Quantum Harmonic Oscillator

The quantum mechanical version of this harmonic oscillator problem may be written as

$$\left( -\frac {\hbar^2}{2 \mu} \frac{d^2}{dx^2} + \frac{1}{2} k x^2 \right) \Psi(x) = E \Psi(x).$$ (14)

By considering the limiting behavior as $x \rightarrow \infty$ and as $x \rightarrow 0$, one finds that only certain energies $E_n$ yield reasonable solutions $\Psi(x)$. The eigenvalues and associated eigenvectors are

$$E_n = \hbar \omega \left( n + \frac{1}{2} \right) = h \nu \left( n + \frac{1}{2} \right),$$ (15)

$$\Psi_n(x) = \left( \frac{\mu \omega}{\pi \hbar 2^{2n} (n!)^2} \right)^{1/4} e... ...r} \right) H_n \left[ \left( \frac{ \mu \omega}{\hbar} \right)^{1/2} x \right],$$ (16)

where $H_n$ are the Hermite polynomials of order $n$. The Hermite polynomials have a number of useful properties such as

$$\frac{d}{dx} H_n(x) = 2n H_{n-1}(x)$$ (17)

$$H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)$$ (18)

$$\int_{-\infty}^{\infty} H_n(x) H_{m}(x) e^{-x^2} dx = \delta_{nm} \sqrt{\pi} 2^n n!.$$ (19)