Expected value of variance of Hamiltonian in coherent states

AI Thread Summary
The discussion focuses on calculating the expected value of the variance of energy in coherent states, specifically addressing challenges with non-hermitian and non-commutative raising and lowering operators. Participants are unsure about their calculations of <H>² and <H²>, seeking clarification on the correct approach. A solution involving the commutation relation between the operators is proposed, and feedback indicates that the approach appears valid. The conversation emphasizes the importance of correctly applying operator algebra in quantum mechanics. Clear steps for resolving the calculations are requested to identify any potential mistakes.
graviton_10
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Homework Statement
Find the variance of the energy in coherent state |ɑ>.
Relevant Equations
<ΔH> = <ɑ| HH |ɑ>
I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation.

Here is my solution:
 

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Check the step circled in orange. ##a^\dagger## and ##a## don't commute.
 
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Likes vanhees71 and Terrakron
Yes, but how to do it the right way?
 
graviton_10 said:
Yes, but how to do it the right way?
Please post the steps for how you reduced ##\langle \alpha | (a^{\dagger} a)^2|\alpha \rangle## to ##|\alpha^*\alpha|^2 \langle \alpha | \alpha \rangle##. That way, we can help you see where you made a mistake.
 
So, I used the fact that the commutator of a and a dagger is 1. Does it look good now?
 

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That looks good.
 
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