, such that Pij exchanges particle i and particle j. If the particles are identical,the Hamiltonian must commute with all the exchange operators, so the parities (±1) areindividually conserved.

We now invoke the following postulates:

1. A multi-particle state of identical particles is an eigenstate of every Pij.

2. For each Pij, the exchange parity pij has the same value: i.e., all +1 or all −1.

3. The exchange parity pij is determined solely by the type of particle involved.

You should not think of these as statements as being “derived” from more fundamental facts!Rather, they are hypotheses about the way particles behave—hypotheses that physicists havemanaged to deduce by looking at a variety of empirical facts. Our task here is to explorethe consequences of these hypotheses.

Particles that have symmetric states (pij = +1) are called bosons. It turns out that the ele-mentary particles that “carry” the fundamental forces are all bosons: these are the photons

2


(elementary particles of light, which carry the electromagnetic force), gluons (elementaryparticles that carry the strong nuclear force, responsible for binding protons and neutronstogether), and W and Z bosons (particles that carry the weak nuclear force responsiblefor beta decay). Other bosons include particles that carry non-fundamental forces, such asphonons (particles of sound), as well as certain composite particles such as alpha particles(helium-4 nuclei).

Particles that have antisymmetric states (pij = −1) are called fermions. All the elementaryparticles of “matter” are fermions: electrons, muons, tauons, quarks, neutrinos, and theiranti-particles (positrons, anti-neutrinos, etc.). Certain composite particles are also fermions,the most notable being protons and neutrons, which are each composed of three quarks.

II. BOSONS

A state of N bosons must be symmetric under every possible exchange operator:

Pij |ψ〉 = |ψ〉 ∀ i, j ∈ {1, . . . , N}, i 6= j. (5)

There is a standard way to construct multi-particle states obeying this symmetry condition.First, consider a two-boson system (N = 2). If both bosons occupy the same single-particlestate, |ϕ〉 ∈H (1), the two-boson state is simply

|ϕ, ϕ〉 = |ϕ〉|ϕ〉. (6)

This evidently satisfies P12|ϕ, ϕ〉 = |ϕ, ϕ〉, as required. Next, suppose the two bosons occupydifferent single-particle states, |ϕ1〉 and |ϕ2〉. It would be wrong to write the two-bosonstate as |ϕ1〉|ϕ2〉, because the particles would not be symmetric under exchange. Instead,we construct the multi-particle state

|ϕ1, ϕ2〉 =1√2

(|ϕ1〉|ϕ2〉+ |ϕ1〉|ϕ2〉

). (7)

This has the appropriate exchange symmetry:

P12 |ϕ1, ϕ2〉 =1√2

(|ϕ2〉|ϕ1〉+ |ϕ1〉|ϕ2〉

)= |ϕ1, ϕ2〉. (8)

This construction can be generalized to arbitrary N . Suppose the N particles occupy single-particle states enumerated by

|φ1〉, |φ2〉, |φ3〉, . . . , |φN〉. (9)

Here, we use the symbol φ, rather than ϕ, to indicate that the states in this list need notbe unique. For example, we could have |φ1〉 = |φ2〉 = |ϕ1〉, meaning there is a certainsingle-particle state |ϕ1〉 occupied by two particles.

The N -boson state can now be written as

|φ1, φ2, . . . , φN〉 = N∑p

(|φp(1)〉|φp(2)〉|φp(3)〉 · · · |φp(N)〉

). (10)

3


The sum is taken over each of the N ! permutations acting on {1, 2, . . . , N}. For eachpermutation p, we let p(j) denote the integer that j is permuted into. The prefactor N is anormalization constant, and it can be shown that its appropriate value is

N =

√1

N !na!nb! · · ·, (11)

where nµ denotes the number of particles in state µ, and N = nα + nβ + · · · is the totalnumber of particles. The proof of this is left as an exercise (Exercise 3).

To see that the above N -particle state is symmetric under exchange, apply an arbitraryexchange operator Pij:

Pij|φ1, φ2, . . . , φN〉 = N∑p

Pij

(· · · |φp(i)〉 · · · |φp(j)〉 · · ·

)= N

∑p

(· · · |φp(j)〉 · · · |φp(i)〉 · · ·

).

(12)

In each term of the sum, two states i and j are interchanged. But since the sum runs throughall permutations of the states, the result is the same with or without the exchange, so westill end up with |φ1, φ2, . . . , φN〉. Therefore, the multi-particle state is symmetric underevery possible exchange operation.

As an example, consider a three-boson system with two particles in state |ϕ1〉, and oneparticle in state |ϕ2〉. To express the three-particle state, we define {|φ1〉, |φ2〉, |φ3〉} suchthat |φ1〉 = |φ2〉 = |ϕ1〉 and |φ3〉 = |ϕ2〉. Then

|φ1, φ2, φ3〉 = N(|φ1〉|φ2〉|φ3〉+ |φ2〉|φ3〉|φ1〉+ |φ3〉|φ1〉|φ2〉

+ |φ1〉|φ3〉|φ2〉+ |φ3〉|φ2〉|φ1〉+ |φ2〉|φ1〉|φ3〉)

= 2N(|ϕ1〉|ϕ1〉|ϕ2〉+ |ϕ1〉|ϕ2〉|ϕ1〉+ |ϕ2〉|ϕ1〉|ϕ1〉

).

(13)

The three distinct exchange symmetry operators have the following effects:

P12|φ1, φ2, φ3〉 = 2N(|ϕ1〉|ϕ1〉|ϕ2〉+ |ϕ2〉|ϕ1〉|ϕ1〉+ |ϕ1〉|ϕ2〉|ϕ1〉

)= |φ1, φ2, φ3〉

P23|φ1, φ2, φ3〉 = 2N(|ϕ1〉|ϕ2〉|ϕ1〉+ |ϕ1〉|ϕ1〉|ϕ2〉+ |ϕ2〉|ϕ1〉|ϕ1〉

)= |φ1, φ2, φ3〉

P13|φ1, φ2, φ3〉 = 2N(|ϕ2〉|ϕ1〉|ϕ1〉+ |ϕ1〉|ϕ2〉|ϕ1〉+ |ϕ1〉|ϕ1〉|ϕ2〉

)= |φ1, φ2, φ3〉.

