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Physics LibreTexts

8.2: Creation and Annihilation Operators

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The second, and particularly powerful way to implement the description of identical particles is via creation and annihilation operators. To see how this description arises, consider some single-particle Hermitian operator A with eigenvalues aj. On physical grounds, and regardless of distinguishability, we require that nj particles in the eigenstate |aj of A must have a total physical value nj×aj for the observable A. We can repeat this for all eigenvalues aj, and obtain a potentially infinite set of basis vectors

|n1,n2,n3,,

for all integer values of nj, including zero. You should convince yourself that this exhausts all the possible ways any number of particles can be distributed over the eigenvalues aj. The spectrum of A can be bounded or unbounded, and discrete or continuous. It may even be degenerate. For simplicity we consider here an unbounded, non-degenerate discrete spectrum.

A special state is given by

|=|0,0,0,

which indicates the state of no particles, or the vacuum. The numbers nj are called the occupation number, and any physical state can be written as a superposition of these states:

|Ψ=n1,n2,n3,=0cn1,n2,n3,|n1,n2,n3,.

The basis states |n1,n2,n3, span a linear vector space called a Fock space F. It is the direct sum of the Hilbert spaces for zero particles H0, one particle H1, two particles, etc.:

F=H0H1H2H3

Since |Ψ is now a superposition over different particle numbers, we require operators that change the particle number. These are the creation and annihilation operators, ˆa and ˆa respectively. Up to a proportionality constant that we will determine later, the action of these operators is defined by

ˆaj|n1,n2,,nj,|n1,n2,,nj+1,ˆaj|n1,n2,,nj,|n1,n2,,nj1,

So the operator ˆaj creates a particle in a state with eigenvalue aj, and the operator ˆaj removes a particle in a state with eigenvalue aj. These operators are each others’ Hermitian adjoint, since removing a particle is the time reversal of adding a particle. Clearly, when an annihilation operator attempts to remove particles that are not there, the result must be zero:

ˆaj|n1,n2,,nj=0,=0

The vacuum is then defined as the state that gives zero when acted on by any annihilation operator: ˆaj|=0 for any j. Notice how we have so far sidestepped the problem of particle swapping; we exclusively used aspects of the total particle number.

What are the basic properties of these creation and annihilation operators? In particular, we are interested in their commutation relations. We will now derive these properties from what we have determined so far. First, note that we can create two particles with eigenvalues ai and aj in the system in any order, and the only difference this can make is in the normalization of the state:

ˆaiˆaj|Ψ=λˆajˆai|Ψ,

where λ is some complex number. Since state |Ψ is certainly not zero, we require that

ˆakˆalλˆalˆak=0.

Since k and l are just dummy variables, we equally have

ˆalˆakλˆakˆal=0.

We now substitute Eq. (8.15) into Eq. (8.14) to eliminate ˆalˆak. This leads to

(1λ2)ˆakˆal=0,

and therefore

λ=±1.

The relation between different creation operators can thus take two forms. They can obey a commutation relation when λ=+1:

ˆalˆakˆakˆal=[ˆal,ˆak]=0,

or they can obey an anti-commutation relation when λ=1:

ˆalˆak+ˆakˆal={ˆal,ˆak}=0

While creating the particles in different temporal order is not the same as swapping two particles, it should not come as a surprise that there are two possible situations (the commutation relation and the anti-commutation relation). We encountered two possibilities in our previous approach as well, where we found that many-particle states are either symmetric or anti-symmetric. In fact, creation operators that obey the commutation relation produce symmetric states, while creation operators that obey the anti-commutation relation produce anti-symmetric states. We also see that the creation operators described by the anti-commutation relations naturally obey Pauli’s exclusion principle. Suppose that we wish to create two identical particles in the same eigenstate |aj. The anti-commutation relations say that {ˆaj,ˆaj}=0, so

ˆa2j=0.

Any higher powers of ˆaj will also be zero, and we can create at most one particle in the state |aj.

Taking the adjoint of the commutation relations for the creation operators gives us the corresponding relations for the annihilation operators

ˆalˆakˆakˆal=[ˆal,ˆak]=0,

or

ˆalˆak+ˆakˆal={ˆal,ˆak}=0.

The remaining question is now what the (anti-) commutation relations are for products of creation and annihilation operators.

We proceed along similar lines as before. Consider the operators ˆaj and ˆak with jk, and apply them in different orders to a state |Ψ.

ˆaiˆaj|Ψ=μˆajˆai|Ψ.

The same argumentation as before leads to μ=±1. For different j and k we therefore find

[ˆaj,ˆak]=0 or {ˆaj,ˆak}=0

Now let’s consider the case j=k. For the special case where |Ψ=|, we find

(ˆajˆakμˆajˆak)|=ˆajˆak|=δjk|,

based on the property that ˆaj|=0. When l=k,

(ˆakˆakμˆakˆak)|=|,

we find for the two possible values of μ

ˆakˆakˆakˆak=1 or ˆakˆak+ˆakˆak=1

which is equivalent to

[ˆak,ˆak]=1 or {ˆak,ˆak}=1.

To summarise, we have two sets of algebras for the creation and annihilation operators. The algebra in terms of the commutation relations is given by

[ˆak,ˆal]=[ˆak,ˆal]=0 and [ˆak,ˆal]=δkl.

This algebra describes particles that obey Bose-Einstein statistics, or bosons. The algebra in terms of anti-commutation relations is given by

{ˆak,ˆal}={ˆak,ˆal}=0 and {ˆak,ˆal}=δkl.

This algebra describes particles that obey Fermi-Dirac statistics, or fermions.

Finally, we have to determine the constant of proportionality for the creation and annihilation operators. We have already required that ˆajˆak|=δjk|. To determine the rest, we consider a new observable that gives us the total number of particles in the system. We denote this observable by ˆn, and we see that it must be additive over all particle numbers for the different eigenvalues of A:

ˆn=jˆnj,

where ˆnj is the number of particles in the eigenstate |aj. The total number of particles does not change if we consider a different observable (although the distribution typically will), so this relation is also true when we count the particles in the states |bj. Pretty much the only way we can achieve this is to choose

ˆn=jˆnj=jˆajˆaj=jˆbjˆbj.

For the case of nj particles in state |aj this gives

ˆajˆaj|nj=nj|nj,

where we ignored the particles in other states |ak with kj for brevity. For the Bose-Einstein case this leads to the relations

ˆaj|nj=nj|nj1 and ˆaj|nj=nj+1|nj+1.

For Fermi-Dirac statistics, the action of the creation and annihilation operators on number states becomes

ˆaj|0j=0 and ˆaj|0j=eiα|1j,ˆaj|1j=eiα|0j and ˆaj|1j=0.

The phase factor eiα can be chosen ±1.


This page titled 8.2: Creation and Annihilation Operators is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform.

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