- #1
naele
- 202
- 1
Homework Statement
Show that, for all complexe numbers [itex]\alpha, a[/itex] has a unique eigenvector [itex]|\alpha\rangle[/itex] that is nothing else but the coherent state
[tex]
\psi(x)=\frac{e^{-\frac{i}{2\hbar}\langle X\rangle\langle P\rangle}}{(\pi\ell^2)^{1/4}}e^{-\frac{(X-\langle X\rangle)^2}{2\ell^2}+\frac{i\langle P\rangle X}{\hbar}}
[/tex]
with
[tex]
\alpha=\langle a \rangle=\frac{1}{\sqrt{2}}\left(\frac{\langle X\rangle}{\ell}+\frac{i\ell\langle P\rangle}{\hbar}\right)
[/tex]
Homework Equations
[tex]
a=\frac{1}{\sqrt{2}}\left(\frac{X}{\ell}+\frac{i\ell P}{\hbar}\right)
[/tex]
[tex] \ell=\sqrt{2}\Delta X[/tex]
The Attempt at a Solution
Ok, so I think I have a game plan. Since [itex]a[/itex] isn't Hermitian then its eigenvalues can be complex. So I should solve the eigenvalue problem for [itex]a|\alpha\rangle=\alpha|\alpha\rangle[/itex]. But since I already showed that when the equality is valid in the Heisenberg inequality we get a gaussian like [itex]\psi(x)[/itex] so if I can show that the eigenvalue problem admits a differential equation similar to what I already showed then that should be sufficient?