(14)

III. FERMIONS

A state of N fermions must be antisymmetric under every possible exchange operator:

Pij |ψ〉 = −|ψ〉 ∀ i, j ∈ {1, . . . , N}, i 6= j. (15)

Similar to the bosonic case, we can explicitly construct multi-fermion states based on theoccupancy of single-particle states. Let us consider the N = 2 case first. If the fermions

4


occupy states |ϕ1〉 and |ϕ2〉, the appropriate two-particle state is

|ϕ1, ϕ2〉 =1√2

(|ϕ1〉|ϕ2〉 − |ϕ2〉|ϕ1〉

). (16)

We can easily check that

P12|ϕ1, ϕ2〉 =1√2

(|ϕ2〉|ϕ1〉 − |ϕ1〉|ϕ2〉

)= −|ϕ1, ϕ2〉. (17)

This fails if we try to let |ϕ1〉 and |ϕ2〉 be the same state: in that case, the two terms cancelto give the zero vector, which is not a valid quantum state. From this, we can deduce thePauli exclusion principle, which states that two fermions cannot occupy the same state;or, equivalently, that each single-particle state is either unoccupied, or occupied by exactlyone fermion.

For general N , let us enumerate the occupied single-particle states by {|φ1〉, |φ2〉, . . . , |φN〉}.Then the appropriate N -fermion state is

|φ1, . . . , φN〉 =1√N !

∑p

s(p) |φp(1)〉|φp(2)〉 · · · |φp(N)〉. (18)

The 1/√N ! prefactor is a normalization constant (as you can verify). The expression involves

a sum over every permutation p of the sequence {1, 2, . . . , N}. Each term in the sum hasa coefficient s(p), denoting the parity of the permutation. The parity of a permutationp is defined as +1 if p involves an even number of exchanges starting from the sequence{1, 2, . . . , N}, and −1 if p involves an odd number of exchanges.

A couple of examples are helpful for understanding these definitions. First, consider N = 2;the sequence {1, 2} has 2! = 2 permutations:

p1 : {1, 2} → {1, 2}, s(p1) = +1

p2 : {1, 2} → {2, 1}, s(p2) = −1.(19)

Next, consider N = 3. The sequence {1, 2, 3} has 3! = 6 permutations:

p1 : {1, 2, 3} → {1, 2, 3}, s(p1) = +1

p2 : {1, 2, 3} → {2, 1, 3}, s(p2) = −1

p3 : {1, 2, 3} → {2, 3, 1}, s(p3) = +1

p4 : {1, 2, 3} → {3, 2, 1}, s(p4) = −1

p5 : {1, 2, 3} → {3, 1, 2}, s(p5) = +1

p6 : {1, 2, 3} → {1, 3, 2}, s(p6) = −1.

(20)

The permutations can be generated by exchanging pairs of sequence elements; each time weperform such an exchange, the sign of s(p) is reversed.

We can now see why the above N -particle state describes fermions. Let us apply Pij to it:

Pij|φ1, . . . , φN〉 =1√N !

∑p

s(p) Pij[· · · |φp(i)〉 · · · |φp(j)〉 · · ·

]=

1√N !

∑p

s(p)[· · · |φp(j)〉 · · · |φp(i)〉 · · ·

].

(21)

5


Consider each term in the above sum. Here, the single-particle state for p(i) and p(j) haveexchanged places. The resulting term must be a match for another one of the terms in|φ1, . . . , φN〉, because the sum runs over all possible permutations—except for one thing: thecoefficient s(p) must have an opposite sign, since the two permutations are related by an

exchange. It follows that Pij|φ1, . . . , φN〉 = −|φ1, . . . , φN〉, as desired for fermions.

If the occupied single-particle states are non-unique, this construction breaks down, as theterms in the sum over p cancel out pairwise. This is again consistent with the Pauli exclusionprinciple.

IV. SECOND QUANTIZATION

In the usual tensor product notation, it cumbersome to deal with symmetric and antisym-metric states. We will now introduce a more convenient formalism, called second quanti-zation. (The reason for the name “second quantization” will not be apparent until later; itis a bad name, but one we are stuck with for historical reasons.)

We start by defining a convenient way to specify states of multiple identical particles, calledthe occupation number representation. First, let us enumerate a set of single-particlestates, {|ϕ1〉, |ϕ2〉, |ϕ3〉, · · · }, that form a complete orthonormal basis for the single-particleHilbert space H (1). Then, we build multi-particle states by specifying how many particlesare in state |ϕ1〉, denoted n1; how many are in state |ϕ2〉, denoted n2; and so on. Thus,

|n1, n2, n3, . . . 〉

is a multi-particle state defined as the appropriate symmetric (for bosons) or antisymmetric(for fermions) tensor product state. The multi-particle state is indexed by the occupationnumbers {n1, n2, · · · }, one integer for each single-particle state in {|ϕ1〉, |ϕ2〉, |ϕ3〉, · · · }.Let us run through a couple of examples. First, consider |0, 2, 0, 0, . . . 〉, a two-particle statewith both particles in state |ϕ2〉. (This only makes sense if the particles are bosons, asfermions cannot share the same state.) The symmetric tensor product state is

|0, 2, 0, 0, . . . 〉 ≡ |ϕ2〉|ϕ2〉. (22)

As our second example, consider |1, 1, 1, 0, 0, . . . 〉. This specifies a three-particle state wherethere is one particle each occupying |ϕ1〉, |ϕ2〉, and |ϕ3〉. If the particles are bosons, thiscorresponds to the symmetric state

|1, 1, 1, 0, 0, . . . 〉 ≡ 1√6

(|ϕ1〉|ϕ2〉|ϕ3〉+ |ϕ3〉|ϕ1〉|ϕ2〉+ |ϕ2〉|ϕ3〉|ϕ1〉

+ |ϕ1〉|ϕ3〉|ϕ2〉+ |ϕ2〉|ϕ1〉|ϕ3〉+ |ϕ3〉|ϕ2〉|ϕ1〉).

(23)

And if the particles are fermions, the appropriate antisymmetric state is

|1, 1, 1, 0, 0, . . . 〉 ≡ 1√6

(|ϕ1〉|ϕ2〉|ϕ3〉+ |ϕ3〉|ϕ1〉|ϕ2〉+ |ϕ2〉|ϕ3〉|ϕ1〉

− |ϕ1〉|ϕ3〉|ϕ2〉 − |ϕ2〉|ϕ1〉|ϕ3〉 − |ϕ3〉|ϕ2〉|ϕ1〉).

(24)

6


A. Fock space

At this point, we have to make a technical detour regarding what Hilbert space the statevectors reside in. The state |0, 2, 0, 0, . . . 〉 is a bosonic two-particle state, which is a vectorin the two-particle Hilbert space H (2) = H (1) ⊗H (1). However, H (2) also contains two-particle states that are not symmetric under exchange, which is not allowed for bosons.Thus, it would be more rigorous for us to narrow the Hilbert space to the space of state

vectors that are symmetric under exchange. We denote this reduced space by H (2)S .

