- #1
snoopies622
- 846
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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 [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.
" . . . 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.
Last edited: