Questions: Schrödinger Equation: Time-Dependent Form

Question 1

A quantum particle is prepared in an energy eigenstate with energy E. Over time, according to the TDSE, what happens to the probability distribution |ψ(r, t)|²?

• A. It spreads out over time because quantum mechanics requires all localized states to delocalize
• B. It oscillates periodically with frequency E/h, changing shape over time
• C. It remains unchanged — the time evolution is a pure phase factor e^{−iEt/ℏ} that does not affect |ψ|²
• D. It decays exponentially unless the particle is in the ground state

Answer: (C)

For an energy eigenstate, the TDSE gives ψ(r, t) = ψ(r, 0) e^{−iEt/ℏ}. The probability density is |ψ(r, t)|² = |ψ(r, 0)|² · |e^{−iEt/ℏ}|² = |ψ(r, 0)|², since |e^{iθ}| = 1 for any real θ. The phase factor rotates in the complex plane but has unit magnitude, so it cancels completely in the probability density. This is why energy eigenstates are called 'stationary states' — all measurable probabilities are time-independent, even though the wavefunction itself is changing.

Question 2

A wavepacket — a spatially localized quantum particle — is a superposition of energy eigenstates, each oscillating at its own frequency E/ℏ. What happens to the wavepacket over time?

• A. Nothing — superpositions of stationary states are themselves stationary, since each component is unchanged
• B. The wavepacket oscillates but maintains its shape, since energy eigenstates are the natural modes of the system
• C. The wavepacket spreads and can move, because components with different energies acquire different phases over time, altering their interference pattern
• D. The wavepacket immediately collapses to one of the energy eigenstates it contains

Answer: (C)

Each energy eigenstate in the superposition acquires its own phase e^{−iEt/ℏ}. Since components with different energies evolve at different rates, the relative phases between components change over time. This changes the interference pattern among the components, which determines the shape of |ψ|². The result: the wavepacket spreads and moves. The term 'stationary' applies only to individual energy eigenstates, not to superpositions. Option A is the key misconception — stationarity of each component does not make the superposition stationary.

Question 3

For a particle in an energy eigenstate with energy E, the probability distribution |ψ(r, t)|² is independent of time, even though the wavefunction ψ(r, t) itself changes.

• T. True
• F. False

Answer: True

The time evolution of an energy eigenstate is ψ(r, t) = ψ(r, 0) e^{−iEt/ℏ}. The wavefunction changes (its complex phase rotates), but |ψ(r, t)|² = |ψ(r, 0)|² · |e^{−iEt/ℏ}|² = |ψ(r, 0)|² since |e^{iθ}| = 1. All measurable probabilities and expectation values of time-independent observables are constant. The global phase rotation of the wavefunction is unobservable — no measurement can detect it.

Question 4

A quantum state that is a superposition of two energy eigenstates is itself a stationary state, since it is composed mostly of stationary-state wavefunctions.

• T. True
• F. False

Answer: False

A superposition of energy eigenstates is not stationary. Consider ψ = c₁ψ₁ e^{−iE₁t/ℏ} + c₂ψ₂ e^{−iE₂t/ℏ}. The probability density |ψ|² contains a cross-term proportional to e^{−i(E₁−E₂)t/ℏ}, which oscillates at frequency (E₁ − E₂)/h. This oscillating interference term causes the probability distribution to change in time — the state is emphatically not stationary. Stationarity of individual components does not sum to stationarity of the superposition.

Question 5

In what sense is the time-dependent Schrödinger equation iℏ ∂ψ/∂t = Ĥψ the quantum analog of Newton's second law, and why does this analogy matter for understanding quantum dynamics?

Model answer: Newton's second law F = ma is a dynamical equation: given the forces acting on a system (encoded by F) and the initial state (position and momentum), it determines how the state evolves in time. The TDSE plays exactly the same role: given the Hamiltonian Ĥ (which encodes the system's kinetic and potential energy) and the initial wavefunction ψ(r, 0), it determines ψ(r, t) for all future times. The analogy matters because it establishes the TDSE as the complete and fundamental law of quantum dynamics — not a special-case tool but the equation governing all quantum evolution, from simple two-state systems to complex many-body dynamics.

The analogy also highlights what is different: Newton's law is deterministic in phase space (position and momentum); the TDSE governs a complex-valued field whose squared magnitude gives probabilities. The 'state' in quantum mechanics is the wavefunction — far richer than a phase-space point — and its evolution is governed by a linear partial differential equation, which is why superposition and interference are possible.