Likewise, |1, 1, 1, 0, . . . 〉 is a three-particle state lying in H (3). If the particles are bosons, we

can narrow the space to H (3)S . If the particles are fermions, we can narrow it to the space

of three-particle states that are antisymmetric under exchange, denoted by H (3)A . Thus,

|1, 1, 1, 0, . . . 〉 ∈ H (3)S/A, where the subscript S/A depends on whether we are dealing with

symmetric states (S) or antisymmetric states (A).

We can make the occupation number representation more convenient to work with bydefining an “extended” Hilbert space, called the Fock space, that is the space ofbosonic/fermionic states for arbitrary numbers of particles. In the formal language of linearalgebra, the Fock space can be written as

H FS/A = H (0) ⊕H (1) ⊕H (2)

S/A ⊕H (3)S/A ⊕H (4)

S/A ⊕ · · · (25)

Here, ⊕ represents the direct sum operation, which combines vector spaces by directlygrouping their basis vectors into a larger basis set; if H1 has dimension d1 and H2 hasdimension d2, then H1 ⊕H2 has dimension d1 + d2. (By contrast, the space H1 ⊗H2,defined via the tensor product, has dimension d1d2.) Once again, the subscript S/A dependson whether we are dealing with bosons (S) or fermions (A).

The upshot is that any multi-particle state that we can write down in the occupation numberrepresentation, |n1, n2, n3, . . . 〉, is guaranteed to lie in the Fock space H F

S/A. Moreover, these

states form a complete basis set for H FS/A.

You may have noticed that the definition of the Fock space included H (0), the “Hilbertspace of 0 particles”. This Hilbert space contains only one distinct state vector, denoted by

|∅〉 ≡ |0, 0, 0, 0, . . . 〉. (26)

This is called the vacuum state, and represents a state in which there are literally noparticles. Note that |∅〉 is not the same thing as a zero vector, but instead satisfies theusual normalization for state vectors, 〈∅|∅〉 = 1. The concept of a “state of no particles”may seem silly, but we will see that there are very good technical reasons for including it inthe formalism. (It will also turn out to have an important physical meaning; see Section V.)

Another subtle consequence of introducing the Fock space concept is that it is now legitimateto write down quantum states that lack definite particle numbers. For example,

1√2

(|1, 0, 0, 0, 0, . . . 〉+ |1, 1, 1, 0, 0, . . . 〉

)is a valid state vector in H F

S/A, consisting of the superposition of a one-particle state and athree-particle state. Such states are not mere curiosities. For instance, in the next chapter

7


we will show that the optical mode of a laser beam can be characterized by a quantum statewith an indefinite number of photons.

B. Second quantization for bosons

After this lengthy prelude, we are ready to introduce the formalism of second quantization.Let us concentrate on bosons first.

We define an operator called the particle creation operator, denoted by a†µ and actingin the following way:

a†µ∣∣n1, n2, . . . , nµ, . . .

⟩=√nµ + 1

∣∣n1, n2, . . . , nµ + 1, . . .⟩. (27)

In this definition, there is one particle creation operator for each state in the single-particlebasis {|ϕ1〉, |ϕ2〉, . . . }. Each creation operator is defined as an operator acting on statevectors in the Fock space H F

S , and has the effect of incrementing the occupation number ofits single-particle state by one. The prefactor of

√nµ + 1 is defined for later convenience.

In particular, applying a creation operator to the vacuum state yields a single-particle state:

a†µ|∅〉 = |0, . . . , 0, 1, 0, 0, . . . 〉.�µ

(28)

The Hermitian conjugate of the creation operator, denoted by aµ, is called the particleannihilation operator. To observe its effects, let us first take the Hermitian conjugate ofthe equation that defines a†µ:⟨

n1, n2, . . .∣∣aµ =

√nµ + 1

⟨n1, n2, . . . , nµ + 1, . . .

∣∣. (29)

Right-multiplying by another occupation number state |n′1, n′2, . . . 〉 results in⟨n1, n2, . . .

∣∣aµ∣∣n′1, n′2, . . . ⟩ =√nµ + 1

⟨. . . , nµ + 1, . . .

∣∣ . . . , n′µ, . . . ⟩=√nµ + 1 δn1

n′1δn2

n′2· · · δnµ+1

n′µ. . .

=√n′µ δ

n1

n′1δn2

n′2· · · δnµ+1

n′µ· · ·

(30)

From this, we can deduce that

aµ∣∣n′1, n′2, . . . , n′µ, . . . ⟩ =

{√n′µ∣∣n′1, n′2, . . . , n′µ − 1, . . .

⟩, if n′µ > 0

0, if n′µ = 0.(31)

In other words, the annihilation operator reduces the occupation number of one of the single-particle states by one (hence its name). As a special exception, if the given single-particlestate is unoccupied (nµ = 0), then applying aµ results in a zero vector (note that this is notthe same thing as the vacuum state |∅〉).The boson creation/annihilation operators obey the following commutation relations:

[aµ, aν ] = [a†µ, a†ν ] = 0,

[aµ, a†ν ] = δµν

for all µ, ν.

8


These can be derived by taking the matrix elements with respect to the occupation numberbasis. We will go through the derivation of the last commutation relation; the others areleft as an exercise (Exercise 4).

To prove that [aµ, a†ν ] = δµν , we first consider the case where the creation/annihilation

operators act on the same single-particle state:⟨n1, n2, . . .

∣∣aµa†µ∣∣n′1, n′2 . . . ⟩ =√

(nµ + 1)(n′µ + 1)⟨. . . , nµ + 1, . . .

∣∣ . . . , n′µ + 1, . . .⟩

=√

(nµ + 1)(n′µ + 1)δn1

n′1δn2

n′2· · · δnµ+1

n′µ+1 · · ·

= (nµ + 1)δn1

n′1δn2

n′2· · · δnµn′µ · · ·⟨

n1, n2, . . .∣∣a†µaµ∣∣n′1, n′2 . . . ⟩ =

√nµn′µ

⟨. . . , nµ − 1, . . .

∣∣ . . . , n′µ − 1, . . .⟩

=√nµn′µδ

n1

n′1δn2

n′2· · · δnµ−1n′µ−1 · · ·

= nµδn1

n′1δn2

n′2· · · δnµn′µ · · ·

(32)

In the second equation, we were a bit sloppy in handling the nµ = 0 and n′µ = 0 cases, butyou can check for yourself that the result on the last line remains correct. Upon taking thedifference of the two equations, we get⟨

n1, n2, . . .∣∣ (aµa†µ − a†µaµ) ∣∣n′1, n′2 . . . ⟩ = δn1

n′1δn2

n′2· · · δnµn′µ · · · =

⟨n1, n2, . . .

∣∣n′1, n′2 . . . ⟩. (33)

Since the occupation number states form a basis set for H FS , we conclude that

aµa†µ − a†µaµ = I . (34)

Next, consider the case where µ 6= ν:⟨n1, . . .

∣∣aµa†ν∣∣n′1, . . . ⟩ =√

(nµ + 1)(n′ν + 1) 〈. . . , nµ + 1, . . . , nν , . . . | . . . , n′µ, . . . , n′ν + 1, . . . 〉

=√n′µnν δ

n1

n′1· · · δnµ+1

n′µ· · · δnνn′ν+1 · · ·⟨

n1, . . .∣∣a†ν aµ∣∣n′1, . . . ⟩ =

√n′µnν 〈. . . , nµ, . . . , nν − 1, . . . | . . . , n′µ − 1, . . . , n′ν . . . 〉

=√n′µnν δ

n1

n′1· · · δnµn′µ−1 · · · δ

nν−1n′ν· · ·

=√n′µnν δ

n1

n′1· · · δnµ+1

n′µ· · · δnνn′ν+1 · · ·

Hence,aµa

†ν − a†ν aµ = 0 for µ 6= ν. (35)

Combining these two results gives the desired commutation relation, [aµ, a†ν ] = δµν . Another

useful result which emerges from the first part of the proof is that⟨n1, n2, . . .

∣∣a†µaµ∣∣n′1, n′2 . . . ⟩ = nµ⟨n1, n2, . . .

∣∣n′1, n′2 . . . ⟩. (36)

Hence, we can define the Hermitian operator

nµ ≡ a†µaµ, (37)

9


whose eigenvalue is the occupation number of single-particle state µ.

If you are familiar with the method of creation/annihilation operators for solving the quan-tum harmonic oscillator, you will have noticed the striking similarity with the particle cre-ation/annihilation operators for bosons. This is no mere coincidence. We will examine therelationship between harmonic oscillators and bosons in the next chapter.

C. Second quantization for fermions

For fermions, the multi-particle states are antisymmetric. The fermion creation operatorcan be defined as follows:

c†µ|n1, n2, . . . , nµ, . . . 〉 =

{(−1)n1+n2+···+nµ−1|n1, n2, . . . , nµ−1, 1, . . . 〉 if nµ = 0

0 if nµ = 1.

= (−1)n1+n2+···+nµ−1 δnµ0

∣∣n1, n2, . . . , nµ−1, 1, . . .⟩.

(38)

In other words, if state µ is unoccupied, then c†µ increments the occupation number to 1,and multiplies the state by an overall factor of (−1)n1+n2+···+nµ−1 (i.e, +1 if there is an evennumber of occupied states preceding µ, and −1 if there is an odd number). The role of thisfactor will be apparent later. Note that this requires us to order the single-particle states;otherwise, it would not make sense to speak of the states “preceding” µ. It does not matterwhich ordering we choose, so long as we make some choice, and stick to it consistently.

According to this definition, if µ is occupied, applying c†µ gives the zero vector. The occu-pation numbers are therefore forbidden from being larger than 1.

The conjugate operator, cµ, is the fermion annihilation operator. To see what it does, takethe Hermitian conjugate of the definition of the creation operator:

〈n1, n2, . . . , nµ, . . . |cµ = (−1)n1+n2+···+nµ−1 δnµ0

⟨n1, n2, . . . , nµ−1, 1, . . .

∣∣. (39)

Right-multiplying this by |n′1, n′2, . . . 〉 gives

〈n1, n2, . . . , nµ, . . . |cµ|n′1, n′2, . . . 〉 = (−1)n1+···+nµ−1 δn1

n′1· · · δnµ−1

n′µ−1

(δnµ0 δ1n′µ

)δnµ+1

n′µ+1. . . (40)

Hence, we can deduce that

cµ|n′1, . . . , n′µ, . . . 〉 =

{0 if n′µ = 0

(−1)n′1+···+n′µ−1|n′1, . . . , n′µ−1, 0, . . . 〉 if n′µ = 1.

= (−1)n′1+···+n′µ−1 δ1n′µ

∣∣n′1, . . . , n′µ−1, 0, . . . ⟩.(41)

In other words, if state µ is unoccupied, then applying cµ gives the zero vector; if state µ isoccupied, applying cµ decrements the occupation number to 0, and multiplies the state bythe aforementioned factor of ±1.

With these definitions, the fermion creation/annihilation operators can be shown to obeythe following anticommutation relations:{

cµ, cν}

={c†µ, c

†ν

}= 0,{

cµ, c†ν

}= δµν

for all µ, ν.

10


Here, {·, ·} denotes an anticommutator, which is defined as follows:{A, B

}≡ AB + BA. (42)

Similar to the bosonic commutation relations, these anticommutation relations can be de-rived by taking matrix elements with occupation number states. We will only go over thelast one,

{cµ, c

†ν

}= δµν ; the other derivations follow similar lines.

Firstly, consider the case where the creation and annihilation operators act on the samesingle-particle state µ:⟨

. . . , nµ, . . .∣∣cµc†µ∣∣ . . . , n′µ, . . . ⟩ = (−1)n1+···+nµ−1(−1)n

′1+···+n′µ−1 δ

nµ0 δ0n′µ

×⟨n1, . . . , nµ−1, 1, . . .

∣∣n′1, . . . , n′µ−1, 1, . . . ⟩= δ0n′µ · δ

n1

n′1δn2

n′2· · · δnµn′µ · · ·

(43)

And by a similar calculation,⟨. . . , nµ, . . .

∣∣c†µcµ∣∣ . . . , n′µ, . . . ⟩ = δ1n′µ · δn1

n′1δn2

n′2· · · δnµn′µ · · · (44)

By adding these two equations, and using the fact that δ0n′µ + δ1n′µ = 1, we get⟨. . . , nµ, . . .

∣∣ {cµ, c†µ} ∣∣ . . . , n′µ, . . . ⟩ =⟨. . . , nµ, . . .

