22.3 The Bargmann-Fock representation

22.3 The Bargmann-Fock representation

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · spectral theorem

Working with the operators a and $a^{†}$ and their commutation relation

$$[a,a^{†}]=1$$

makes it clear that there is a simpler way to represent these operators than the Schr¨odinger representation as operators on position space functions that we have been using, while the Stone-von Neumann theorem assures us that this will be unitarily equivalent to the Schr¨odinger representation. This representation appears in the literature under a large number of diferent names, depending on the context, all of which refer to the same representation:

Definition (Bargmann-Fock or oscillator or holomorphic or Segal-Shale-Weil representation)

The Bargmann-Fock (etc.) representation is given by taking as state space $\mathcal{H}=\mathcal{F}$ , where $\mathcal{F}$ is the space of holomorphic functions (satisfying $\begin{array}{r}\frac{d}{d\overline{z}}\psi =0)\end{array}$ on $C$ with finite norm in the inner product

$$⟨{\psi }_1|{\psi }_2⟩=\frac{1}{\pi }{\int }_{\mathbf{C}}\overline{{\psi }_1(z)}{\psi }_2(z)e^{-|z{|}^2}{d}^{2}z\text{\hspace{1em}(22.4)}$$

where $d^{2}z=dRe(z)dIm(z)$ . The space $\mathcal{F}$ is sometimes called “Fock space”. We define the following two operators acting on this space:

$$a=\frac{d}{dz},a^{†}=z$$

Since

$$[a,a^{†}]z^n=\frac{d}{dz}(z{z}^n)-z\frac{d}{dz}{z}^n=(n+1-n)z^n=z^n$$

the commutator is the identity operator on polynomials

$$[a,a^{†}]=1$$

One finds

Theorem

The Bargmann-Fock representation has the following properties

• The elements

$$\frac{z^n}{\sqrt{n!}}$$

of $\mathcal{F}$ for $n=0,1,2,...$ . are orthonormal.

• The operators a and $a^{†}$ are adjoints with respect to the given inner product on $\mathcal{F}$

• The basis

$$\frac{z^n}{\sqrt{n!}}$$

of $\mathcal{F}forn=0,1,2,...$ is complete.

Proof. The proofs of the above statements are not dificult, in outline they are

• For orthonormality one can compute the integrals

$${\int }_{\mathbf{C}}{\overline{z}}^m{z}^n{e}^{-|z{|}^2}{d}^{2}z$$

in polar coordinates.

• To show that z and $\frac{d}{dz}$ are adjoint operators, use integration by parts.

• For completeness, assume $⟨n|\psi ⟩=0$ for all $n.$ . The expression for the $|n⟩$ as Hermite polynomials times a Gaussian then implies that

$$\int F(q)e^{-\frac{q^2}{2}}\psi (q)dq=0$$

for all polynomials $F(q)$ . Computing the Fourier transform of $\psi (q)e^{-\frac{q^2}{2}}$ gives

$$\int e^{-ikq}{e}^{-\frac{q^2}{2}}\psi (q)dq=\int \sum _{j=0}^{\infty }\frac{(-ikq{)}^j}{j!}{e}^{-\frac{q^2}{2}}\psi (q)dq=0$$

So $\psi (q)e^{-\frac{q^2}{2}}$ has Fourier transform 0 and must be 0 itself. Alternatively, one can invoke the spectral theorem for the self-adjoint operator $H$ , which guarantees that its eigenvectors form a complete and orthonormal set.

Since in this representation the number operator $N=a^{†}a$ satisfies

$$N{z}^n=z\frac{d}{dz}{z}^n=n{z}^n$$

the monomials in $z$ diagonalize the number and energy operators, so one has

$$\frac{z^n}{\sqrt{n!}}$$

for the normalized energy eigenstate of energy $\begin{array}{r}\hslash \omega (n+\frac{1}{2})\end{array}$

Note that we are here taking the state space $\mathcal{F}$ to include infinite linear combinations of the states $|n⟩$ , as long as the Bargmann-Fock norm is finite. We will sometimes want to restrict to the subspace of finite linear combinations of the $|n⟩$ , which we will denote ${\mathcal{F}}^{fin}$ . This is the space $\mathbf{C}[\mathbf{z}]$ of polynomials, and $\mathcal{F}$ is its completion for the Bargmann-Fock norm.

Previous section: 22.2 Creation and annihilation operators · Next section: 22.4 Quantization by annihilation and creation operators · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 244、245、246、247、248、249、250、251、252、253

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：461cd8acacffadb41f056f9b4158f2b227594f8538628d1a31493d321b2ecd97

OCR 产物 SHA-256：461cd8acacffadb41f056f9b4158f2b227594f8538628d1a31493d321b2ecd97