In this lesson's particle model, a potential U(x) that does not depend on time permits stationary energy eigenstates to be found with the time-independent equation.

In -(ħ²/2m)ψ'' + Uψ = Eψ, what does -(ħ²/2m)ψ'' represent?

• The potential-energy operator acting on ψ.
• The total-energy eigenvalue multiplied by ψ.
• The kinetic-energy operator acting on ψ.

An infinite well has walls at x = 0 and x = L. What boundary values must ψ satisfy?

• ψ must have equal nonzero values at both walls.
• ψ must have zero values at both walls.
• ψ must have opposite nonzero values at the walls.

In constant U, ψ = A sin(kx) has kinetic eigenvalue T = ħ²k²/(2m). For the same mass, how does T change when k doubles?

• It becomes four times its previous value.
• It becomes two times its previous value.
• It becomes one quarter of its previous value.

For ψ_n = √(2/L) sin(nπx/L) in the infinite well, how many half-waves lie between the walls when n = 2?

For a stationary energy eigenstate Ψ(x,t) = ψ(x) exp(-iEt/ħ), what stays unchanged with time?

• Its position probability density |Ψ(x,t)|².
• Its full complex wavefunction Ψ(x,t).
• Its position measurement result in every trial.

An infinite well has U = 0 for 0 < x < L. Each candidate is normalised on that interval. Which satisfies both the stationary equation for one constant E and the wall conditions?

• ψ(x) = √(2/L) cos(πx/L)
• ψ(x) = √(2/L) sin(πx/L)
• ψ(x) = √(30/L)(x/L)(1-x/L)

In the family ψ_n = √(2/L) sin(nπx/L), n = 0 gives the infinite well's ground state.