- #1
Patrick McBride
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Homework Statement
Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is:
[; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;]
[; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;]
where P, X are the momentum and position operators, respectively, and [; X= \sqrt{m}x ;].
Homework Equations
[; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;]
[; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;]
[; X= \sqrt{m}x ;] [; P = p/ \sqrt{m}} ;]
Sch. Eqn.: [;-\frac{\hbar}{2m}\frac{d^2\Psi }{dx^2}+\frac{1}{2}m\omega^2x^2\Psi = E\Psi ;].
The Attempt at a Solution
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So as from what I can tell [; [P+i\omega X] ;] refers to the +x direction and [;
[P-i\omega X] ;] is in the -x direction. The momentum operator is going to be [; P = p/ \sqrt{m}} ;].Sch. Eqn.: [;-\frac{\hbar}{2m}\frac{d^2\Psi }{dx^2}+\frac{1}{2}m\omega^2x^2\Psi = E\Psi ;].
So what I assume is Sch.Eqn. must be solved algebraically. From there we can get our Hamiltonian.
Any suggestions?