Quantum Mathematics Learning Guide 4: Unitary Evolution, Schrödinger’s Equation and Quantum Dynamics

Quantum dynamics connects a state at one time to the state at another time. Mathematically, that connection is built from a Hermitian Hamiltonian and a unitary time-evolution operator.

The first three guides built the language: complex amplitudes, vectors, inner products, operators, eigenvalues and tensor products. This final guide turns that static language into motion. We will see how the Schrödinger equation acts as the differential equation of quantum evolution, why Hermitian generators produce unitary motion, how energy eigenstates evolve and how a two-level system can be solved with matrices.

State now + Hamiltonian + elapsed time → state later.

1. The time-dependent Schrödinger equation

In standard non-relativistic quantum mechanics, the time-dependent Schrödinger equation is

iℏ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩.

Here i is the imaginary unit, ℏ is the reduced Planck constant, |ψ(t)⟩ is the time-dependent state vector and H is the Hamiltonian operator. The Hamiltonian represents the total energy observable of the system and acts as the generator of time evolution.

MIT OpenCourseWare’s Quantum Physics I develops exactly this chain: linearity, complex numbers, operators and the Schrödinger equation. See MIT OpenCourseWare: Quantum Physics I lecture notes.

2. Why this is a differential equation

A differential equation relates a quantity to its rate of change. In school calculus, dy/dt = ky describes exponential growth or decay. Quantum mechanics uses a vector-valued differential equation: the rate of change of the state vector is determined by the Hamiltonian acting on that state.

Rearranging gives

∂|ψ(t)⟩/∂t = -(i/ℏ)H|ψ(t)⟩.

The factor -i is crucial. It leads to oscillatory, norm-preserving evolution rather than ordinary real exponential growth or decay for a closed system with Hermitian H.

3. Solving the equation when H is time-independent

If the Hamiltonian does not depend explicitly on time, the formal solution is

|ψ(t)⟩ = U(t)|ψ(0)⟩

with

U(t)=exp(-iHt/ℏ).

This matrix exponential is the quantum analogue of the exponential solution to a linear differential equation. Because H is Hermitian, -iHt/ℏ is anti-Hermitian, and its exponential is unitary.

4. What a matrix exponential means

For a matrix A, the exponential is defined by the power series

e^A = I + A + A²/2! + A³/3! + ...

The same definition used for the scalar exponential extends to matrices. If A is diagonal, exponentiation is especially easy: exponentiate each diagonal entry.

Worked example: diagonal Hamiltonian

Suppose

H = [[E₀,0],[0,E₁]].

Then

U(t)=diag(e^{-iE₀t/ℏ}, e^{-iE₁t/ℏ}).

If the initial state is α|0⟩+β|1⟩, then

|ψ(t)⟩ = αe^{-iE₀t/ℏ}|0⟩ + βe^{-iE₁t/ℏ}|1⟩.

The magnitudes of the coefficients stay fixed while their phases rotate at energy-dependent rates.

5. Energy eigenstates evolve by phase

If |E⟩ is an eigenstate of H with eigenvalue E, then H|E⟩=E|E⟩. The time evolution is

|E,t⟩ = e^{-iEt/ℏ}|E⟩.

The state vector acquires a phase. If it is the only energy component present, that phase is global and does not change ordinary measurement probabilities. This is why energy eigenstates are often called stationary states: their probability distributions for time-independent observables compatible with the Hamiltonian can remain unchanged even though the state vector carries a time-dependent phase.

6. Superpositions of energies produce relative phase evolution

Now suppose

|ψ(0)⟩ = a|E₀⟩ + b|E₁⟩.

At time t,

|ψ(t)⟩ = ae^{-iE₀t/ℏ}|E₀⟩ + be^{-iE₁t/ℏ}|E₁⟩.

Factor out the first phase:

|ψ(t)⟩ = e^{-iE₀t/ℏ}[a|E₀⟩ + be^{-i(E₁-E₀)t/ℏ}|E₁⟩].

The leading factor is global phase. The physically relevant changing part is the relative phase governed by the energy difference ΔE=E₁-E₀. This is one reason energy gaps, not only absolute energy labels, matter in quantum dynamics.

7. Unitarity preserves total probability

If |ψ(t)⟩=U(t)|ψ(0)⟩ and U is unitary, then

⟨ψ(t)|ψ(t)⟩ = ⟨ψ(0)|U†U|ψ(0)⟩ = ⟨ψ(0)|ψ(0)⟩.

So a normalised state remains normalised. This is the mathematical conservation law underneath closed-system probability preservation.

IBM Quantum Learning describes quantum operations using unitary matrices precisely because they preserve the Euclidean norm of state vectors. See IBM Quantum Learning: Unitary Operations.

8. The Hamiltonian as generator of motion

For a small time interval Δt, expand the exponential:

U(Δt) ≈ I - iHΔt/ℏ.

This first-order approximation shows how H determines the immediate direction of motion in state space. The Hamiltonian is therefore not merely “the energy matrix”; it is the generator that tells the state how to rotate through Hilbert space as time advances.

