Coherent States - The Wavefunction of A Coherent State

The Wavefunction of A Coherent State

To find the wavefunction of the coherent state, it is easiest to employ the Heisenberg picture of the quantum harmonic oscillator for the coherent state . Now we have that

So the coherent state is an eigenstate of the annihilation operator in the Heisenberg picture. It is easy to show that in the Schrödinger picture the same eigenvalue occurs:

Taking the coordinate representations we obtain the following differential equation

which is easily solved to give

where is a still undetermined phase, which we must fix by demanding that the wavefunction satisfies the Schrödinger equation. We obtain that

where is the initial phase of the eigenvalue, i.e. . The mean position and momentum of the wavepacket are

$\langle \hat{x}(t) \rangle = \sqrt{\frac{2\hbar}{m\omega}}\Re \qquad \qquad \langle \hat{p}(t) \rangle = \sqrt{2m\hbar\omega}\Im$

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