Schrödinger Equation in Higher Dimensions

Phase 3: Advanced Quantum Systems

Syllabus Goal

Transition from 1D to 3D potential problems.

Separation of Variables

For central potentials $V(r)$: $$ \psi(r, \theta, \phi) = R(r) Y_{\ell m}(\theta, \phi) $$

• Radial Equation: Depends on $V(r)$.
• Angular Part: Spherical Harmonics $Y_{\ell m}$, universal for all central potentials.

3D Infinite Well

Particle in a box of dimensions $L_x, L_y, L_z$. $$ E = \frac{\pi^2 \hbar^2}{2m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) $$ Degeneracy arises if lengths are equal (symmetry).