9. A two-level Hamiltonian built from Pauli matrices

Any 2×2 Hermitian matrix can be expressed as a real linear combination of the identity and the Pauli matrices. A simple example is

H=(ℏω/2)Z.

Since Z has eigenvalues +1 and -1, the energy eigenvalues are +ℏω/2 and -ℏω/2. The evolution operator becomes

U(t)=exp(-iωtZ/2).

Because Z²=I, the exponential can be simplified:

U(t)=cos(ωt/2)I - i sin(ωt/2)Z.

This identity is a matrix version of Euler’s formula and appears repeatedly in qubit dynamics.

Worked example: evolving |+⟩ under Z

Start with |+⟩=(|0⟩+|1⟩)/√2. Under H=(ℏω/2)Z,

|ψ(t)⟩ = (e^{-iωt/2}|0⟩ + e^{+iωt/2}|1⟩)/√2.

Factor out e^{-iωt/2}:

|ψ(t)⟩ ∼ (|0⟩ + e^{iωt}|1⟩)/√2,

where the symbol ∼ here means “equivalent up to global phase.” The state’s relative phase rotates continuously.

10. Another two-level example: H proportional to X

Take

H=(ℏΩ/2)X.

Then

U(t)=cos(Ωt/2)I - i sin(Ωt/2)X.

Apply this to |0⟩:

|ψ(t)⟩ = cos(Ωt/2)|0⟩ - i sin(Ωt/2)|1⟩.

The probabilities are

• P(0)=cos²(Ωt/2)
• P(1)=sin²(Ωt/2)

The state oscillates between the two computational basis states. This is the mathematical skeleton behind familiar two-level oscillation phenomena.

11. Time-independent Schrödinger equation

When solving many physical systems, one first solves the eigenvalue equation

H|E⟩ = E|E⟩.

In wave mechanics this becomes the time-independent Schrödinger equation. Solving it finds allowed energy eigenvalues and eigenfunctions for the chosen Hamiltonian and boundary conditions.

Once those energy eigenstates are known, a general initial state can be expanded in that basis and each component evolved by its phase factor e^{-iEt/ℏ}.

Find the energy eigenbasis → expand the initial state → evolve each component → recombine.

12. Wavefunctions are state vectors in a position representation

In introductory wave mechanics, students often meet a complex-valued wavefunction ψ(x,t) before hearing much about abstract vectors. The two pictures fit together: ψ(x,t) is the position-space representation of the state |ψ(t)⟩.

The quantity |ψ(x,t)|² is a probability density in position. Normalisation becomes

∫ |ψ(x,t)|² dx = 1.

This is the continuous analogue of summing squared amplitudes in a finite basis.

13. The Hamiltonian for a particle in one dimension

For a non-relativistic particle moving in one spatial dimension under potential V(x), a standard Hamiltonian is

H = -(ℏ²/2m)d²/dx² + V(x).

The first term represents kinetic energy through a second derivative; the second represents potential energy through multiplication by V(x). Acting on a wavefunction, the time-dependent Schrödinger equation becomes a partial differential equation.

This is one point where quantum mathematics expands beyond finite matrices. Operators can act on functions, and the relevant vector space may be infinite-dimensional. The finite-dimensional qubit model remains valuable because it isolates the same linear structure in a simpler setting.

14. Boundary conditions quantise allowed solutions

Differential equations usually need boundary conditions. In quantum mechanics, those conditions can restrict which energy eigenfunctions are physically admissible. This is one route by which discrete energy levels emerge.

For an ideal particle in a one-dimensional box, the wavefunction must satisfy boundary conditions at the walls. The allowed standing-wave solutions correspond to a discrete set of wave numbers and therefore a discrete set of energies.

The important mathematical lesson is general: quantisation often appears because a differential equation plus boundary conditions admits only particular eigenvalues.

15. Conservation of energy for a time-independent Hamiltonian

For a closed system with a time-independent Hamiltonian, the expectation value of energy is conserved. In operator language, H commutes with itself, so its expectation does not change under the evolution it generates.

More generally, an observable A with no explicit time dependence has conserved expectation when it commutes with H under the usual closed-system dynamics. This connects conservation laws to commutators.

16. Time-dependent Hamiltonians

If H=H(t), the simple formula U(t)=e^{-iHt/ℏ} generally cannot be used with one fixed H. The difficulty becomes especially important when Hamiltonians at different times do not commute.

The formal solution uses a time-ordered exponential. At an introductory level, the useful lesson is that changing the generator during evolution adds an ordering problem: the sequence of infinitesimal transformations matters.

17. Continuous-time evolution and quantum circuits

Quantum computing often presents a sequence of discrete gates: U₁, U₂, U₃. Quantum physics often presents continuous time evolution generated by H. These are compatible viewpoints.

A gate is a unitary transformation. A physical device implements that gate through some controlled Hamiltonian acting for some interval. In mathematical modelling, we can sometimes ignore the lower-level dynamics and treat the gate as the unitary object itself.

This is a useful example of abstraction: different layers can be correct at the same time, provided we remember which layer owns which question.

