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Module 09

Time evolution and energy eigenstates

Settle one thing: eigenstates do nothing, superpositions move, and the rate at which they move is the level spacing.

What you will see

  • A single eigenstate: the tube turns, the probability density does not budge
  • Two levels superposed: the density sloshes with period exactly 2πħ/ΔE
  • Widen the gap and it sloshes faster; keep one level and everything freezes

Assumed background

  • Stationary states and energy levels (module 03)
  • Complex phases and beats

“Stationary states do not change” and “quantum systems evolve” are usually stated in the same breath, which sounds contradictory to a beginner.

Both are right; they simply have different subjects.

Superposition: why eigenstates stand still and superpositions slosh

The third dimension here is time. The present |ψ|² is at the front and its history stacks up behind it into a space-time surface: eigenstates carve straight grooves, superpositions ripple.

Loading 3D scene…

Potential well
Superposition coefficients cₙ
1.00
1.00
0.00
0.00

With a single non-zero coefficient nothing moves; two or more and it starts to slosh.

Evolution
t = 0.00
Speed
⟨x⟩ versus time

This curve is the swing of the atom as an antenna. Leave only one level and it flatlines — eigenstates do not radiate, which is why atoms are stable.

∣Ψ(t)⟩=0.71 ∣0⟩+0.71 ∣1⟩(each term also carries e−iEnt/ℏ)\ket{\Psi(t)} = 0.71\,\ket{0} + 0.71\,\ket{1}\quad(\text{each term also carries } \ee^{-\ii E_n t/\hbar})
⟨x⟩ = 0.000⟨E⟩ = 0.771Beat period T = 2πħ/ΔE = 6.79
  • ψ (complex tube)
  • current |ψ|²
  • history of |ψ|² (time recedes)

What to look for

  • Start with only c₀ (press "Ground state"): the tube turns but |ψ|² does not move at all, and the history surface is a set of straight grooves. That is what "stationary" means.
  • Press "0+1": the density starts swinging left and right and the history surface turns into slanted ripples. The period is set by the level spacing — you can check the T = 2πħ/ΔE in the readout with a stopwatch.
  • Press "0+2": the frequency changes (ΔE changed) and the motion becomes a breathing rather than a swing — parity decides how it moves.
  • Switch to the oscillator and press "Coherent state": the packet keeps its shape and swings like a classical ball. That is a privilege of evenly spaced levels; in a square well it falls apart.
  • Switch to the history view and look down on the waterfall: the whole evolution is flattened into one still image, and the period is easier to read there than in the animation.

✕ "The energy of a superposition is the average of the levels"

The expectation value is indeed a weighted average, but any single measurement returns some Eₙ. The average itself is never a measurement outcome.

✕ "An eigenstate does not change with time"

The state vector changes constantly — the phase turns. What stays fixed is every observable. The difference is that a global phase is unobservable.

✕ "The sloshing frequency is E/ħ"

It is the level *difference* over ħ. The zero of energy can be shifted at will without affecting any observation, which is exactly why only differences carry physical meaning.

Think it through

  1. Set c₀ = 0.99 and c₁ = 0.14. Does the frequency change? Does the amplitude? Guess first, then try.
  2. In the infinite well E ∝ n², so 0+1 and 2+3 slosh at different rates. Which is faster?
  3. If the two superposed states have the same parity (say 0 and 2), does ⟨x⟩ oscillate? Look at the curve, then think about the parity of the matrix element ⟨0|x|2⟩.

The general solution is a weighted sum

Any initial state can be expanded in energy eigenstates:

∣Ψ(0)⟩=∑ncn∣n⟩,⟹∣Ψ(t)⟩=∑ncn e−iEnt/ℏ∣n⟩.\ket{\Psi(0)} = \sum_n c_n \ket{n}, \qquad\Longrightarrow\qquad \ket{\Psi(t)} = \sum_n c_n\,\ee^{-\ii E_n t/\hbar}\ket{n}.

In this basis time evolution is trivial: every coefficient just rotates in phase, its modulus untouched. That is why finding energy eigenstates is worth so much effort — once you have them, evolving any initial state is a matter of multiplying term by term.

The probability density

∣Ψ(x,t)∣2=∑n,mcn∗cm ψn∗ψm e−i(Em−En)t/ℏ|\Psi(x,t)|^2 = \sum_{n,m} c_n^* c_m\,\psi_n^*\psi_m\,\ee^{-\ii(E_m-E_n)t/\hbar}

therefore has time-independent diagonal terms, and only the cross terms oscillate, at

ωnm=Em−Enℏ.\omega_{nm} = \frac{E_m - E_n}{\hbar}.

Three statements worth separating

ClaimVerdict
“An eigenstate does not change in time”Sloppy. The state vector changes (the phase turns); the observables do not.
“The global phase of ∣Ψ⟩\ket{\Psi} is meaningless”Correct. But the relative phase between terms does mean something, and it runs.
“A superposition has the average of the two energies”Dangerous. The expectation value is $\sum

Beats: the difference of two frequencies

Superposing two eigenstates makes the density oscillate at the difference frequency. It is exactly the mathematics of two tuning forks beating.

T=2πℏ∣E2−E1∣.T = \frac{2\pi\hbar}{|E_2-E_1|}.

You can check this directly in the scene: change which levels are superposed, or change the well parameters to move the gap, and time the period against the formula. The larger the gap, the faster it sloshes.

Coherent states: the most classical quantum statesoptional

The oscillator has a special family of superpositions:

∣α⟩=e−∣α∣2/2∑nαnn!∣n⟩.\ket{\alpha} = \ee^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}\ket{n}.

Because the oscillator’s levels are evenly spaced (En=ℏω(n+12)E_n = \hbar\omega(n+\tfrac12)), every cross term oscillates at an integer multiple of ω\omega, and the superposition does not disperse: it keeps its Gaussian shape and swings back and forth like a classical ball, saturating the uncertainty relation at ΔxΔp=ℏ/2\Delta x\Delta p=\hbar/2 throughout.

This is the quantum state of a laser field and the cleanest example of a classical limit. The “Coherent state” preset in the scene loads it — note that it is the only superposition whose shape is preserved. Do the same in a square well and the packet falls apart, then partially reassembles much later (a quantum revival).

Key formulas

  • General solution

    ∣Ψ(t)⟩=∑ncne−iEnt/ℏ∣n⟩\ket{\Psi(t)}=\sum_n c_n\ee^{-\ii E_n t/\hbar}\ket{n}

    In the eigenbasis, evolution is phase rotation term by term

  • Bohr frequency

    ωnm=(Em−En)/ℏ\omega_{nm}=(E_m-E_n)/\hbar

    Only differences are observable

  • Beat period

    T=2πℏ/∣E2−E1∣T = 2\pi\hbar/|E_2-E_1|

    Time it with a stopwatch in the scene

  • Mean energy

    ⟨E⟩=∑n∣cn∣2En\langle E\rangle = \sum_n |c_n|^2 E_n

    But a single measurement returns some Eₙ

Think it through

  1. Set c1=0.99c_1 = 0.99 and c2=0.14c_2 = 0.14. Does the frequency of the sloshing change? Does the amplitude? Guess first.
  2. In the infinite well En∝n2E_n\propto n^2, so the gaps are not constant. Which sloshes faster, 1+2 or 3+4?
  3. An atom in an energy eigenstate does not radiate. Why, then, does a real excited atom decay spontaneously? (Hint: quantise the electromagnetic field too; the stationary state belongs to atom plus field.)

Go deeper · matching textbook sections

The 3D scenes build the picture; the full derivations and exercises live in the textbook.

Having finished this module