Stationary and non-stationary states

Stationary statestationary stateAn energy eigenstate. Time evolution contributes only a global phase e^{−iEt/ħ}, so every expectation value is constant in time — what is stationary is the probability distribution, not the particle.See 2.5 is one of the most misread terms in the subject. The word stationary invites the thought “at rest”. The particle is not at rest.

What does not change in a stationary state

For $\Psi_n(x,t)=\psi_n(x)\ee^{-\ii E_nt/\hbar}$:

$$|\Psi_n(x,t)|^2=\psi_n^*(x)\psi_n(x)\underbrace{\ee^{+\ii E_nt/\hbar}\ee^{-\ii E_nt/\hbar}}_{=1}=|\psi_n(x)|^2\tag{2.5.1}$$

The probability density does not depend on time. More generally, for any observable $\hat A$ with no explicit time dependence, the expectation value

$$\langle \hat A\rangle_n=\int\Psi_n^*\hat A\Psi_n\,\dd x=\int\psi_n^*\hat A\psi_n\,\dd x\tag{2.5.2}$$

is time-independent too — the phase factors appear in pairs, in $\Psi^*$ and $\Psi$, and always cancel.

Non-stationary states: the cross term does the work

As soon as two or more coefficients are non-zero, the probability density moves. The previous section computed the general two-state result:

$$|\Psi|^2=|c_1|^2|\psi_1|^2+|c_2|^2|\psi_2|^2 +2|c_1||c_2|\,\psi_1\psi_2\cos\!\left(\omega_{21}t+\delta\right)\tag{2.5.6}$$

(taking real $\psi_n$; $\delta$ is the initial phase difference between $c_1$ and $c_2$).

The first two terms are the individual densities, and they do not move. All of the time dependence sits in the cross term, oscillating at the Bohr frequency

$$\omega_{21}=\frac{E_2-E_1}{\hbar}\tag{2.5.7}$$

See for yourself

The simulation below has a stationary/superposition toggle. This is the one thing in this section you should confirm with your own hands:

◈The infinite square well

Drag the width L and the quantum number n and watch the energy, the waveform and the probability distribution move together. Units: ħ = m = 1.

Energy Eₙ (n=1)

4.935

relative to E₁(L=1)

1.00 ×

⟨x⟩ / L

0.500

Δx / L

0.180

Well width L1.00

Shrink the box and every level is pushed up together: E ∝ 1/L²

Quantum number n1

nodes = n − 1 = 0 (endpoints excluded)

State

Stationary ψₙSuperposition

Show ψShow |ψ|²

In stationary mode, change $n$ freely — the probability density (purple) is a curve that never moves, however long you wait. Switch to superposition with $c_1=c_2$ and press play: the purple curve starts sloshing. Finally drag $c_2$ to zero, leaving only $c_1$, and the sloshing stops immediately.

Why atoms do not collapse

Classical electrodynamics makes a fatal prediction: an electron orbiting a nucleus is accelerating, must therefore radiate, and would lose all its energy and spiral into the nucleus in about $10^{-11}$ seconds. Matter should not exist.

Stationary states are the answer.

∎Key formulas

• Stationary state

  $$\Psi_n(x,t)=\psi_n(x)\ee^{-\ii E_nt/\hbar}\ \Longrightarrow\ |\Psi_n|^2=|\psi_n|^2$$

  Every observable is time-independent

• Density of a two-state superposition

  $$|\Psi|^2=|c_1\psi_1|^2+|c_2\psi_2|^2+2|c_1c_2|\psi_1\psi_2\cos(\omega_{21}t+\delta)$$

  All time dependence lives in the cross term

• Characteristic evolution time

  $$\tau\sim\frac{\hbar}{\Delta E}$$

  Bigger gap, faster motion; excellent for estimates

?Self-check3 questions

1. 1.

   Which statements about a particle in a stationary state ψₙ are correct? (Select all that apply.)

   Select all that apply

   • The probability density |Ψ|² does not change with time
   • The wavefunction Ψ does not change with time
   • ⟨p⟩ vanishes for stationary states of the infinite well
   • The kinetic energy of the particle is zero

   Submit
2. 2.

   An infinite well starts in an equal superposition of ψ₁ and ψ₃. Compared with an equal superposition of ψ₁ and ψ₂, the oscillation frequency of the probability density changes by a factor of:

   • 8/3
   • 3/8
   • 2
   • 9

   Submit
3. 3.

   Classical electrodynamics predicts that an orbiting electron radiates and collapses. How does quantum mechanics avoid this?

   • The ground-state charge distribution is time-independent, and a static distribution does not radiate; besides, there is no lower level to fall to
   • The electron is not actually charged in quantum mechanics
   • The electron moves too fast to radiate
   • The radiation is absorbed by the nucleus

   Submit

What comes next

We have been talking about probability distributions without ever extracting numbers from them systematically. The next section defines expectation values and variances, and writes down the uncertainty principle rigorously for the first time.