22.2 Creation and annihilation operators

22.2 Creation and annihilation operators

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

It turns out that there is a quite easy method which allows one to explicitly find eigenfunctions and eigenvalues of the harmonic oscillator Hamiltonian (although it’s harder to show it gives all of them). This also leads to a new representation of the Heisenberg group (of course unitarily equivalent to the Schr¨odinger one by the Stone-von Neumann theorem). Instead of working with the self-adjoint operators $Q$ and P that satisfy the commutation relation

$$[Q,P]=i\hslash 1$$

we define

$$a=\sqrt{\frac{m\omega }{2\hslash }}Q+i\sqrt{\frac{1}{2m\omega \hslash }}P,a^{†}=\sqrt{\frac{m\omega }{2\hslash }}Q-i\sqrt{\frac{1}{2m\omega \hslash }}P$$

which satisfy the commutation relation

$$[a,a^{†}]=1$$

Since

$$Q=\sqrt{\frac{\hslash }{2m\omega }}(a+a^{†}),P=-i\sqrt{\frac{m\omega \hslash }{2}}(a-a^{†})$$

the Hamiltonian operator is

$$\begin{array}{rl}H& =\frac{P^2}{2m}+\frac{1}{2}m{\omega }^2{Q}^2\\ & =\frac{1}{4}\hslash \omega (-(a-a^{†}{)}^2+(a+a^{†}{)}^2)\\ & =\frac{1}{2}\hslash \omega (a{a}^{†}+a^{†}a)\\ & =\hslash \omega \left(a^{†}a+\frac{1}{2}\right)\end{array}$$

The problem of finding eigenvectors and eigenvalues for $H$ is seen to be equivalent to the same problem for the operator

$$N=a^{†}a$$

Such an operator satisfies the commutation relations

$$[N,a]=[a^{†}a,a]=a^{†}[a,a]+[a^{†},a]a=-a$$

and

$$[N,a^{†}]=a^{†}$$

If |c⟩ is a normalized eigenvector of $N$ with eigenvalue $c,$ one has

$$c=⟨c|a^{†}a|c⟩=|a|c⟩{|}^2\ge 0$$

so eigenvalues of $N$ must be non-negative. Using the commutation relations of $N,a,a^{†}$ gives

$$Na|c⟩=([N,a]+aN)|c⟩=a(N-1)|c⟩=(c-1)a|c⟩$$

and

$$N{a}^{†}|c⟩=([N,a^{†}]+a^{†}N)|c⟩=a^{†}(N+1)|c⟩=(c+1)a^{†}|c⟩$$

This shows that $a|c⟩$ will have eigenvalue $c-1$ for $N.$ , and a normalized eigenfunction for N will be

$$|c-1⟩=\frac{1}{\sqrt{c}}a|c⟩$$

Similarly, since

$$|a^{†}|c⟩{|}^2=⟨c|a{a}^{†}|c⟩=⟨c|(N+1)|c⟩=c+1$$

we have

$$|c+1⟩=\frac{1}{\sqrt{c+1}}{a}^{†}|c⟩$$

If we start of with a state |0⟩ that is a non-zero eigenvector for $N$ with eigenvalue $0,$ we see that the eigenvalues of $N$ will be the non-negative integers, and for this reason $N$ is called the “number operator”.

We can find such a state by looking for solutions to

$$a|0⟩=0$$

|0⟩ will have energy eigenvalue $\frac{1}{2}\hslash \omega$ , and this will be the lowest energy eigenstate. Acting by $a^{†}$ n-times on |0⟩ gives states with energy eigenvalue $(n+\frac{1}{2})\hslash \omega$ . The equation for |0⟩ is

$$a|0⟩=\left(\sqrt{\frac{m\omega }{2\hslash }}Q+i\sqrt{\frac{1}{2m\omega \hslash }}P\right){\psi }_0(q)=\sqrt{\frac{m\omega }{2\hslash }}\left(q+\frac{\hslash }{m\omega }\frac{d}{dq}\right){\psi }_0(q)=0$$

One can check that this equation has a single normalized solution

$${\psi }_0(q)=(\frac{m\omega }{\pi \hslash }{)}^{\frac{1}{4}}{e}^{-\frac{m\omega }{2\hslash }{q}^2}$$

which is the lowest-energy eigenfunction.

