Guided reading
Follow the algebraic route from the compatible pair \(\hat L^2,\hat L_z\) to the ladder operators and the allowed projection values.
Compatible pair
Since the components do not commute with one another, the total angular-momentum operator is introduced.
\[\hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2\]
\[[\hat L^2,\hat L_u]=0\]
By convention, \(\hat L_z\) is chosen as the component to be known together with \(\hat L^2\).
Direct eigenvalue result
\[\hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle\]
\[\hat L_z|l,m\rangle=m\hbar|l,m\rangle\]
\[-l\le m\le +l,\qquad m=-l,-l+1,\ldots,l-1,l\]
Thus, for a fixed \(l\), there are \(2l+1\) allowed values of \(m\).
Raising and lowering operators
\[\hat L_\pm=\hat L_x\pm i\hat L_y\]
\[\hat L_x={1\over2}(\hat L_++\hat L_-),\qquad \hat L_y={1\over2i}(\hat L_+-\hat L_-)\]
\[[\hat L_\pm,\hat L_\mp]=\pm2\hbar\hat L_z,\qquad [\hat L_z,\hat L_\pm]=\pm\hbar\hat L_\pm\]
\[[\hat L^2,\hat L_\pm]=0,\qquad \hat L_\pm^\dagger=\hat L_\mp\]
Ladder action
The ladder changes \(m\) by one unit and keeps \(l\) unchanged.
\[\hat L_\pm|l,m\rangle=\sqrt{l(l+1)-m(m\pm1)}\,\hbar |l,m\pm1\rangle\]
\[\hat L_\pm\hat L_\mp=\hat L^2-\hat L_z^2\pm\hbar\hat L_z\]
The ladder must stop at the maximum and minimum allowed projections.
\[\hat L_+|l,l\rangle=0,\qquad \hat L_-|l,-l\rangle=0\]
Associated physics
The algebra says that the magnitude and one projection can be specified together, while the other projections cannot be fixed at the same time. The finite ladder gives the allowed projections and the allowed integer or half-integer values of \(l\).
This chain supplies the operational rules used in the following representations; the detailed proofs remain part of the long-form treatment.
Calculator link
The matrix calculator for this chapter uses the algebra above to build \(\hat L_z,\hat L_x,\hat L_y,\hat L_\pm\) and \(\hat L^2\) in the \(\hat L_z\) basis.
Open the angular-momentum matrix calculator.
Exercise-ready boundary
This page supports guided exercises on: the angular-momentum algebra from the compatible pair through ladder quantization and normalized ladder action.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed for a compact calculation or explanation.
- Long-form material: complete proofs, long demonstrations, extended examples and the full exercise set remain in the textbook.
- Boundary: do not introduce results, notation or physical claims outside the sequence presented here and in Chapter 6.
Practice anchors
Use these anchors only within the material introduced on this page.
- Focus: the angular-momentum algebra from the compatible pair through ladder quantization and normalized ladder action.
- Conceptual check: connect the allowed \(m\) values to projection measurements.
- Key equation 1:
\[\hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle,\qquad \hat L_z|l,m\rangle=m\hbar|l,m\rangle\]
- Key equation 2:
\[\hat L_\pm|l,m\rangle=\sqrt{l(l+1)-m(m\pm1)}\,\hbar |l,m\pm1\rangle\]
- Typical task: apply the ladder formula, identify endpoints, or build a matrix element.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 6 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, proofs, examples and exercises.