Chapter 6 · Sections 6.3 and 6.5

Angular Momentum Matrix Calculator

Choose \(l\), build \(\hat L_z,\hat L_x,\hat L_y,\hat L_+,\hat L_-\) and \(\hat L^2\) in the \(\hat L_z\) basis, then choose \(x\), \(y\), or \(z\) as the reference basis and inspect the transformed matrices and the basis decomposition.

Original calculator · Quantum Mechanics book-app · Prof. Mario Reis

Inputs

Angular momentum \(l\)

Reference basis

The first calculator is fixed in the \(\hat L_z\) basis. The second uses the unitary matrix \(U\) to express the same operators in the chosen \(x\), \(y\), or \(z\) basis.

Book equations used

\[\hat L_z|l,m\rangle=m\hbar|l,m\rangle,\qquad \hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle\]

\[\hat L_\pm|l,m\rangle=\sqrt{l(l+1)-m(m\pm1)}\,\hbar |l,m\pm1\rangle\]

\[\hat L_x={1\over2}(\hat L_++\hat L_-),\qquad \hat L_y={1\over2i}(\hat L_+-\hat L_-)\]

\[(\hat O)_{\rm basis}=U(\hat O)_zU^\dagger\]

Angular Momentum Matrix Calculator

Calculator 1 · matrices in the \(\hat L_z\) basis

Calculator 2 · chosen basis and \(U\)

Decomposition of the chosen basis in the \(\hat L_z\) basis