Guided reading
The Hermite differential equation has a solution for every value of \(\epsilon\). The book then imposes the physical condition that the complete wave function be integrable; this is what restricts the energy to a discrete set of values.
Hermite equation and its solutions
The equation obtained in the preceding section is
\[\frac{d^2\chi}{d\xi^2}-2\xi\frac{d\chi}{d\xi}+(2\epsilon-1)\chi=0\]
Its solutions are the Hermite functions \(\chi_n(\xi)=A_nH_n(\xi)\). The corresponding wave function must satisfy the normalization condition
\[\int_{-\infty}^{+\infty}|\psi(x)|^2\,dx=1\]
Condition for an integrable solution
For the wave function to be integrable, the parameter in the Hermite equation must obey
\[2\epsilon=2n+1,\qquad n=0,1,2,\ldots\]
Since \(\epsilon=E/(\hbar\omega)\), this mathematical condition selects the allowed energies.
Energy eigenvalues
\[E_n=\hbar\omega\left(n+\frac12\right),\qquad n=0,1,2,\ldots\]
These are the energy eigenvalues of the quantum harmonic oscillator.
Discrete stationary states
- Each nonnegative integer \(n\) labels one allowed stationary state.
- The energy difference between successive levels is \(E_{n+1}-E_n=\hbar\omega\).
- The lowest allowed energy is \(E_0=\hbar\omega/2\).
From differential equation to spectrum
The differential equation supplies the Hermite solutions. Requiring the associated wave function to be integrable imposes \(2\epsilon=2n+1\); substituting \(\epsilon=E/(\hbar\omega)\) gives the energy eigenvalues above.
Exercise-ready boundary
This page is designed to support short guided exercises on: the normalization condition for the QHO, \(2\epsilon=2n+1\), and the energy eigenvalues \(E_n\).
- 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: the normalization condition for the QHO, \(2\epsilon=2n+1\), and the energy eigenvalues \(E_n\).
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\int_{-\infty}^{+\infty}|\psi(x)|^2\,dx=1\]
- Equation: \[2\epsilon=2n+1,\qquad n=0,1,2,\ldots\]
- Equation: \[E_n=\hbar\omega\left(n+\frac12\right),\qquad n=0,1,2,\ldots\]
- 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.