Guided reading
This closing item is a compact map for reviewing Chapter 6 and preparing exercises. It does not add new results beyond the chapter sequence.
Core algebra
- Angular momentum begins as \(\vec L=\vec r\times\vec p\).
- The components obey \([\hat L_u,\hat L_v]=i\hbar\epsilon_{uvw}\hat L_w\).
- \(\hat L^2\) commutes with each component, and \(\hat L_z\) is chosen as reference.
\[\hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle,\qquad \hat L_z|l,m\rangle=m\hbar|l,m\rangle\]
Ladder result
\[\hat L_\pm|l,m\rangle=\sqrt{l(l+1)-m(m\pm1)}\,\hbar |l,m\pm1\rangle\]
The finite ladder is the route to the allowed \(l\) and \(m\) values.
Representations
- In matrices, a fixed \(l\) subspace has dimension \(2l+1\).
- Changing the measured reference axis is described by a unitary change of basis.
- In position space, \(\vec{\hat L}=(-i\hbar)\vec r\times\vec\nabla\).
\[\hat L^2Y_l^m=l(l+1)\hbar^2Y_l^m,\qquad \hat L_zY_l^m=m\hbar Y_l^m\]
Uncertainty
The generalized uncertainty relation becomes an angular-momentum statement through the same commutator algebra.
\[\Delta L_u\Delta L_v\ge {\hbar\over2}\left|\epsilon_{uvw}\langle\hat L_w\rangle\right|\]
Calculator support
The angular-momentum matrix calculator checks the matrix forms of \(\hat L_z,\hat L_x,\hat L_y,\hat L_\pm\) and \(\hat L^2\), and shows how a chosen \(x\), \(y\), or \(z\) basis is decomposed in the \(\hat L_z\) basis.
Open the angular-momentum matrix calculator.
Exercise map
The exercise set groups conceptual, text-based, practice, development and advanced tasks. The HTML chapter supports first-step reasoning and compact checks; the complete derivations, proofs and worked-out exercise style remain in the textbook and solution manual.
Exercise-ready boundary
This page supports guided exercises on: the complete Chapter 6 toolkit: commutators, compatible observables, ladders, representations and uncertainty.
- 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 complete Chapter 6 toolkit: commutators, compatible observables, ladders, representations and uncertainty.
- Conceptual check: connect the equation to what can be measured, represented or inferred physically.
- Key equation 1:
\[[\hat L_u,\hat L_v]=i\hbar\epsilon_{uvw}\hat L_w\]
- Key equation 2:
\[\hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle,\qquad \hat L_z|l,m\rangle=m\hbar|l,m\rangle\]
- Typical task: verify one relation, apply it to a specified state or matrix, and state the measurement consequence.
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.