Guided reading
After discussing measurement, the chapter returns to dynamics. In the Schrodinger picture, the state vector carries the time dependence. Operators are usually fixed unless the observable itself has explicit time dependence, such as a time-dependent external field.
TDSE in vector form
The time-dependent Schrodinger equation can be written directly in Hilbert-space notation:
\[i\hbar\frac{d}{dt}|\Psi(t)\rangle=\hat H|\Psi(t)\rangle.\]
This is the same dynamical postulate introduced in wave mechanics, but now the state is a vector rather than a position-space wave function. The Hamiltonian \(\hat H\) is the energy operator.
Evolution operator
The chapter writes the time-evolved state as
\[|\Psi(t)\rangle=\hat U(t)|\Psi(0)\rangle.\]
Substituting this into the TDSE gives a differential equation for the evolution operator:
\[i\hbar\frac{d\hat U(t)}{dt}=\hat H\hat U(t).\]
For a time-independent Hamiltonian, the solution is
\[\hat U(t)=e^{-i\hat Ht/\hbar}.\]
Why the evolution must be unitary
The state must remain normalized. If \(|\Psi(t)\rangle=\hat U(t)|\Psi(0)\rangle\), then
\[\langle\Psi(t)|\Psi(t)\rangle=\langle\Psi(0)|\hat U^\dagger(t)\hat U(t)|\Psi(0)\rangle.\]
For this to equal \(\langle\Psi(0)|\Psi(0)\rangle\) for every initial state, the operator must satisfy
\[\hat U^\dagger(t)\hat U(t)=\hat 1.\]
Unitary time evolution is therefore the mathematical statement that probability is conserved between measurements.
Spin example
For a spin Hamiltonian proportional to \(\hat\sigma_z\), the \(z\)-basis diagonalizes the Hamiltonian. A balanced initial state evolves as
\[|\Psi(t)\rangle=\frac{1}{\sqrt2}\left(e^{-i\alpha t/\hbar}|+\rangle+e^{i\alpha t/\hbar}|-\rangle\right).\]
An SG(Z) device would still see equal probabilities, but an SG(X) device can be sensitive to the changing relative phase.
Time-evolution postulate
The chapter states the postulate in this language: the vector representing the quantum state evolves in a way that preserves normalization, and its evolution is ruled by the time-dependent Schrodinger equation.
Exercise-ready boundary
This page is designed to support short guided exercises on: Time-dependent state vectors, the unitary evolution operator and phase evolution in the energy basis.
- 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: Time-dependent state vectors, the unitary evolution operator and phase evolution in the energy basis.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[i\hbar\frac{d}{dt}|\Psi(t)\rangle=\hat H|\Psi(t)\rangle.\]
- Equation: \[|\Psi(t)\rangle=\hat U(t)|\Psi(0)\rangle.\]
- Equation: \[i\hbar\frac{d\hat U(t)}{dt}=\hat H\hat U(t).\]
- 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 3 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. Cropped figures are shown here with source indication for educational review; consult the original book for the complete presentation.