∣∣ . . . , n′µ, . . . ⟩ (45)

And hence, {cµ, c

†µ

}= I . (46)

Next, we must prove that{cµ, c

†ν

}= 0 for µ 6= ν. We will show this for µ < ν (the µ > ν

case follows by Hermitian conjugation). This is, once again, by taking matrix elements:⟨. . . , nµ, . . . , nν , . . .

∣∣cµc†ν∣∣ . . . , n′µ, . . . , n′ν , . . . ⟩ = (−1)n1+···+nµ−1(−1)n′1+···+n′ν−1 δ

nµ0 δ0n′ν

×⟨. . . , 1, . . . , nν , . . .

∣∣ . . . , n′µ, . . . , 1, . . . ⟩= (−1)n

′µ+···+n′ν−1 δn1

n′1δn2

n′2· · ·(δnµ0 δ1n′µ

)· · ·(δnν1 δ0n′ν

)· · ·

= (−1)1+nµ+1+···+nν−1 δn1

n′1δn2

n′2· · ·(δ0nµδ

1n′µ

)· · ·(δ0n′νδ

1nν

)· · ·⟨

. . . , nµ, . . . , nν , . . .∣∣c†ν cµ∣∣ . . . , n′µ, . . . , n′ν , . . . ⟩ = (−1)n1+···+nν−1(−1)n

′1+···+n′µ−1 δnν1 δ1n′µ

×⟨. . . , nµ, . . . , 0, . . .

∣∣ . . . , 0, . . . , n′ν , . . . ⟩= (−1)nµ+···+nν−1 δn1

n′1δn2

n′2· · ·(δnµ0 δ1n′µ

)· · ·(δnν1 δ0n′ν

)· · ·

= (−1)0+nµ+1+···+nν−1 δn1

n′1δn2

n′2· · ·(δnµ0 δ1n′µ

)· · ·(δnν1 δ0n′ν

)· · ·

The two equations differ by a factor of −1, so adding them gives zero. Putting everythingtogether, we conclude that

{cµ, c

†ν

}= δµν , as stated above.

As you can see, the derivation of the fermionic anticommutation relations gets pretty hairy,in large part due to the (−1)(··· ) factors in the definitions of the creation and annihilationoperators. But once these relations have been derived, we can deal entirely with the creation

11


and annihilation operators, without worrying about the underlying occupation number rep-resentation and its (−1)(··· ) factors. By the way, if we had chosen to omit the (−1)(··· ) factorsin the definitions, the creation and annihilation operators would still satisfy the anticom-mutation relation {cµ, c†ν} = δµν , but two creation operators or two annihilation operatorswould commute rather than anticommute. During subsequent calculations, the “algebra” ofcreation and annihilation operators ends up being much harder to deal with.

D. Second-quantized operators

One of the key benefits of second quantization is that it allows us to express multi-particlequantum operators clearly and succinctly, using the creation and annihilation operatorsdefined in Sections IV B–IV C as “building blocks”.

We start by considering a system of non-interacting particles. When there is just one particle(N = 1), let H1 denote the single-particle Hamiltonian, which is a Hermitian operator acting

on the single-particle Hilbert space H (1). For general N , the multi-particle Hamiltonian His a Hermitian operator acting on the Fock space H F . Given H1, what is H?

For bosons, the answer is:

H =∑µν

a†µHµν aν , where Hµν = 〈ϕµ|H1|ϕν〉, (47)

where aµ and a†µ are the boson creation and annihilation operators defined in Section IV B.

For fermions, the expression is exactly the same, but with the aµ and a†µ operators replaced

by the fermion creation and annihilation operators, cµ and c†µ, defined in Section IV C. Inthe remainder of this subsection, we will use the notation for bosonic operators, but theresults apply equally well to fermions.

To understand why this expression for H makes sense, consider its matrix elements withrespect to various states. Firstly, for the vacuum state |∅〉,

〈∅|H|∅〉 = 0. (48)

This makes sense. Secondly, single-particle states can be written as |ϕµ〉 = a†µ|∅〉, so

〈ϕµ|H|ϕν〉 = 〈∅|aµ(∑µ′ν′

a†µ′Hµ′ν′ aν′)a†ν |∅〉

=∑µ′ν′

Hµ′ν′ 〈∅|aµa†µ′ aν′a†ν |∅〉

=∑µ′ν′

Hµ′ν′ δµµ′δ

νν′

= Hµν .

(49)

This exactly matches the definition of Hµν in the original expression for H.

Thirdly, consider the case where {|ϕ1〉, |ϕ2〉, . . . } forms an eigenbasis of H1. Then,

H1|ϕµ〉 = Eµ|ϕµ〉 ⇒ Hµν = Eµ δµν . ⇒ H =∑µ

Eµ nµ. (50)

12


As previously noted, nµ = a†µaµ is the number operator, a Hermitian operator which cor-responds to the occupation number of single-particle state µ. Therefore, the total energyis the sum of the single-particle energies, which is exactly what we expect for a system ofnon-interacting particles.

Another way to interpret H is to think of the Hamiltonian as the generator of time evolution.Then H =

∑µν a

†µHµν aν says that each infinitesimal time step consists of a superposition of

alternative evolution processes. Each term in the superposition, a†µHµν aν , describes a particlebeing annihilated in state ν, and immediately re-created in state µ, which is equivalent to“transferring” a particle from ν to µ. The quantum amplitude for this process is describedby the matrix element Hµν . This description of time evolution is applicable not just tosingle-particle states, but also to multi-particle states containing any number of particles.

Note also that the number of particles does not change during time evolution. Whenever aparticle is annihilated in a state ν, it is immediately re-created in some state µ. This impliesthat the Hamiltonian commutes with the total particle number operator:

[H, N ] = 0, where N ≡∑µ

a†µaµ. (51)

The formal proof for this is left as an exercise (see Exercise 5). It follows from the creationand annihilation operators’ commutation relations (for bosons) or anticommuattion relations(for fermions).

Apart from the total energy, other kinds of observables—such as the total momentum, totalangular momentum, etc.—can be expressed in a similar way. Let A1 be a Hermitian operatoracting on single-particle states, corresponding to some physical observable A. Then, for amulti-particle system, the operator corresponding to the “total A” is

A =∑µν

a†µAµν aν , where Aµν = 〈ϕµ|A1|ϕν〉. (52)

Hermitian operators can also be constructed out of other kinds of groupings of creation andannihilation operators. For example, a pairwise (two-particle) potential can be describedwith a superposition of creation and annihilation operator pairs, of the form

V =∑µνλσ

a†µa†ν Vµν;λσ aλaσ. (53)

We can ensure that V is Hermitian by imposing a constraint on the coefficents:

V † =∑µνλσ

a†σa†λ V∗µν;λσ aν aµ = V ⇐ V ∗σλ;νµ = Vµν;λσ. (54)

In the next section, we will show how to determine the Vµν;λσ coefficients for any given two-

particle interaction, such as the electrostatic potential function. The V operator can thenbe added to a Hamiltonian to describe a system of interacting particles. In terms of timeevolution, the V operator “transfers” (annihilates and then re-creates) a pair of particlesduring each infinitesimal time step. Since the number of annihilated particles is always equalto the number of created particles, the interaction conserves the total particle number.

