Coherent states in quantum mechanics (Schrödinger Cat)

[CharlieCW]

Hello. I've been struggling for a day with the following problem on Quantum coherent states, so I was wondering if you could tell me if I'm going in the right direction (I've read the books of Sakurai and Weinberg but can't seem to find an answer)

1. Homework Statement

*Suppose a Schrödinger cat state given by $|\alpha_+>:=N(|\alpha\rangle+|-\alpha\rangle)$, where $|\alpha\rangle$ are the coherent states of the harmonic oscillator and $N$ is a constant of proportionality. Verify the relationship of Robertson-Schrödinger for $X$ and $P$ operators.

2. Relevant expressions

$$a|\alpha\rangle=\alpha|\alpha\rangle$$

$$x=x_0(a+a^{\dagger})$$

$$p=p_0(a-a^{\dagger})$$

3. The attemp at a solution

The solution is simple as we only have to calculate $\langle x \rangle$, $\langle p \rangle$, $\langle x^2 \rangle$, $\langle p^2 \rangle$, and $rel(x,p)$.

From the definition of coherent states, $a|\alpha\rangle=\alpha|\alpha\rangle$ and with $x=x_0(a+a^{\dagger})$, I began by calculating $\langle x \rangle$ in the following way:

$$\langle x \rangle=\langle\alpha_+|x|\alpha_+\rangle=(\langle\alpha|+\langle-\alpha|)x(|\alpha\rangle+|-\alpha\rangle)$$
$$\langle x \rangle=\langle\alpha|x|\alpha\rangle+\langle\alpha|x|-\alpha\rangle+\langle-\alpha|x|\alpha\rangle+\langle-\alpha|x|-\alpha\rangle=x_0(\langle\alpha|a+a^\dagger|\alpha\rangle+\langle\alpha|a+a^\dagger|-\alpha\rangle+\langle-\alpha|a+a^\dagger|\alpha\rangle+\langle-\alpha|a+a^\dagger|-\alpha\rangle)$$

From the definition $a|\alpha\rangle=\alpha|\alpha\rangle$, we deduce $|\alpha\rangle a^\dagger=|\alpha\rangle \alpha^{*}$.

Moreover (I'm not entirely sure of this step), by replacing $\alpha'\rightarrow-\alpha$ we deduce that $a|-\alpha\rangle=-\alpha|\alpha\rangle$ and $|-\alpha\rangle a^\dagger=|\alpha\rangle -\alpha^{*}$.

Applying the operators to the above expression, we finally get:

$$\langle x \rangle=x_0 ((a+a^{*})+(-a+a{*})+(a-a^{*})+(-a-a^{*}))=0$$

In an analogous manner, $\langle p \rangle=0$. If this procedure is correct, the calculation of the rest of the problem is straightforward.

My main doubt comes with the eigenvalues of the coherent states for $|-\alpha\rangle$.

 

[Asim Riaz]

Is it true that $|\alpha\rangle a^\dagger=|\alpha\rangle \alpha^{*}$ and $|-\alpha\rangle a^\dagger=|\alpha\rangle -\alpha^{*}$?Any help would be much appreciated!Yes, it is true that $|\alpha\rangle a^\dagger=|\alpha\rangle \alpha^{*}$ and $|-\alpha\rangle a^\dagger=|\alpha\rangle -\alpha^{*}$. Your calculation of the expectation values for $\langle x \rangle$ and $\langle p \rangle$ is correct. To calculate the remaining expectation values and the relation between $x$ and $p$, you can use the following properties of the coherent states: $$ \langle \alpha | a^{\dagger}a | \alpha \rangle = \langle \alpha | \alpha \rangle \alpha^* \alpha = | \alpha |^2 $$ $$ \langle \alpha | aa^{\dagger} | \alpha \rangle = \langle \alpha | \alpha \rangle \alpha \alpha^* = | \alpha |^2 $$ $$ \langle \alpha | a^{\dagger} a + aa^{\dagger} | \alpha \rangle = | \alpha |^2 ( \alpha^* + \alpha ) = 2| \alpha |^2 \cos \theta $$ and $$ \langle \alpha | a^{\dagger} a - aa^{\dagger} | \alpha \rangle = | \alpha |^2 ( \alpha^* - \alpha ) = 2| \alpha |^2 \sin \theta , $$ where $\theta$ is the phase of the coherent state. From these equations, you can easily calculate the remaining expectation values and the relation between $x$ and $p$. I hope this helps.