The Time-Evolution Operator and Energy-Basis Dynamics
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You should be comfortable with the time-dependent Schrödinger equation, the concept of Hermitian operators, and the expansion of a state vector in an orthonormal basis of energy eigenstates.
Why this matters
The time-evolution operator provides the definitive link between the static structure of a quantum system and its dynamic behavior.
In classical mechanics, we often solve for trajectories by integrating equations of motion. In quantum mechanics, the state of the system is a vector in Hilbert space, and its evolution is governed by a unitary transformation. Understanding this operator allows us to predict how probabilities change over time without solving differential equations for every observable, providing a powerful algebraic framework for dynamics.
The Core Idea: Rotation in State Space
Imagine the state of a quantum system as a vector pointing in a specific direction within an abstract space. The Hamiltonian, which represents the total energy of the system, acts as the generator of time translations. If the Hamiltonian does not change with time, the evolution of the state vector is analogous to a rotation in this complex vector space. The "speed" of this rotation is determined by the energy of the state. Because the evolution is unitary, the length of the state vector (and thus the total probability) remains constant. This perspective shifts the focus from solving differential equations to applying a single, well-defined operator that maps the initial state to the final state.
Definition of the Time-Evolution Operator
For a time-independent Hamiltonian \(\hat{H}\), the time-evolution operator \(\hat{U}(t, t_0)\) is defined as the exponential of the Hamiltonian scaled by time and the reduced Planck constant. This operator is unitary, meaning its inverse is its Hermitian conjugate, which ensures that probability is conserved during the evolution.
The unitarity condition \(\hat{U}^\dagger \hat{U} = \hat{I}\) guarantees that the norm of the state vector is preserved. If we set \(t_0 = 0\), the operator simplifies to \(\hat{U}(t) = e^{-i\hat{H}t/\hbar}\). This form highlights that the Hamiltonian is the generator of time translations, a fundamental symmetry in physics.
Dynamics in the Energy Basis
To see how this operator acts on a general state, we expand the state in terms of the energy eigenstates \(|E_n\rangle\), which satisfy \(\hat{H}|E_n\rangle = E_n|E_n\rangle\). Since the energy eigenstates are eigenvectors of the Hamiltonian, they are also eigenvectors of the time-evolution operator. The eigenvalue associated with each state is a pure phase factor.
Consequently, if the initial state is a superposition \(|\psi(t_0)\rangle = \sum_n c_n |E_n\rangle\), the state at a later time \(t\) is obtained by multiplying each coefficient by its corresponding phase factor:
Notice that the coefficients \(c_n\) themselves do not change in magnitude. Only the relative phases between the components of the superposition evolve. This is the key to understanding quantum interference and oscillations in observables.
Expectation Values and Stationary States
The physical consequences of this phase evolution are most visible in the expectation values of observables. For an energy eigenstate (a stationary state), the probability density \(|\psi_n(x,t)|^2\) is time-independent because the phase factor cancels out in the modulus squared. However, for a superposition of states with different energies, the relative phases change with time, leading to time-dependent probability densities and expectation values.
Consider an observable \(\hat{A}\). Its expectation value is \(\langle \psi(t) | \hat{A} | \psi(t) \rangle\). Substituting the evolved state, we find that the time dependence arises from terms involving \(e^{-i(E_m - E_n)(t-t_0)/\hbar}\). If the energy difference \(E_m - E_n\) is non-zero, the expectation value oscillates with a frequency \(\omega_{mn} = (E_m - E_n)/\hbar\). This is the quantum mechanical origin of atomic transition frequencies and other periodic phenomena.
The derivation
The Time-Evolution Operator and Energy Basis Dynamics
For a system whose Hamiltonian Ĥ is Hermitian and time-independent, the state at time t is obtained from the state at t0 by a unitary operator Û(t, t0) = exp[−i Ĥ (t − t0) / ħ], so that if |ψ(t0)⟩ = Σn cn |En⟩ then |ψ(t)⟩ = Σn cn e−iEnt/ħ |En⟩: each energy-eigenstate amplitude merely accumulates a phase at rate En/ħ.