13


Another way to construct a Hermitian operator from creation and annihilation operators is

A =∑µ

(αµa

†µ + α∗µaµ

). (55)

If such a term is added to a Hamiltonian, it breaks the conservation of total particle number.Each infinitesimal time step will contain a superposition of processes that decrease theparticle number (due to aµ), and processes that increase the particle number (due to a†µ).Even if we start out with a state with a fixed number of particles, such as the vacuumstate |∅〉, the system subsequently evolves into a superposition of states of different particlenumbers. In the theory of quantum electrodynamics, this type of operator is used to describethe emission and absorption of photons caused by moving charges.

Incidentally, the name “second quantization” comes from this process of using creation andannihilation operators to define Hamiltonians. The idea is that single-particle quantum me-chanics is derived by “quantizing” classical Hamiltonians, via the imposition of commutationrelations like [x, p] = i~. Then, to extend the theory to multi-particle systems, we use thesingle-particle states to define creation/annihilation operators obeying a set of commutationor anticommutation relations. This can be viewed as a “second” quantization step.

V. QUANTUM FIELD THEORY

A. Field operators

So far, we have been agnostic about the nature of the single-particle states {|ϕ1〉, |ϕ2〉, . . . }used to construct the creation and annihilation operators. Let us now consider the specialcase where these quantum states are representable by wavefunctions. Let |r〉 denote aposition eigenstate for a d-dimensional space. A single-particle state |ϕµ〉 has a wavefunction

ϕµ(r) = 〈r|ϕµ〉. (56)

Due to the completeness and orthonormality of the basis, these wavefunctions satisfy∫ddr ϕ∗µ(r)ϕν(r) = 〈ϕµ|

(∫ddr |r〉〈r|

)|ϕν〉 = δµν ,

∑µ

ϕ∗µ(r)ϕµ(r′) = 〈r′|

(∑µ

|ϕµ〉〈ϕµ|

)|r〉 = δd(r− r′).

(57)

We can use the wavefunctions and the creation/annihilation operators to construct a newand interesting set of operators. For simplicity, suppose the particles are bosons, and let

ψ(r) =∑µ

ϕµ(r) aµ, ψ†(r) =∑µ

ϕ∗µ(r) a†µ. (58)

Using the aforementioned wavefunction properties, we can derive the inverse relations

aµ =

∫ddr ϕ∗µ(r) ψ(r), a†µ =

∫ddr ϕµ(r) ψ†(r). (59)

14


From the commutation relations for the bosonic aµ and a†µ operators, we can show that[ψ(r), ψ(r′)

]=[ψ†(r), ψ†(r′)

]= 0,

[ψ(r), ψ†(r′)

]= δd(r− r′). (60)

In the original commutation relations, the operators for different single-particle states com-mute; now, the operators for different positions commute. A straightforward interpretionfor the operators ψ†(r) and ψ(r) is that they respectively create and annihilate one particleat a point r (rather than one particle in a given eigenstate).

It is important to note that r here does not play the role of an observable. It is an index, inthe sense that each r is associated with distinct ψ(r) and ψ†(r) operators. These r-dependentoperators serve to generalize the classical concept of a field. In a classical field theory, eachpoint r is assigned a set of numbers corresponding to physical quantities, such as the electricfield components Ex(r), Ey(r), and Ez(r). In the present case, each r is assigned a set ofquantum operators. This kind of quantum theory is called a quantum field theory.

We can use the ψ(r) and ψ†(r) operators to write second quantized observables in a waythat is independent of the choice of single-particle basis wavefunctions. As discussed in theprevious section, given a Hermitian single-particle operator A1 we can define a many-particleobservable A =

∑µν a

†µAµν aν , where Aµν = 〈ϕµ|A1|ϕν〉. This many-particle observable can

be re-written as

A =

∫ddr ddr′ ψ†(r) 〈r|A1|r′〉 ψ(r′), (61)

which makes no explicit reference to the single-particle basis states.

For example, consider the familiar single-particle Hamiltonian describing a particle in apotential V (r):

H1 = T1 + V1, T1 =|p|2

2m, V1 = V (r), (62)

where r and p are position and momentum operators (single-particle observables). Thecorresponding second quantized operators for the kinetic energy and potential energy are

T =~2

2m

∫ddr ddr′ ψ†(r)

(∫ddk

(2π)d|k|2 eik·(r−r′)

)ψ(r′)

=~2

2m

∫ddr ∇ψ†(r) · ∇ψ(r)

V =

∫ddr ψ†(r) V (r) ψ(r).

(63)

(In going from the first to the second line, we performed integrations by parts.) This resultis strongly reminiscent of the expression for the expected kinetic and potential energies insingle-particle quantum mechanics:

〈T 〉 =~2

2m

∫ddr |∇ψ(r)|2 , 〈V 〉 =

∫ddr V (r) |ψ(r)|2, (64)

where ψ(r) is the single-particle wavefunction.

How are the particle creation and annihilation operators related to the classical notion of“the value of a field at point r”, like an electric field E(r) or magnetic field B(r)? Field

15


variables are measurable quantities, and should be described by Hermitian operators. Aswe have just seen, Hermitian operators corresponding to the kinetic and potential energycan be constructed via products of ψ†(r) with ψ(r). But there is another type of Hermitian

operator that we can construct by taking linear combinations of of ψ†(r) with ψ(r). Oneexample is

ψ(r) + ψ(r)†.

Other possible Hermitian operators have the form

F (r) =

∫ddr′

(f(r, r′)ψ(r) + f ∗(r, r′)ψ†(r′)

), (65)

where f(r, r′) is some complex function. As we shall see, it is this type of Hermitian operatorthat corresponds to the classical notion of a field variable like an electric or magnetic field.

In the next two sections, we will try to get a better understanding of the relationship betweenclassical fields and bosonic quantum fields. (For fermionic quantum fields, the situation ismore complicated; they cannot be related to classical fields of the sort we are familiar with,for reasons that lie outside the scope of this course.)

