Physical interpretation of this coherent state

• I
• ergospherical
In summary, the conversation discusses the coherent state for a quantum harmonic oscillator. It is shown that the state is an eigenvector of the raising and lowering operators, and the inner product of two states is calculated. The physical interpretation of the state is also discussed, with the mean value of momentum being proportional to the imaginary part of ##\alpha##. The position and momentum representations of the coherent state can be seen to oscillate like in a classical system using Ehrenfest's theorem.
ergospherical
Given the usual raising & lowering operators ##A^{\dagger}## & ##A## for a quantum harmonic oscillator, consider a coherent state ##|\alpha\rangle \equiv e^{\alpha A^{\dagger} - \bar{\alpha} A} |0\rangle##. I first check that ##|\alpha\rangle## is an eigenvector of ##A##. I already proved that if ##X##, ##Y## commute with ##[X,Y]## then ## e^{X+Y} = e^{X} e^{Y} e^{-\frac{1}{2}[X,Y]}##, which is applicable here because both ##A^{\dagger}## & ##A## clearly commute with ##[A^{\dagger}, A] = 1##, therefore\begin{align*}
|\alpha \rangle &= e^{\alpha A^{\dagger}} e^{-\bar{\alpha} A} e^{-\frac{1}{2}[\alpha A^{\dagger},-\bar{\alpha} A]} |0\rangle = e^{\frac{1}{2}|\alpha|^2} e^{\alpha A^{\dagger}} |0\rangle
\end{align*}where I used that ##e^{-\bar{\alpha} A}|0 \rangle = (1 - \bar{\alpha} A + \dots)|0\rangle = |0 \rangle##. Upon application of ##A##,\begin{align*}
A|\alpha\rangle = e^{\frac{1}{2}|\alpha|^2} (A e^{\alpha A^{\dagger}}) |0 \rangle = e^{\frac{1}{2}|\alpha|^2} ([A , e^{\alpha A^{\dagger}}] + e^{\alpha A^{\dagger}} A) |0 \rangle &= e^{\frac{1}{2}|\alpha|^2} (\alpha e^{\alpha A^{\dagger}} + e^{\alpha A^{\dagger}} A) |0 \rangle \\
&= \alpha e^{\frac{1}{2}|\alpha|^2} e^{\alpha A^{\dagger}} |0\rangle \\
&= \alpha |\alpha \rangle
\end{align*}which means that ##|\alpha \rangle## is of eigenvalue ##\alpha##. I make use of a similar operator identity ##e^X e^Y = e^Y e^X e^{[X,Y]}## to calculate the inner product of two general states:\begin{align*}
\langle \alpha | \beta \rangle = e^{\frac{1}{2} (|\alpha|^2 + |\beta|^2)}\langle 0 | e^{\bar{\beta} A} e^{\alpha A^{\dagger}}| 0 \rangle &= e^{\frac{1}{2} (|\alpha|^2 + |\beta|^2)}\langle 0 | e^{\alpha A^{\dagger}} e^{\bar{\beta} A} e^{[\bar{\beta} A, \alpha A^{\dagger}]} | 0 \rangle \\
&= e^{\frac{1}{2}(|\alpha|^2 + |\beta|^2)}e^{-2\alpha \bar{\beta}} \langle 0 | e^{\alpha A^{\dagger}} e^{\bar{\beta} A} | 0 \rangle \\
&= e^{\frac{1}{2}(|\alpha|^2 -2\alpha \bar{\beta} + |\beta|^2))}
\end{align*}where I used the fact that ##\langle 0 | e^{\alpha A^{\dagger}} \leftrightarrow e^{\bar{\alpha} A} |0\rangle##. It is therefore also the case that ##|\langle \alpha | \beta \rangle|^2 = e^{|\alpha - \beta|^2}##, that the set ##\{ |\alpha \rangle \}_{\alpha \in \mathbf{C}}## spans the space and that one can select a basis from a suitable subset of ##\{ |\alpha \rangle \}_{\alpha \in \mathbf{C}}##.

To consider the physical interpretation of ##|\alpha(t) \rangle## for a general complex ##\alpha \in \mathbf{C}##, it is advised to calculate ##\langle \alpha | P | \alpha \rangle##,\begin{align*}
\langle \alpha | P | \alpha \rangle = \frac{i}{\sqrt{2}} \langle \alpha | (A^{\dagger} - A) | \alpha \rangle &= \frac{i}{\sqrt{2}} \langle \alpha | (\bar{\alpha} - \alpha) |\alpha \rangle \\
&= \frac{i}{\sqrt{2}} (\bar{\alpha} - \alpha) \\
&= \sqrt{2} \mathrm{Im}(\alpha)
\end{align*}How am I supposed to interpret that the mean value of the momentum is proportional to the imaginary part of ##\alpha##? Also, how would I use this result to describe, qualitatively, how the position and momentum space wavefunctions evolve (I already worked out that ##|\alpha(t) \rangle = e^{-i\omega t/2} | e^{-i\omega t} \alpha \rangle##?

Last edited:
vanhees71
Thanks for the link. Is there a way to see that the position and momentum basis wavefunctions oscillate like in a classical system without further calculation apart from the OP? I only ask because my problem sheet suggests that it is not necessary to work anything else out explicitly.

vanhees71
You can refer to Ehrenfest's theorem. Since the harmonic oscillator has a linear set of equations of motion for ##x## and ##p## the expectation values fulfill precisely the same equations of motion as the classical harmonic oscillator. A coherent state is one of minimal uncertainty product, ##\Delta x \Delta p=\hbar/2##, and thus are closest to a classical description of the system, especially for large ##|\alpha|##.

Demystifier and ergospherical

1. What is a coherent state?

A coherent state is a quantum state of a system that is described by a wave function that does not change over time. It is often referred to as a "classical-like" state because it exhibits properties similar to those of classical systems, such as a well-defined position and momentum.

2. How is the coherent state different from other quantum states?

Unlike other quantum states, the coherent state has a minimum uncertainty in both position and momentum, making it a highly localized state. It also exhibits a stable phase relationship between the position and momentum, leading to properties that resemble those of a classical oscillator.

3. What is the physical interpretation of a coherent state?

The physical interpretation of a coherent state is that it represents a state of a quantum system that is highly localized in space and has a well-defined phase relationship between position and momentum. It is often used to describe the behavior of macroscopic systems, such as lasers and superconductors.

4. How are coherent states used in practical applications?

Coherent states have many practical applications, including in quantum optics, quantum computing, and quantum information processing. They are also used in engineering applications, such as in the design of lasers and other coherent light sources.

5. Can coherent states be observed in experiments?

Yes, coherent states have been observed in various experiments, such as in the behavior of superconducting circuits and in the properties of light in a laser. They have also been observed in experiments involving quantum entanglement and quantum teleportation.

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