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Thread: Quantum Harmonic Oscillator

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    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:


     x^{2}
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    Re: Quantum Harmonic Oscillator

    Quote Originally Posted by Natalie2016 View Post
    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:


     x^{2}
    Start with the Hamiltonian: 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.)

    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
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