Index
Chapter 6 · Item 6.4
Consequences of the algebra
Means, variances and the angular-momentum cone
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Guided reading

After \(\hat L^2\) and \(\hat L_z\) have definite eigenvalues, the next question is what can still be said about the \(x\) and \(y\) components.

Mean transverse components

Writing \(\hat L_x\) and \(\hat L_y\) through \(\hat L_\pm\), the ladder action makes the relevant inner products proportional to \(\langle l,m|l,m\pm1\rangle\), which vanish.

\[\langle \hat L_x\rangle={1\over2}\left(\langle l,m|\hat L_+|l,m\rangle+\langle l,m|\hat L_-|l,m\rangle\right)\]
\[\langle \hat L_y\rangle={1\over2i}\left(\langle l,m|\hat L_+|l,m\rangle-\langle l,m|\hat L_-|l,m\rangle\right)\]
\[\langle \hat L_x\rangle=\langle \hat L_y\rangle=0\]

This result is true for any allowed \(-l\le m\le +l\).

Quadratic averages

The next step is to calculate the quadratic operator. The terms \(\langle \hat L_+^2\rangle\) and \(\langle \hat L_-^2\rangle\) vanish by the same orthogonality argument.

\[\langle \hat L_x^2\rangle={1\over4}\left(\langle \hat L_+^2\rangle+\langle \hat L_-^2\rangle+\langle \hat L_+\hat L_-\rangle+\langle \hat L_-\hat L_+\rangle\right)\]
\[\hat L_\pm\hat L_\mp=\hat L^2-\hat L_z^2\pm\hbar\hat L_z\]
\[\langle \hat L_x^2\rangle=\langle \hat L_y^2\rangle={\hbar^2\over2}\left[l(l+1)-m^2\right]\]
Variances and standard deviations

The variance is the quadratic spread around the mean value. For any component, it is defined as

\[\operatorname{var}(L_u)=\langle \hat L_u^2\rangle-\langle \hat L_u\rangle^2\]

The standard deviation is the positive square root of the variance:

\[\Delta L_u=\sqrt{\operatorname{var}(L_u)}\]

Using the quadratic averages above, the transverse variances follow as

\[\operatorname{var}(L_x)=\operatorname{var}(L_y)={\hbar^2\over2}\left[l(l+1)-m^2\right]\]
\[\Delta L_x=\Delta L_y={\hbar\over\sqrt2}\left[l(l+1)-m^2\right]^{1/2}\]
\[\operatorname{var}(L_z)=0,\qquad \Delta L_z=0\]

The \(z\) component is precise in the state \(|l,m\rangle\), but the transverse components are not.

Maximum projection

Even for the maximum projection \(m=l\), the transverse spread does not vanish:

\[\Delta L_x(m=l)=\Delta L_y(m=l)=\hbar\sqrt{l\over2}\]

Therefore the angular momentum is not absolutely aligned with the \(z\)-axis, even at maximum projection.

Cone representation
Angular momentum cone for l equals 3
For \(l=3\), the magnitude \(\sqrt{l(l+1)}\hbar\) and projection \(m\hbar\) are definite, while the orthogonal components are not.
Physical consequence

After a measurement of \(L_z\), the value of that component is precise. Since \(L_x\) and \(L_y\) retain finite variances, the angular momentum cannot be pictured as a classical vector pointing in one sharply defined direction.

The cone is a visualization aid: the magnitude \(\sqrt{l(l+1)}\hbar\) and the height \(m\hbar\) are fixed, while any position on the cone surface is equally probable.

Exercise-ready boundary

This page supports guided exercises on: expectation values, transverse variances and the physical cone representation of angular momentum.

  • 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: expectation values, transverse variances and the physical cone representation of angular momentum.
  • Conceptual check: explain why \(\Delta L_z=0\) does not make \(\Delta L_x\) and \(\Delta L_y\) vanish.
  • Key equation 1:
    • \[\langle \hat L_x\rangle=\langle \hat L_y\rangle=0\]
  • Key equation 2:
    • \[\Delta L_x=\Delta L_y={\hbar\over\sqrt2}\left[l(l+1)-m^2\right]^{1/2}\]
  • Key equation 3:
    • \[\operatorname{var}(L_z)=0,\qquad \Delta L_z=0\]
  • Typical task: compute a variance for given \(l,m\) and interpret the cone picture.
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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.