1. The problem statement, all variables and given/known data
Find the expectation values of x and p for the state
[itex]\vert \alpha \rangle = e^{-\frac{1}{2}\vert\alpha\vert^2}exp(\alpha a^{\dagger})\vert 0 \rangle[/itex], where ##a## is the destruction operator.
2. Relevant equations
Destruction and creation operators
##a=Ax+Bp##
##a^{\dagger}=Ax-Bp##
For some constants A (real) and B (imaginary) whose value is not important now.
3. The attempt at a solution
I've found a solution, but it is so simple it looks dumb. State the expectation value:
##\langle x \rangle = \langle 0 \vert e^{- \frac{1}{2}\vert\alpha \vert ^2} e^{\alpha^*a}\hat{x}e^{\alpha a^{\dagger}}e^{- \frac{1}{2}\vert\alpha \vert ^2} \vert 0\rangle = ##
Now if we descompose the first exponential ##e^{\alpha^*a}## there will be an annihilation operator acting on the ground state, which is equal to 0 and therefore ANY expectation value for this state is zero...
That seems wrong. Where did I go wrong?
Find the expectation values of x and p for the state
[itex]\vert \alpha \rangle = e^{-\frac{1}{2}\vert\alpha\vert^2}exp(\alpha a^{\dagger})\vert 0 \rangle[/itex], where ##a## is the destruction operator.
2. Relevant equations
Destruction and creation operators
##a=Ax+Bp##
##a^{\dagger}=Ax-Bp##
For some constants A (real) and B (imaginary) whose value is not important now.
3. The attempt at a solution
I've found a solution, but it is so simple it looks dumb. State the expectation value:
##\langle x \rangle = \langle 0 \vert e^{- \frac{1}{2}\vert\alpha \vert ^2} e^{\alpha^*a}\hat{x}e^{\alpha a^{\dagger}}e^{- \frac{1}{2}\vert\alpha \vert ^2} \vert 0\rangle = ##
Now if we descompose the first exponential ##e^{\alpha^*a}## there will be an annihilation operator acting on the ground state, which is equal to 0 and therefore ANY expectation value for this state is zero...
That seems wrong. Where did I go wrong?
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