B. Revisiting the harmonic oscillator

Before delving into the links between classical fields and bosonic quantum fields, it is firstnecessary to revisit the harmonic oscillator, to see how the concept of a mode of oscillationcarries over from classical to quantum mechanics.

A classical harmonic oscillator is described by the Hamiltonian

H(x, p) =p2

2m+

1

2mω2x2, (66)

where x is the “position” of the oscillator, which we call the oscillator variable; p isthe corresponding momentum variable; m is the mass; and ω is the natural frequency ofoscillation. We know that the classical equation of motion has the general form

x(t) = A e−iωt +A∗ eiωt. (67)

This describes an oscillation of frequency ω. It is parameterized by the mode amplitudeA, a complex number that determines the magnitude and phase of the oscillation.

For the quantum harmonic oscillator, x and p are replaced by the Hermitian operators xand p. From these, the operators a and a† can be defined:

a =

√mω

2~

(x+

ip

mω

),

a† =

√mω

2~

(x− ip

mω

).

⇔

x =

√~

2mω

(a+ a†

)p = −i

√mω~

2

(a− a†

).

(68)

We can then show that [a , a†

]= 1, H = ~ω

(a†a+

1

2

), (69)

16


and from these the energy spectrum of the quantum harmonic oscillator can be derived.These facts should have been covered in an earlier course.

Here, we are interested in how the creation and annihilation operators relate to the dynamicsof the quantum harmonic oscillator. In the Heisenberg picture, with t = 0 as the referencetime, we define the time-dependent operator

x(t) = U †(t) x U(t), U(t) ≡ exp

(− i~Ht

). (70)

We will adopt the convention that all operators written with an explicit time dependenceare Heisenberg picture operators, while operators without an explicit time dependence areSchrodinger picture operators; hence, x ≡ x(0). The Heisenberg picture creation and anni-hilation operators, a†(t) and a(t), are related to x(t) by

x(t) =

√~

2mω

(a(t) + a†(t)

). (71)

The Heisenberg equation for the annihilation operator is

da(t)

dt=i

~

[H, a(t)

]=i

~U †(t)

[H, a

]U(t)

=i

~U †(t)

(− ~ωa

)U(t)

= −iωa(t).

(72)

Hence, solution for this differential equation is

a(t) = a e−iωt, (73)

and Eq. (71) becomes

x(t) =

√~

2mω

(a e−iωt + a† eiωt

). (74)

This has exactly the same form as the classical oscillatory solution (67)! Comparing the

two, we see that a times the scale factor√

~/2mω plays the role of the mode amplitude A.

Now, suppose we come at things from the opposite end. Let’s say we start with creation andannihilation operators satisfying Eq. (69), from which Eqs. (72)–(73) follow. Using the cre-ation and annihilation operators, we would like to construct an observable that correspondsto a classical oscillator variable. A natural ansatz is

x(t) = C a e−iωt + C∗ a† eiωt, (75)

which is manifestly Hermitian.

How might the constant C be chosen? A convenient method is to identify how the oscilla-tor variable relates to the oscillator energy in the classical limit. For a standard classicalharmonic oscillator, the energy satisfies

E = mω2 x2, (76)

17


where x2 denotes the mean squared displacement over an oscillation cycle. This can beproven rigorously from Eqs. (66)–(67); one way to remember it is that the average kineticenergy of a harmonic oscillator equals its average potential energy (this is an instance of thevirial theorem of classical mechanics).

The classical limit of a quantum harmonic oscillator is defined using a coherent state, whichis a quantum state |α〉 labelled by a complex number α, such that

a|α〉 = α|α〉. (77)

How this state is defined need not concern us for now; for details, see Appendix E. In thisstate, the expectation value of the energy is

〈E〉 = 〈α|H|α〉 = ~ω(|α|2 +

1

2

). (78)

We can use Eq. (77) with Eq. (75) to compute the expectated mean squared displacementin the coherent state. We moreover assume that C is real (this is a matter of convention;the creation and annihilation operators can always be re-defined so that this holds), so

〈α|x2(t)|α〉 = C2(α2 e−2iωt + 2|α|2 + α∗2 e2iωt

). (79)

Averaging over an oscillation cycle,

〈α|x2(t)|α〉 = 2 C2|α|2. (80)

Finally, the classical limit is reached by taking |α| → ∞ (so that the energy is much largerthan the energy quantum ~ω). Combining Eqs. (76), (78), and (80) in this limit gives

~ω|α|2 = mω2 · 2C2|α|2 (81)

⇒ C =

√~

2mω. (82)

This is precisely the scale factor found in Eq. (74).

C. A scalar boson field

We now have the tools available to understand the connection between a very simple classicalfield and its quantum counterpart. Consider a classical scalar field variable f(x, t), definedin one spatial dimension, whose classical equation of motion is the wave equation:

∂2f(x, t)

∂x2=

1

c2∂2f(x, t)

∂t2. (83)

The constant c is a wave speed. This sort of classical field arises in many physical contexts,including the propagation of sound through air, in which case c is the speed of sound.

For simplicity, let us first assume that the field is defined within a finite interval of lengthL, with periodic boundary conditions: f(x, t) ≡ f(x+L, t). Solutions to the wave equationcan be described by the following ansatz:

f(x, t) =∑n

(An ϕn(x) e−iωnt +A∗n ϕ∗n(x) eiωnt

). (84)

18


This ansatz describes a superposition of normal modes. Each normal mode (labelled n)varies harmonically in time with a mode frequency ωn, and varies in space according toa complex mode profile ϕn(x); its overall magnitude and phase is specified by the modeamplitude An. The mode profiles are normalized according to some fixed convention, e.g.∫ L

0

dx |ϕn(x)|2 = 1. (85)

By substituting Eq. (84) into Eq. (83), and using the periodic boundary conditions, we find

ϕn(x) =1√L

exp (iknx) , ωn = ckn =2πcn

L, n ∈ Z. (86)

These mode profiles are orthonormal:∫ L

0

dxϕ∗m(x)ϕn(x) = δmn. (87)

Each normal mode also carries energy. We assume that the time-averaged energy densityfor normal mode n has the form

U(x) = ρ |f(x)|2, (88)

where ρ is some parameter that has to be derived from the underlying physical context,and (· · · ) denotes a time-average. (For example, for acoustic modes, ρ is the mass densityof the underlying acoustic medium; in the next chapter, we will see a concrete exampleinvolving the energy density of an electromagnetic mode.) Hence, using Eq. (84), we findthe time-averaged total energy

E =

∫ L

0

dx U(x)

=

∫ L

0

dx ρ

(2∑n

|An|2 |ϕn(x)|2)

= 2ρ∑n

|An|2.

