# Thread: Quantum Harmonic Oscillator

1. ## Quantum Harmonic Oscillator

For a quantum harmonic oscillator, write down a differential equation for the wavefunction of the ground state, by directly deriving it from the defining property.

a|0> = 0 Solve such a differential equation (up to normalisation).

Attempt:

$\displaystyle x^{2}$

2. ## Re: Quantum Harmonic Oscillator

Originally Posted by Natalie2016
For a quantum harmonic oscillator, write down a differential equation for the wavefunction of the ground state, by directly deriving it from the defining property.

a|0> = 0 Solve such a differential equation (up to normalisation).

Attempt:

$\displaystyle x^{2}$
Start with the Hamiltonian: $\displaystyle H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x$. Now substitute in the operator form for p. (The x in the Hamiltonian becomes the operator x, of course.)

$\displaystyle E |x> = - \frac{\hbar ^2}{2m} \frac{d^2}{dx^2} |x> + ~ \frac{1}{2} m \omega ^2 x^2 |x>$

Can you proceed from there?

-Dan