The Harmonic Oscillator

The connection of the above algebraic tour de force with the harmonic oscillator is very simple: The Hamiltonian (4.38) can be written as

$$\fbox{$ \begin{array}{rcl} \displaystyle \hat{H}=\frac{\hat{p}^2}... ...{2} \right) = \hbar \omega \left( \hat{N} + \frac{1}{2} \right) \end{array}$\ }$$ (272)

which you can check by inserting the definitions of $a$ and $a^{\dagger }$. The eigenvectors of $\hat{H}$ are the eigenvectors of $\hat{N}$:

$$\hat{H}\vert n\rangle = \hbar \omega \left( \hat{N} + \frac{1}{2}... ...t) \vert n\rangle = \hbar \omega \left( n + \frac{1}{2} \right) \vert n\rangle,$$ (273)

from which we can read off the eigenvalues of the harmonic oscillator, $E_n= \hbar\omega(n+1/2)$. The corresponding eigenfunctions are, of course, the eigenfunctions of the harmonic oscillator,

$$\vert n\rangle \leftrightarrow \psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{n! 2^n}}H_n\left(\sqrt{\frac{m\omega}{\hbar}}\right)e^{-\frac{m\omega}{2\hbar}x^2}.$$ (274)

This is not so easy to see directly; it is proofed for the ground state $\vert\rangle$ in the problems.