3d Oscillator

Quantum Mechanics · Part 4

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Table of Contents

  1. 1.The Hamiltonian
  2. 2.Cartesian Approach
  3. 3.Total Quantum Number
  4. 4.Degeneracy
  5. 5.Spherical Approach
  6. 6.Quantum Numbers in Spherical Form
  7. 7.Selection Rules
  8. 8.Ladder Operators
  9. 9.Ground State
  10. 10.Applications

Key Equations

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2r^2$$
$$E_{n_x,n_y,n_z} = \hbar\omega\left(n_x + n_y + n_z + \frac{3}{2}\right)$$
$$g_N = \frac{(N+1)(N+2)}{2}$$
$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[\frac{1}{2}m\omega^2r^2 + \frac{\hbar^2\ell(\ell+1)}{2mr^2}\right]u = Eu$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Quantum Mechanics course on CoursesHub.World. Master quantum mechanics from mathematical foundations to advanced topics. Covers Hilbert spaces, wave functions, angular momentum, perturbation theory, scattering, and path integrals with 450+ pages ...

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