Seeking a phase angle operator for the QHO

[snoopies622]

According to Daniel Gillespie in A Quantum Mechanics Primer (1970),

" . . . any observable which in classical mechanics is some well behaved function of position and momentum, f(x,p), is represented in quantum mechanics by the operator $f ( \hat{x} , \hat {p} )$. That is,

$$<br /> a = f (x,p) . . . implies . . . \hat{a} = f ( \hat{x} , \hat {p} ) = f ( x , -i \hbar \frac {d}{dx}) ."<br />$$

Apparently this works for finding the angular momentum operators, for example in classical mechanics

$$<br /> L_z = xp_y - yp_x<br />$$

and in quantum mechanics

$$<br /> \hat{L}_z = \hat{x} \hat{p_y}- \hat{y} \hat{p_x} =<br /> <br /> x(-i\hbar \frac {\partial}{\partial y}) - y(-i\hbar \frac {\partial}{\partial x} )= -i\hbar (x \frac {\partial}{\partial y} - y \frac {\partial}{\partial x})<br />$$

Now I am wondering if this idea can be applied to the harmonic oscillator. Specifically, since phase angle

$$<br /> \theta = arc tan ( \frac {p}{ x \sqrt {km} } )<br />$$

(1) can I make a quantum phase angle operator by replacing the p and x above with their corresponding quantum operators?

and

(2) what are the phase angle eigenvalues for a QHO?

Thanks.

 

[snoopies622]

a bump and a follow-up:

Last night I worked out that for the classical harmonic oscillator, if we define

$<br /> N=H/ \hbar \omega<br />$

then the Poisson bracket between N and phase angle $\theta$ is $2 / \omega$.

Is there any connection between this and the photon number - angle phase uncertainly relation?

 

[andresB]

I think this could be useful

http://arxiv.org/pdf/hep-th/9304036v1.pdf

 

[snoopies622]

Thanks AndresB, looks very interesting. Will try to take it in this weekend when I have a few free minutes.