According to Daniel Gillespie in A Quantum Mechanics Primer (1970),(adsbygoogle = window.adsbygoogle || []).push({});

" . . . 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 [itex] f ( \hat{x} , \hat {p} ) [/itex]. That is,

[tex]

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

[/tex]

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

[tex]

L_z = xp_y - yp_x

[/tex]

and in quantum mechanics

[tex]

\hat{L}_z = \hat{x} \hat{p_y}- \hat{y} \hat{p_x} =

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})

[/tex]

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

[tex]

\theta = arc tan ( \frac {p}{ x \sqrt {km} } )

[/tex]

(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.

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# Seeking a phase angle operator for the QHO

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