Guided reading
Separation of variables is not just a trick. It isolates states with a definite energy. The spatial equation selects the energy spectrum, while the time equation supplies a phase that leaves \(|\Psi|^2\) unchanged.
When separation is allowed
If the potential depends only on position, \(V=V(x)\), the time-dependent equation can be solved by trying a product form:
\[\Psi(x,t)=\psi(x)\,\phi(t).\]
The assumption is not arbitrary decoration. It asks whether the same state can be described by a fixed spatial shape multiplied by a time-dependent phase.
Separate the variables
Substituting \(\Psi=\psi\phi\) in the TDSE and dividing by \(\psi\phi\) gives
\[-\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2\psi}{dx^2}+V(x)=i\hbar\frac{1}{\phi}\frac{d\phi}{dt}.\]
The left side depends only on \(x\), and the right side depends only on \(t\). Therefore both must be the same constant, identified as the energy \(E\).
The spatial eigenvalue equation
\[\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x).\]
\[\hat H\psi_n=E_n\psi_n.\]
This is the time-independent Schrodinger equation. It is an eigenvalue problem: boundary conditions select allowed functions \(\psi_n(x)\) and allowed energies \(E_n\).
Why stationary states are stationary
The time part satisfies \(i\hbar\,d\phi/dt=E\phi\), so
\[\phi(t)=e^{-iEt/\hbar},\qquad \Psi(x,t)=\psi(x)e^{-iEt/\hbar}.\]
The phase changes in time, but the probability density does not:
\[|\Psi(x,t)|^2=|\psi(x)|^2.\]
Exercise-ready boundary
This page is designed to support short guided exercises on: Separation of variables, time-independent potentials and the energy eigenvalue equation.
- 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: Separation of variables, time-independent potentials and the energy eigenvalue equation.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\Psi(x,t)=\psi(x)\,\phi(t).\]
- Equation: \[-\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2\psi}{dx^2}+V(x)=i\hbar\frac{1}{\phi}\frac{d\phi}{dt}.\]
- Equation: \[\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x).\]
- 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 2 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. No original book figure is reproduced on this page.