23.2 Coherent states and the Bargmann-Fock state space

23.2 Coherent states and the Bargmann-Fock state space

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

One thing coherent states provide is an alternate complete set of norm one vectors in $\mathcal{H}.$ , so any state can be written in terms of them. However, these states are not orthogonal (they are eigenvectors of a non-self-adjoint operator so the spectral theorem for self-adjoint operators does not apply). The inner product of two coherent states is

$$\begin{array}{rl}⟨\beta |\alpha ⟩& =⟨0|e^{-\frac{1}{2}|\beta {|}^2}{e}^{\overline{\beta }a}{e}^{-\frac{1}{2}|\alpha {|}^2}{e}^{\alpha a^{†}}|0⟩\\ & =e^{-\frac{1}{2}(|\alpha {|}^2+|\beta {|}^2)}{e}^{\overline{\beta }\alpha }⟨0|e^{\alpha a^{†}}{e}^{\overline{\beta }a}|0⟩\\ & =e^{\overline{\beta }\alpha -\frac{1}{2}(|\alpha {|}^2+|\beta {|}^2)}\end{array}\text{\hspace{1em}(23.3)}$$

and

$$|⟨\beta |\alpha ⟩{|}^2=e^{-|\alpha -\beta {|}^2}$$

The Dirac formalism used for representing states as position space or momentum space distributions with a continuous basis |q⟩ or $|p⟩$ can also be adapted to the Bargmann-Fock case. In the position space case, with states functions of $q,$ the delta-function distribution $\delta (q-q^{\prime })$ provides an eigenvector $|q^{\prime }⟩$ for the $Q$ operator, with eigenvalue $q^{\prime }$ . As discussed in chapter 12, the position space wavefunction of a state |$\psi$⟩ can be thought of as given by

$$\psi (q)=⟨q|\psi ⟩$$

with

$$⟨q|q^{\prime }⟩=\delta (q-q^{\prime })$$

In the Bargmann-Fock case, there is an analog of the distributional states $|q⟩$ , given by taking states that are eigenvectors for ${}^{a,}$ but unlike the $|\alpha ⟩$ , are not normalizable. We define

$$|{\delta }_w⟩=e^{\overline{w}{a}^{†}}|0⟩=\sum _{n=0}^{\infty }\frac{{\overline{w}}^n}{\sqrt{n!}}|n⟩=e^{\overline{w}z}=e^{\frac{|w{|}^2}{2}}|\overline{w}⟩$$

Instead of equation 23.3, such states satisfy

$$⟨{\delta }_{w_1}|{\delta }_{w_2}⟩=e^{{\overline{w}}_2{w}_1}$$

The $|{\delta }_w$ behave in a manner analogous to the delta-function, since the Bargmann-Fock analog of computing $⟨q|\psi ⟩$ using the function space inner product is, writing

$$\psi (w)=\sum _{n=0}^{\infty }{c}_n\frac{w^n}{\sqrt{n!}}$$

the computation

$$\begin{array}{rl}⟨{\delta }_z|\psi ⟩& =\frac{1}{\pi }{\int }_{\mathbf{C}}{e}^{z\overline{w}}{e}^{-|w{|}^2}\psi (w)d^{2}w\\ & =\frac{1}{\pi }{\int }_{\mathbf{C}}{e}^{z\overline{w}}{e}^{-|w{|}^2}\sum _{n=0}^{\infty }{c}_n\frac{w^n}{\sqrt{n!}}{d}^{2}w\\ & =\frac{1}{\pi }{\int }_{\mathbf{C}}\sum _{m=0}^{\infty }\frac{z^m}{\sqrt{m!}}\frac{{\overline{w}}^m}{\sqrt{m!}}{e}^{-|w{|}^2}\sum _{n=0}^{\infty }{c}_n\frac{w^n}{\sqrt{n!}}{d}^{2}w\\ & =\sum _{n=0}^{\infty }{c}_n\frac{z^n}{\sqrt{n!}}=\psi (z)\end{array}$$

Here we have used the orthogonality relations

$${\int }_{\mathbf{C}}{\overline{w}}^m{w}^n{e}^{-|w{|}^2}{d}^{2}w=\pi n!{\delta }_{n,m}\text{\hspace{1em}(23.4)}$$

It is easily seen that the Bargmann-Fock wavefunction of a coherent state is given by

$$⟨{\delta }_z|\alpha ⟩=e^{-\frac{|\alpha {|}^2}{2}}{e}^{\alpha z}\text{\hspace{1em}(23.5)}$$

while for number operator eigenvector states

$$⟨{\delta }_z|n⟩=\frac{z^n}{\sqrt{n!}}$$

In section 23.5 we will compute the Bargmann-Fock wavefunction $⟨{\delta }_z|q⟩$ for position eigenstates, see equation 23.13.

Like the |$\alpha$⟩ (and unlike the $|q⟩$ or $|p⟩)$ , these states $|{\delta }_w⟩$ are not orthogonal for diferent eigenvalues of ${}^{a,}$ but they span the state space, providing an overcomplete basis, and satisfy the resolution of the identity relation

$$1=\frac{1}{\pi }{\int }_{\mathbf{C}}|{\delta }_w⟩⟨{\delta }_w|e^{-|w{|}^2}{d}^{2}w\text{\hspace{1em}(23.6)}$$

This can be shown using

$$|{\delta }_w⟩⟨{\delta }_w|=\left(\sum _{n=0}^{\infty }\frac{{\overline{w}}^n}{\sqrt{n!}}|n⟩\right)\left(\sum _{m=0}^{\infty }\frac{w^m}{\sqrt{m!}}⟨m|\right)$$

as well as

$$1=\sum _{n=0}^{\infty }|n⟩⟨n|$$

and the orthogonality relations 23.4. Note that the normalized coherent states similarly provide an over-complete basis, with

$$1=\frac{1}{\pi }{\int }_{\mathbf{C}}|\alpha ⟩⟨\alpha |d^2\alpha \text{\hspace{1em}(23.7)}$$

To avoid confusion over the various ways in which complex variables $z$ and w appear here, note that this is just the analog of what happens in the position space representation, where $q$ is variously a coordinate on classical phase space, an argument of a wavefunction, a label of a position operator eigenstate, and a multiplication operator. The analog of the position operator $Q$ here is $a^{†}$ which is multiplication by $z$ (unlike $\mathrm{Q},$ not self-adjoint). The conjugate complex coordinate $\overline{z}$ is analogous to the momentum coordinate, quantized to a diferentiation operator. One confusing aspect of this formalism is that complex conjugation takes elements of $\mathcal{H}$ (holomorphic functions) to antiholomorphic functions, which are in a diferent space. The quantization of $\overline{z}$ is not the complex-conjugate of $z,$ but the adjoint operator.

Previous section: 23.1 Coherent states and the Heisenberg group action · Next section: 23.3 The Heisenberg group action on 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 254、255、256、257、258、259、260、261、262、263、264、265

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：7bcb3e44c7470c65c521cd1e2d9ca20b16ba8429042c359829e553e80d104203

OCR 产物 SHA-256：7bcb3e44c7470c65c521cd1e2d9ca20b16ba8429042c359829e553e80d104203