The rest of the energy eigenfunctions can be found by computing

$$|n⟩=\frac{a^{†}}{\sqrt{n}}\dots \frac{a^{†}}{\sqrt{2}}\frac{a^{†}}{\sqrt{1}}|0⟩=\frac{1}{\sqrt{n!}}{\left(\frac{m\omega }{2\hslash }\right)}^{\frac{n}{2}}{\left(q-\frac{\hslash }{m\omega }\frac{d}{dq}\right)}^n{\psi }_0(q)$$

To show that these are the eigenfunctions of equation 22.2, one starts with the definition of Hermite polynomials as a generating function

$$e^{2qt-t^2}=\sum _{n=0}^{\infty }{H}_n(q)\frac{t^n}{n!}\text{\hspace{1em}(22.3)}$$

and interprets the $H_n(q)$ as the Taylor coeficients of the left-hand side at $t=0$ deriving the identity

$$\begin{array}{rl}{H}_n(q)& ={\left(\frac{d^n}{d{t}^n}{e}^{2qt-t^2}\right)}_{|t=0}\\ & =e^{q^2}{\left(\frac{d^n}{d{t}^n}{e}^{-(q-t{)}^2}\right)}_{|t=0}\\ & =(-1{)}^n{e}^{q^2}{\left(\frac{d^n}{d{q}^n}{e}^{-(q-t{)}^2}\right)}_{|t=0}\\ & =(-1{)}^n{e}^{q^2}\frac{d^n}{d{q}^n}{e}^{-q^2}\\ & =e^{\frac{q^2}{2}}{\left(q-\frac{d}{dq}\right)}^n{e}^{-\frac{q^2}{2}}\end{array}$$

Taking q to $\sqrt{\frac{m\omega }{\hslash }}q$ this can be used to show that $|n⟩={\psi }_n(q)$ is given by 22.2. In the physical interpretation of this quantum system, the state $|n⟩$ , with energy $\hslash \omega (n+\frac{1}{2})$ is thought of as a state describing n “quanta”. The state |0⟩ is the “vacuum state” with zero quanta, but still carrying a “zero-point” energy of $\frac{1}{2}\hslash \omega .$ . The operators $a^{†}$ and a have somewhat similar properties to the raising and lowering operators we used for $SU(2)$ but their commutator is diferent (the identity operator), leading to simpler behavior. In this case they are called “creation” and “annihilation” operators respectively, due to the way they change the number of quanta. The relation of such quanta to physical particles like the photon is that quantization of the electromagnetic field (see chapter 46) involves quantization of an infinite collection of oscillators, with the quantum of an oscillator corresponding physically to a photon with a specific momentum and polarization. This leads to a well known problem of how to handle the infinite vacuum energy corresponding to adding up $\begin{array}{l}\frac{1}{2}\end{array}$ ℏ$\omega$ for each oscillator.

The first few eigenfunctions are plotted below. The lowest energy eigenstate is a Gaussian centered at $q=0,$ with a Fourier transform that is also a Gaussian centered at $p=0$ . Classically the lowest energy solution is an oscillator at rest at its equilibrium point $(q=p=0)$ , but for a quantum oscillator one cannot have such a state with a well-defined position and momentum. Note that the plot gives the wavefunctions, which in this case are real and can be negative. The square of this function is what has an interpretation as the probability density for measuring a given position.

Figure 22.1: Harmonic oscillator energy eigenfunctions.

While we have preserved constants in our calculations in this section, in what follows we will often for simplicity set $\hslash =m=\omega =1$ , which can be done by an appropriate choice of units. Equations with the constants can be recovered by rescaling. In particular, our definition of annihilation and creation operators will be given by

$$a=\frac{1}{\sqrt{2}}(Q+iP),a^{†}=\frac{1}{\sqrt{2}}(Q-iP)$$

Previous section: 22.1 The harmonic oscillator with one degree of freedom · Next section: 22.3 The Bargmann-Fock representation · 中文版本

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