Guided reading
This item begins with the Schwartz inequality, uses it to obtain the Schrödinger generalized uncertainty principle, and then applies the result to angular momentum.
Starting point: Schwartz inequality
Consider a vector written as a superposition of two vectors.
\[|\psi\rangle=|n\rangle+\alpha |m\rangle,\qquad \langle\psi|\psi\rangle\ge0\]
\[\langle\psi|\psi\rangle=\langle n|n\rangle+\alpha\langle n|m\rangle+\alpha^\ast\langle m|n\rangle+\alpha^\ast\alpha\langle m|m\rangle\ge0\]
Minimizing the positive inner product with respect to \(\alpha\) and \(\alpha^\ast\) gives
\[{\partial\langle\psi|\psi\rangle\over\partial\alpha}=\langle n|m\rangle+\alpha^\ast\langle m|m\rangle=0,\qquad \alpha^\ast=-{\langle n|m\rangle\over\langle m|m\rangle}\]
\[{\partial\langle\psi|\psi\rangle\over\partial\alpha^\ast}=\langle m|n\rangle+\alpha\langle m|m\rangle=0,\qquad \alpha=-{\langle m|n\rangle\over\langle m|m\rangle}\]
Substitution of these minimizing values leaves the inequality
\[\langle n|n\rangle\langle m|m\rangle\ge |\langle n|m\rangle|^2\]
Fluctuation vectors
For two Hermitian observables \(A\) and \(B\), define deviations from the expectation values.
\[\delta\hat a=\hat A-\langle\hat A\rangle,\qquad \delta\hat b=\hat B-\langle\hat B\rangle\]
\[|a\rangle=\delta\hat a|\psi\rangle,\qquad |b\rangle=\delta\hat b|\psi\rangle\]
\[\langle a|a\rangle=\langle(\hat A-\langle\hat A\rangle)^2\rangle=(\Delta A)^2,\qquad \langle b|b\rangle=(\Delta B)^2\]
Taking these vectors into the Schwartz inequality gives
\[(\Delta A)^2(\Delta B)^2=\langle a|a\rangle\langle b|b\rangle\ge |\langle a|b\rangle|^2\]
Commutator and anti-commutator split
The inner product on the right is written as an expectation value of \(\delta\hat a\,\delta\hat b\).
\[\langle a|b\rangle=\langle\delta\hat a\,\delta\hat b\rangle\]
\[\delta\hat a\,\delta\hat b={1\over2}[\delta\hat a,\delta\hat b]+{1\over2}\{\delta\hat a,\delta\hat b\}\]
\[[\delta\hat a,\delta\hat b]=[\hat A,\hat B]\]
\[[\hat A,\hat B]=i\hat C\]
\[\delta\hat a\,\delta\hat b={i\over2}\hat C+{1\over2}\{\delta\hat a,\delta\hat b\}\]
The covariance is introduced from the anti-commutator term.
\[\operatorname{cov}(A,B)={1\over2}\langle\{\delta\hat a,\delta\hat b\}\rangle={1\over2}\langle\{\hat A,\hat B\}\rangle-\langle\hat A\rangle\langle\hat B\rangle\]
\[|\langle a|b\rangle|^2={1\over4}|\langle[\hat A,\hat B]\rangle|^2+|\operatorname{cov}(A,B)|^2\]
Schrödinger uncertainty principle
Combining the Schwartz inequality with the last expression gives the generalized uncertainty relation. This equation is the Schrödinger uncertainty relation:
\[(\Delta A)^2(\Delta B)^2\ge {1\over4}|\langle[\hat A,\hat B]\rangle|^2+|\operatorname{cov}(A,B)|^2\]
The same content can be written through the covariance matrix.
\[\Sigma=\begin{pmatrix}\operatorname{var}(A)&\operatorname{cov}(A,B)\\ \operatorname{cov}(A,B)&\operatorname{var}(B)\end{pmatrix},\qquad \det\Sigma\ge {1\over4}|\langle[\hat A,\hat B]\rangle|^2\]
Application to angular momentum
For angular momentum, the generalized relation becomes
\[(\Delta L_u)^2(\Delta L_v)^2\ge {1\over4}|\langle[\hat L_u,\hat L_v]\rangle|^2+|\operatorname{cov}(L_u,L_v)|^2\]
The expectation values obtained in the chapter give
\[\langle\hat L_z\rangle=m\hbar,\qquad \langle\hat L_x\rangle=\langle\hat L_y\rangle=0\]
\[\langle\hat L_x\hat L_y+\hat L_y\hat L_x\rangle=0,\qquad \langle\hat L_z\hat L_x\rangle=\langle\hat L_y\hat L_z\rangle=0\]
Therefore the covariance term vanishes for the angular-momentum components considered here:
\[\operatorname{cov}(L_u,L_v)=0\]
Using the angular-momentum commutator then gives
\[\Delta L_u\Delta L_v\ge {\hbar\over2}\left|\epsilon_{uvw}\langle\hat L_w\rangle\right|\]
\[\Delta L_x\Delta L_y\ge {\hbar^2\over2}|m|\]
Physical meaning
The standard deviation \(\Delta L_z\) can be zero for a state with definite \(m\). This does not violate the uncertainty principle because the corresponding products involving \(L_z\) have a zero lower bound in this state.
The transverse product is different: \(\Delta L_x\Delta L_y\) is controlled by the commutator and by the projection quantum number \(m\). This is the uncertainty-principle version of the cone discussion in the previous item.
Exercise-ready boundary
This page supports guided exercises on: Schwartz inequality, the Schrödinger uncertainty relation and the angular-momentum uncertainty product.
- 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: Schwartz inequality, the Schrödinger uncertainty relation and the angular-momentum uncertainty product.
- Conceptual check: explain the role of the covariance term before setting it to zero for angular momentum.
- Key equation 1:
\[\langle n|n\rangle\langle m|m\rangle\ge |\langle n|m\rangle|^2\]
- Key equation 2:
\[(\Delta A)^2(\Delta B)^2\ge {1\over4}|\langle[\hat A,\hat B]\rangle|^2+|\operatorname{cov}(A,B)|^2\]
- Key equation 3:
\[\Delta L_x\Delta L_y\ge {\hbar^2\over2}|m|\]
- Typical task: identify the commutator, evaluate the covariance statement and interpret the uncertainty product.
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.