Established physics · Quantum Mechanics 201 · Lesson 8 of 15
How quantum states evolve
The Schrödinger equation governs state evolution; moving and spreading probability distributions follow from the evolving amplitudes.
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Picture it
Release a localized free-particle wave packet. Its center moves according to its average momentum, while the packet can broaden because its momentum components evolve with different phases. The prediction is a changing distribution of possible position detections across repeated preparations. It is not a movie showing one particle’s measured position at every instant.
The Hamiltonian organizes the evolution
The Hamiltonian represents the system’s energy and specifies its dynamics. For a nonrelativistic particle it includes kinetic energy and the potential-energy landscape. Given an initial state and a specified Hamiltonian, the Schrödinger equation determines the evolving state. Outcome probabilities can change because the relevant amplitudes and their relative phases change.
Closed evolution preserves probability
For a closed system, Schrödinger evolution is unitary: it preserves normalization and the inner-product relationships between states. This is not a claim that the visible shape of every wave packet stays fixed. A normalized packet can move and spread while its integrated probability remains one. The state of a subsystem interacting with an environment can have a different, nonunitary reduced description.
Stationary does not mean every amplitude freezes
An energy eigenstate of a time-independent Hamiltonian gains an overall phase. Its probability distribution for a fixed measurement stays unchanged. A superposition of different energies gains relative phases, which can change observable distributions. Stationary probability densities and changing amplitude phases are therefore compatible descriptions, rather than competing claims.
The free Gaussian is one solvable example
The demonstration follows a one-dimensional, nonrelativistic, initially unchirped Gaussian with no external potential. It uses ℏ = m = 1 and average momentum 2. The center moves; the position spread grows; the momentum spread stays constant. Bound systems and externally driven systems can behave differently, so this example is not a universal movie of every quantum state.
- Specify the initial state
- Evolve with the Hamiltonian
- Predict a later measurement
The graph displays a normalized Gaussian density, not a measured particle trajectory.
Try the example on paper
For σ₀ = 1, p̄ = 2, and ℏ = m = 1: t = 0 gives center 0 and width 1; t = 2 gives center 4 and width √2; t = 4 gives center 8 and width √5. The momentum width stays 0.5.
Make it concrete
Start with σ₀ = 1, x̄(0) = 0, and p̄ = 2 in the chosen units. At t = 2, the center is x̄ = 4 and the position standard deviation is √2 ≈ 1.414. The momentum standard deviation remains 0.5. The later spread product is larger than the initial minimum, without violating normalization or the uncertainty bound.
The precise version
iℏ ∂ψ/∂t = Hψ. For H = p²/(2m) + V(x), the position form is iℏ ∂ψ/∂t = [−ℏ²/(2m) ∂²/∂x² + V(x)]ψ. For this free Gaussian, x̄(t) = x̄(0) + p̄t/m and σx(t) = σ₀√[1 + (ℏt/(2mσ₀²))²], while σp = ℏ/(2σ₀) stays fixed.
Watch the trap
Unitary does not mean unchanging, and wave-packet spreading does not mean probability is leaking away. Also, a plot of a probability density is not a recorded trajectory of a single system.
Keep this
Dynamics evolves the amplitudes. Normalization survives while distributions can move, spread, or interfere.
Words for this lesson: Schrödinger equation · Hamiltonian · unitary evolution · wave packet · stationary state · momentum
Check your understanding
Reason through each question before opening its answer and explanation.
1. What specifies Schrödinger evolution along with an initial state?
- The number of people watching
- The final answer chosen by the detector operator
- The Hamiltonian
Show answer and explanation
The Hamiltonian
The Hamiltonian specifies the dynamics, including the relevant kinetic terms, potentials, and interactions.
2. What does unitary closed-system evolution preserve?
- Every amplitude’s complex value
- Normalization and inner products
- The exact position-density shape in every system
Show answer and explanation
Normalization and inner products
Unitarity preserves total probability and state-overlap relationships while allowing nontrivial evolution.
3. A stationary energy eigenstate gains an overall phase. Must its position probability density change?
- Yes, every phase rotation changes that density
- No; an overall phase leaves that density unchanged
- Only after someone inspects it
Show answer and explanation
No; an overall phase leaves that density unchanged
The squared magnitude removes a common overall phase.
4. In the stated free example, p̄ = 2, m = 1, x̄(0) = 0. Where is the center at t = 2?
- x̄ = 1
- x̄ = 0 because the packet is quantum
- x̄ = 4
Show answer and explanation
x̄ = 4
The center follows x̄ = p̄t/m = 4 for this free state.
5. For the free Gaussian here, which spread remains constant?
- Both are forced to remain at their initial product
- The momentum spread
- The position spread for all times
Show answer and explanation
The momentum spread
Free evolution preserves the momentum distribution, while position-momentum correlations develop and position width grows.
6. Why does a reduced subsystem description sometimes look nonunitary?
- It can exchange correlations with an environment that is not included in that description
- All closed-system probability must be destroyed
- The Schrödinger equation contains a human-awareness term
Show answer and explanation
It can exchange correlations with an environment that is not included in that description
Ignoring environmental degrees of freedom produces reduced dynamics; the larger modeled closed system can still evolve unitarily.