Introduction: Entering the Real World

Real particles live in three dimensions. The extension from 1D to 3D is straightforward in Cartesian coordinates, though more interesting physics emerges in spherical coordinates for central potentials.

The 3D TISE

\[-\frac{\hbar^2}{2m}\nabla^2\psi + V(\vec{r})\psi = E\psi\]

In Cartesian coordinates:

\[\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\]

Separable Potentials

If \(V(x, y, z) = V_x(x) + V_y(y) + V_z(z)\), the 3D problem separates into three 1D problems:

\[\psi(x, y, z) = X(x)Y(y)Z(z)\] \[E = E_x + E_y + E_z\]

Worked Example: 3D Infinite Box

Particle in a box \(0 < x, y, z < L\):

\[\psi_{n_x n_y n_z} = \left(\frac{2}{L}\right)^{3/2}\sin\frac{n_x\pi x}{L}\sin\frac{n_y\pi y}{L}\sin\frac{n_z\pi z}{L}\] \[E_{n_x n_y n_z} = \frac{\pi^2\hbar^2}{2mL^2}(n_x^2 + n_y^2 + n_z^2)\]

The Quantum Connection

In 3D, degeneracy becomes common. For the 3D box, states like (2,1,1), (1,2,1), (1,1,2) have the same energy—they're related by symmetry. This degeneracy, absent in 1D, leads to rich physics when broken by perturbations.