A common misconception is that the time-evolution operator \(\hat{U}(t)\) is the same as the time-dependent state \(|\psi(t)\rangle\). The operator is a map that acts on states; it is not a state itself. Furthermore, while \(\hat{U}(t)\) is unitary for a time-independent Hamiltonian, if the Hamiltonian is time-dependent, \(\hat{U}(t)\) is still unitary but is defined by a time-ordered exponential, not a simple exponential. Confusing the operator with the state, or assuming simple exponentiation for time-dependent Hamiltonians, leads to incorrect predictions of interference effects. C
Worked example
Problem. A spin-1/2 particle is in a state \(|\psi(0)\rangle = \frac{1}{\sqrt{2}} (|+\rangle + |-\rangle)\), where \(|+\rangle\) and \(|-\rangle\) are eigenstates of \(\hat{S}_z\) with eigenvalues \(+\hbar/2\) and \(-\hbar/2\) respectively. The Hamiltonian is \(\hat{H} = \omega \hat{S}_z\). Find the state \(|\psi(t)\rangle\) and the expectation value \(\langle S_x \rangle(t)\).
- Identify the energy eigenstates. Here, \(|+\rangle\) and \(|-\rangle\) are the energy eigenstates with energies \(E_+ = \hbar\omega/2\) and \(E_- = -\hbar\omega/2\).
- Apply the time-evolution operator to each basis state. \(|+\rangle(t) = e^{-i(\hbar\omega/2)t/\hbar}|+\rangle = e^{-i\omega t/2}|+\rangle\) and \(|-\rangle(t) = e^{-i(-\hbar\omega/2)t/\hbar}|-\rangle = e^{i\omega t/2}|-\rangle\).
- Construct the evolved state: \(|\psi(t)\rangle = \frac{1}{\sqrt{2}} (e^{-i\omega t/2}|+\rangle + e^{i\omega t/2}|-\rangle)\).
- Calculate \(\langle S_x \rangle(t)\). We know \(\hat{S}_x = \frac{\hbar}{2} (\sigma_x)\). In the \(|+\rangle, |-\rangle\) basis, \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\). The state vector is \(\frac{1}{\sqrt{2}} \begin{pmatrix} e^{-i\omega t/2} \\ e^{i\omega t/2} \end{pmatrix}\). The expectation value is \(\langle \psi(t) | \hat{S}_x | \psi(t) \rangle = \frac{\hbar}{2} \left( \frac{1}{\sqrt{2}} e^{i\omega t/2}, \frac{1}{\sqrt{2}} e^{-i\omega t/2} \right) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} e^{-i\omega t/2} \\ e^{i\omega t/2} \end{pmatrix}\).
- Compute the matrix multiplication: The middle term is \(\frac{1}{\sqrt{2}} \begin{pmatrix} e^{i\omega t/2} \\ e^{-i\omega t/2} \end{pmatrix}\). The dot product is \(\frac{1}{2} (e^{i\omega t/2} e^{i\omega t/2} + e^{-i\omega t/2} e^{-i\omega t/2}) = \frac{1}{2} (e^{i\omega t} + e^{-i\omega t}) = \cos(\omega t)\).
- Multiply by \(\hbar/2\) to get the final result.
Units check. \( \hbar \) is in \( \mathrm{J\,s} \), the unit of angular momentum, and \( \cos(\omega t) \) is dimensionless, so \( \langle S_x \rangle \) is an angular momentum. Also \( \hat{H} = \omega \hat{S}_z \) is \( \mathrm{s^{-1}} \times \mathrm{J\,s} = \mathrm{J} \), an energy. The spin precesses about \( z \) at angular frequency \( \omega \) (Larmor precession).
Three quick tests
- If a system is in an energy eigenstate, does the probability density change with time?
Answer
No, the probability density is time-independent for stationary states. - What is the condition for the time-evolution operator to be unitary?
Answer
The Hamiltonian must be Hermitian. - How does the relative phase between two energy eigenstates evolve?
Answer
It evolves as \(e^{-i(E_1 - E_2)t/\hbar}\), oscillating with frequency \((E_1 - E_2)/\hbar\).
Problem set for PU-202 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.