Guided reading
For the quantum harmonic oscillator, the position moments follow from the parity of the Hermite-polynomial eigenfunctions. The momentum moments, variances, and uncertainty product then follow from the corresponding expectation values.
Expectation values for position
For the \(q\)-th position expectation value, use
\[\langle\hat x^q\rangle=\int_{-\infty}^{\infty}\psi_n^*(x)\hat x^q\psi_n(x)\,dx=\int_{-\infty}^{\infty}x^q\psi_n^2(x)\,dx.\]
The QHO wave functions are real, and \(\psi_n^2(x)\) is even. Therefore, for odd \(q\),
\[\langle\hat x^q\rangle=0.\]
Dimensionless position moments
Introduce
\[\xi=\alpha x,\qquad \alpha=\sqrt{\frac{m\omega}{\hbar}},\qquad \psi_n(z)=\sqrt\alpha\,f(z).\]
Then define the dimensionless moments by
\[\langle q\rangle=\langle\xi^q\rangle=\int_{-\infty}^{\infty}\xi^q f^2(\xi)\,d\xi.\]
For even \(q\), they obey
\[\frac q2(q-1)\langle q-2\rangle+2(2n+1)\langle q\rangle-2\frac{q+2}{q+1}\langle q+2\rangle=0.\]
The derivation of this recurrence is not trivial; it is developed in detail in the book.
First even moments
Normalization gives \(\langle0\rangle=1\). Taking \(q=0\) in the relation above gives
\[\langle\hat x^2\rangle=\frac{\hbar}{m\omega}\left(n+\frac12\right).\]
The next even moment is obtained by taking \(q=2\):
\[\langle\hat x^4\rangle=\left(\frac{\hbar}{m\omega}\right)^2\frac38\left[1+4\left(n+\frac12\right)^2\right].\]
Expectation values for momentum
Since \(\langle\hat x\rangle=0\), the Ehrenfest theorem gives
\[\langle\hat p\rangle=0.\]
The virial theorem gives
\[\langle\hat p^2\rangle=m^2\omega^2\langle\hat x^2\rangle=m\hbar\omega\left(n+\frac12\right).\]
Variance and standard deviation
Using the expectation values above,
\[\operatorname{var}(x)=\langle\hat x^2\rangle-\langle\hat x\rangle^2=\frac{\hbar}{m\omega}\left(n+\frac12\right),\]
\[\operatorname{var}(p)=\langle\hat p^2\rangle-\langle\hat p\rangle^2=m\hbar\omega\left(n+\frac12\right).\]
The standard deviation is \(\Delta q=\sqrt{\operatorname{var}(q)}\).
Uncertainty product
The resulting product of standard deviations is
\[\Delta x\,\Delta p=\hbar\left(n+\frac12\right).\]
For the ground state, \(n=0\), this gives \(\Delta x\,\Delta p=\hbar/2\).
Role of parity
The even parity of \(\psi_n^2(x)\) makes every odd position moment vanish. The remaining even moments determine the variances, while the momentum results are obtained from the Ehrenfest and virial theorems.
Exercise-ready boundary
This page is designed to support short guided exercises on: parity of position moments, \(\langle\hat x^2\rangle\), \(\langle\hat p\rangle\), \(\langle\hat p^2\rangle\), and the uncertainty product.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Practice anchors
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: expectation values and uncertainty for a QHO stationary state.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\Delta x\,\Delta p=\hbar\left(n+\frac12\right)\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 4 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, derivations and exercises.