6. 3D Harmonic Oscillator
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Isotropic harmonic oscillator in three dimensions: important for nuclear and molecular physics.
The Hamiltonian
$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2r^2$$
Isotropic: same frequency in all directions
Cartesian Approach
Hamiltonian separates:
$$\hat{H} = \hat{H}_x + \hat{H}_y + \hat{H}_z$$
Energy eigenvalues:
$$E_{n_x,n_y,n_z} = \hbar\omega\left(n_x + n_y + n_z + \frac{3}{2}\right)$$
where $n_x, n_y, n_z = 0, 1, 2, \ldots$
Total Quantum Number
Define $N = n_x + n_y + n_z$:
$$E_N = \hbar\omega\left(N + \frac{3}{2}\right), \quad N = 0, 1, 2, \ldots$$
Degeneracy
Number of ways to write $N = n_x + n_y + n_z$:
$$g_N = \frac{(N+1)(N+2)}{2}$$
Examples:
- $N=0$: $g_0 = 1$ (ground state)
- $N=1$: $g_1 = 3$ (three states)
- $N=2$: $g_2 = 6$
Spherical Approach
In spherical coordinates, wave functions factor:
$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi)$$
Radial equation:
$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[\frac{1}{2}m\omega^2r^2 + \frac{\hbar^2\ell(\ell+1)}{2mr^2}\right]u = Eu$$
Quantum Numbers in Spherical Form
Energy levels labeled by radial quantum number $n_r$:
$$E_{n_r,\ell} = \hbar\omega(2n_r + \ell + 3/2)$$
where $n_r = 0, 1, 2, \ldots$ and $\ell = 0, 1, 2, \ldots$
Relation: $N = 2n_r + \ell$
Selection Rules
For given $N$, allowed $\ell$ values have same parity:
- Even $N$: $\ell = 0, 2, 4, \ldots$
- Odd $N$: $\ell = 1, 3, 5, \ldots$
Ladder Operators
Can define six ladder operators (three pairs):
$$\hat{a}_{i\pm} = \frac{1}{\sqrt{2m\hbar\omega}}(m\omega\hat{x}_i \mp i\hat{p}_i), \quad i = x,y,z$$
Energy increases/decreases by $\hbar\omega$ with each operator
Ground State
$$\psi_0(\vec{r}) = \left(\frac{m\omega}{\pi\hbar}\right)^{3/4}e^{-m\omega r^2/(2\hbar)}$$
Spherically symmetric, non-degenerate
Applications
- Nuclear physics: Shell model (nucleons in average potential)
- Molecular physics: Vibrations near equilibrium
- Quantum dots: Electrons confined in parabolic potential
- Trapped ions: Paul and Penning traps