angular momentum energy levels
angular momentum energy levels

https://stationarystates.com/basic-quantum-theory/angular-momentum…article-in-a-box/

 


Energy Levels of a Particle in a Box with Angular Momentum

1. Particle in a 1D Box

In 1D, angular momentum doesn’t exist in the usual sense because rotation requires at least two dimensions.
Energy levels remain:

    \[ E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}, \quad n = 1,2,3,\dots \]

2. Particle in a 2D or 3D Box

For a 2D rectangle or 3D cube, the Schrödinger equation separates in Cartesian coordinates:

    \[ \psi(x,y,z) = X(x) Y(y) Z(z) \]

Angular momentum is not conserved in a cubical box. Energy depends on quantum numbers along each axis:

    \[ E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2 m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) \]

3. Particle in a Spherical Box

If the box is spherically symmetric, angular momentum L is a good quantum number.
The Schrödinger equation in spherical coordinates:

    \[ -\frac{\hbar^2}{2m} \nabla^2 \psi(r,\theta,\phi) = E \psi(r,\theta,\phi) \]

Separate variables:

    \[ \psi(r,\theta,\phi) = R_{n\ell}(r) Y_\ell^m(\theta,\phi) \]

where Y_\ell^m are spherical harmonics and \ell is the angular momentum quantum number.
Energy levels include a centrifugal term:

    \[ E_{n\ell} = \frac{\hbar^2}{2m} \left( \frac{\alpha_{n\ell}}{R} \right)^2 \]

Here, \alpha_{n\ell} are the zeros of spherical Bessel functions.
Larger angular momentum (\ell > 0) increases energy because the wavefunction is “pushed outward”.

4. Key Takeaways

  • 1D box: Angular momentum is irrelevant; energy levels are unchanged.
  • Rectangular/cubical box: Energy depends on quantum numbers along each axis, not angular momentum.
  • Spherical box: Higher angular momentum quantum number \ell raises the energy.

Intuition: Higher angular momentum → particle “rotates” more → less probability near the center → higher energy.