18. Approximation and numerical methods

Realistic Hamiltonians are often too complicated to solve exactly. Quantum mathematics therefore relies heavily on approximation and numerical computation: matrix diagonalisation, basis truncation, finite differences, perturbation theory, variational methods and numerical integration.

But approximation introduces a new responsibility: the learner must track what was approximated and how to test whether the approximation is adequate.

• Does the state remain approximately normalised?
• Does refining the discretisation change the answer materially?
• Does increasing the basis size stabilise the prediction?
• Are conserved quantities drifting numerically?
• Is the time step small enough for the chosen method?

These are not merely computational questions. They are examples of mathematical verification.

19. Common misconception: Schrödinger’s equation directly predicts one measurement outcome

The equation evolves the quantum state. It does not normally specify a single deterministic measurement result. The state then determines probabilities for measurement outcomes through the measurement rules.

This separation is essential:

Unitary dynamics evolves amplitudes deterministically; measurement outcomes are sampled according to probabilities derived from the state.

20. Common misconception: probability changes because total probability leaks away

For a closed system under unitary evolution, total probability remains one. Individual basis probabilities can change because amplitude moves and interferes among components, while the state’s total norm is preserved.

21. Common misconception: the Hamiltonian is always diagonal

No. A Hamiltonian can be diagonal in its own energy eigenbasis but non-diagonal in another basis. Choosing a basis that diagonalises H can simplify evolution, but the physical state and operator exist independently of one coordinate representation.

22. Worked synthesis problem

Let

H=(ℏΩ/2)X and |ψ(0)⟩=|0⟩.

Step 1: Use X²=I. Then

e^{-iΩtX/2}=cos(Ωt/2)I-i sin(Ωt/2)X.

Step 2: Apply to |0⟩.

|ψ(t)⟩=cos(Ωt/2)|0⟩-i sin(Ωt/2)|1⟩.

Step 3: Check normalisation.

cos²(Ωt/2)+sin²(Ωt/2)=1.

Step 4: Measurement probabilities.

P(0)=cos²(Ωt/2) and P(1)=sin²(Ωt/2).

Step 5: Interpret. The state evolves continuously and reversibly, while computational-basis probabilities oscillate. Nothing has violated probability conservation; unitary evolution redistributes amplitude while preserving the state norm.

23. The full four-guide chain

The Quantum Mathematics series now forms one connected progression:

• Complex numbers carry magnitude and phase.
• State vectors organise amplitudes in a complex vector space.
• Inner products compare states and produce amplitudes in chosen bases.
• Hermitian operators represent observables.
• Eigenvectors and eigenvalues organise definite-value states and possible outcomes.
• Unitary operators preserve norm and describe reversible closed-system transformations.
• Tensor products construct composite systems.
• Entanglement appears when the joint state cannot factor into subsystem states.
• The Hamiltonian generates continuous time evolution.
• Schrödinger’s equation tells the quantum state how to change with time.

The series is therefore not four isolated topics. It is one mathematical machine.

Represent → normalise → transform → decompose → measure → compose systems → evolve → verify.

24. Practice set

1. State the time-dependent Schrödinger equation in ket notation.
2. For time-independent H, write the formal time-evolution operator.
3. Explain why a Hermitian Hamiltonian generates unitary evolution.
4. If H|E⟩=E|E⟩, how does |E⟩ evolve in time?
5. For H=diag(E₀,E₁), write U(t).
6. Why can a superposition of two energy eigenstates show changing interference even when each amplitude magnitude is constant in the energy basis?
7. For H=(ℏΩ/2)X and initial state |0⟩, find P(1,t).
8. What is the continuous analogue of the finite-dimensional normalisation condition?
9. Give one reason numerical quantum dynamics needs verification tests.
10. Explain the difference between deterministic state evolution and probabilistic measurement outcomes.

Answers

1. iℏ ∂|ψ⟩/∂t=H|ψ⟩.
2. U(t)=e^{-iHt/ℏ}.
3. Because -iH is anti-Hermitian, so its exponential is unitary.
4. e^{-iEt/ℏ}|E⟩.
5. diag(e^{-iE₀t/ℏ},e^{-iE₁t/ℏ}).
6. The relative phase evolves according to the energy difference.
7. sin²(Ωt/2).
8. ∫|ψ(x,t)|²dx=1.
9. Discretisation, truncation and finite time steps can introduce error even when the exact dynamics preserves structure.
10. The Schrödinger equation evolves the state deterministically between measurements; measurement probabilities are derived from the evolved state.

25. Series navigation

• Guide 1: Complex Numbers, State Vectors and Probability Amplitudes
• Guide 2: Matrices, Operators, Eigenvalues and Measurement
• Guide 3: Tensor Products, Qubits and Entanglement
• Guide 4: Unitary Evolution, Schrödinger’s Equation and Quantum Dynamics
• Return to the BTT Mathematics Learning Hub

Educational note: this article gives a mathematical introduction to standard non-relativistic quantum dynamics. Relativistic quantum theory, quantum field theory, open-system dynamics and advanced measurement theory require additional mathematical frameworks.