- #1

Patrick McBride

- 3

- 0

## 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

[/B]

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?