(89)

To quantize the classical field, we treat each normal mode as an independent oscillator, withcreation and annihilation operators a†n and an satisfying[

am , a†n

]= δmn,

[am , an

]=[a†m , a

†n

]= 0. (90)

We then take the Hamiltonian to be that of a set of independent harmonic oscillators:

H =∑n

~ωna†nan + E0, (91)

where E0 is the ground-state energy. Just like in the previous section, we can define aHeisenberg-picture annihilation operator, and solving its Heisenberg equation yields

an(t) = ane−iωnt. (92)

19


We can then define a Schrodinger picture Hermitian operator of the form

f(x) =∑n

Cn(anϕn(x) + a†nϕ

∗n(x)

), (93)

where Cn is a real constant (one for each normal mode). The corresponding Heisenbergpicture operator is

f(x, t) =∑n

Cn(anϕn(x) e−iωnt + a†nϕ

∗n(x) eiωnt

), (94)

which we can identify as the quantum version of the classical solution (84).

To determine the Cn scale factors, we again follow the procedure from the previous section,comparing the energy of the quantum system’s classical limit to the energy of the classicalfield theory; the latter is described by Eq. (89). The resulting field operator is

f(x, t) =∑n

√~ωn2ρ

(anϕn(x) e−iωnt + a†nϕ

∗n(x) eiωnt

). (95)

Returning to the Schrodinger picture, and using the explicit mode profiles from Eq. (86),we get

f(x) =∑n

√~ωn2ρL

(ane

iknx + a†ne−iknx

). (96)

Finally, if we are interested in the infinite-L limit, we can convert the sum over n into anintegral. The result is

f(x) =

∫dk

√~ω(k)

4πρ

(a(k) eikx + a†(k) e−ikx

), (97)

where a(k) denotes a rescaled annihilation operator defined by an →√

2π/L a(k), satisfying[a(k) , a†(k′)

]= δ(k − k′). (98)

D. Looking ahead

In the next chapter, we will use these ideas to formulate a quantum theory of electromag-netism. This is a bosonic quantum field theory in which the creation and annihilationoperators act upon particles called photons—the elementary particles of light. Linear com-binations of these photon operators can be used to define Hermitian field operators thatcorrespond to the classical electromagnetic field variables. In the classical limit, the quan-tum field theory reduces to Maxwell’s theory of the electromagnetic field.

It is hard to overstate the importance of quantum field theories in physics. At a funda-mental level, all elementary particles currently known to humanity can be described usinga quantum field theory called the Standard Model. These particles are roughly divided into

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two categories. The first consists of “force-carrying” particles: photons (which carry theelectromagnetic force), gluons (which carry the strong nuclear force), and the W/Z bosons(which carry the weak nuclear force); these particles are excitations of bosonic quantumfields, similar to the one described in the previous section. The second category consistsof “particles of matter”, such as electrons, quarks, and neutrinos; these are excitations offermionic quantum fields, whose creation and annihilation operators obey anticommutationrelations.

As Wilczek (1999) has pointed out, the modern picture of fundamental physics bears astriking resemblence to the old idea of “luminiferous ether”: a medium filling all of spaceand time, whose vibrations are physically-observable light waves. The key difference, as wenow understand, is that the ether is not a classical medium, but one obeying the rules ofquantum mechanics. (Another difference, which we have not discussed so far, is that modernfield theories can be made compatible with relativity.)

It is quite compelling to think of fields, not individual particles, as the fundamental objectsin the universe. This point of view “explains”, in a sense, why all particles of the sametype have the same properties (e.g., why all electrons in the universe have exactly the samemass). The particles themselves are not fundamental; they are excitations of deeper, morefundamental entities—quantum fields!

Exercises

1. Consider a system of two identical particles. Each single-particle Hilbert space H (1)

is spanned by a basis {|µi}. The exchange operator is defined on H (2) = H (1)⊗H (1)

by

P(∑

ij

ψij|µi〉|µj〉)≡∑ij

ψij|µj〉|µi〉. (99)

Prove that P is linear, unitary, and Hermitian. Moreover, prove that the operation isbasis-independent: i.e., given any other basis {νj} that spans H (1),

P(∑

ij

ϕij|νi〉|νj〉)

=∑ij

ϕij|νj〉|νi〉. (100)

2. Prove that the exchange operator commutes with the Hamiltonian

H = − ~2

2me

(∇2

1 +∇22

)+

e2

4πε0|r1 − r2|. (101)

3. An N -boson state can be written as

|φ1, φ2, . . . , φN〉 = N∑p

(|φp(1)〉|φp(2)〉|φp(3)〉 · · · |φp(N)〉

). (102)

Prove that the normalization constant is

N =

√1

N !∏

µ nµ!, (103)

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where nµ denotes the number of particles occupying the single-particle state µ.

4. Prove that for boson creation and annihilation operators, [aµ, aν ] = [a†µ, a†ν ] = 0.

5. Let A1 be an observable (Hermitian operator) for single-particle states. Given a single-particle basis {|ϕ1〉, |ϕ2〉, . . . }, define the bosonic multi-particle observable

A =∑µν

a†µ 〈ϕµ|A1|ϕν〉 aν , (104)

where a†µ and aµ are creation and annihilation operators satisfying the usual bosonic

commutation relations, [aµ, aν ] = 0 and [aµ, a†ν ] = δµν . Prove that A commutes with

the total number operator: [A,∑µ

a†µaµ

]= 0. (105)

Next, repeat the proof for a fermionic multi-particle observable

A =∑µν

c†µ 〈ϕµ|A1|ϕν〉 cν , (106)

where c†µ and cµ are creation and annihilation operators satisfying the fermionic anti-

commutation relations, {cµ, cν} = 0 and {cµ, c†ν} = δµν . In this case, prove that[A,∑µ

c†µcµ

]= 0. (107)

Further Reading

[1] Bransden & Joachain, §10.1–10.5

[2] Sakurai, §6

[3] F. Wilczek, The Persistence of Ether, Physics Today 52, 11 (1999). [link]

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http://physicstoday.scitation.org/doi/10.1063/1.882562

Documents

Chapter 3: Identical ParticlesY. D. Chong (2018) PH4401: Quantum Mechanics III Chapter 3: Identical Particles These our actors, As I foretold you, were all spirits and Are melted into