Quantum Mechanics 101 · Unit 3: The Mathematical Machinery

Quantum Academy · Chapter 10: The Schrödinger Equation

The engine that moves the wavefunction through time

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Instructions. Work through these on paper before checking the answers. Use h = 6.626×10⁻³⁴ J·s, ħ = 1.055×10⁻³⁴ J·s, electron mass mₑ = 9.109×10⁻³¹ kg, 1 eV = 1.602×10⁻¹⁹ J and hc = 1239.8 eV·nm. For calculation questions, show every step and keep three significant figures. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Schrödinger equation
2.Hamiltonian
3.Operator
4.Partial derivative
5.Curvature
6.Potential energy function
7.Eigenvalue equation
8.Stationary state
9.Superposition principle
10.Group velocity
11.Dispersion
12.Boundary conditions

A.The speed of a wave packet's envelope, dω/dk; for a free particle it equals p/m.
B.Requirements at edges, walls and infinity, such as ψ → 0 or continuity, that select the physically allowed solutions.
C.The total-energy operator Ĥ, kinetic plus potential; it drives the time evolution of ψ.
D.How sharply a graph bends, measured by its second derivative; for ψ it tracks kinetic energy.
E.A definite-energy solution whose probability density and averages do not change in time.
F.The spreading of a wave packet because its component waves travel at different speeds.
G.The rate of change of a function of several variables with respect to one of them while the others are held fixed.
H.An equation like Ĥφ = Eφ, in which an operator returns the same function multiplied by a number, the eigenvalue.
I.A rule that turns one function into another, such as 'take the second derivative' or 'multiply by V(x)'.
J.V(x), the energy a particle would have at each position because of the forces on it: the landscape ψ moves through.
K.The equation of motion for the wavefunction, iħ ∂ψ/∂t = Ĥψ; it fixes how ψ changes in time.
L.Because the equation is linear, any weighted sum of solutions is also a solution.

Part B Questions

1.
In iħ ∂ψ/∂t = Ĥψ, what does the Hamiltonian operator Ĥ represent?Circle one
1. AThe particle's momentum
2. BThe total energy, kinetic plus potential
3. CThe probability of finding the particle
4. DThe measuring apparatus
2.
In the same region, wavefunction A has many tight wiggles while wavefunction B is a long, gentle wave. Which statement is correct?Circle one
1. AA has more potential energy
2. BB has more kinetic energy
3. CA has more kinetic energy
4. DThey must have equal kinetic energy
3.
The time-dependent Schrödinger equation contains a random term that makes ψ jump unpredictably.Circle one
TrueFalse
4.
If ψ₁ and ψ₂ both solve the time-dependent Schrödinger equation for the same potential, then 0.6ψ₁ + 0.8iψ₂ is also a solution.Circle one
TrueFalse
5.
A solution of definite energy whose probability density never changes in time is called a state.Fill in the blank
6.
The centre of a free wave packet moves at the velocity, which equals p/m.Fill in the blank
7.
Explain why multiplying a spatial shape φ(x) by the phase factor e^(−iEt/ħ) leaves the probability density unchanged in time.Short answer
8.
Why does requiring ψ to be normalizable lead to discrete energy levels for a particle bound in a well?Short answer
9.
Hydrogen's ground state is a stationary state. Give two reasons it is nevertheless wrong to say the electron is sitting still.Short answer
10.
An electron wave packet has a central de Broglie wavelength of 0.50 nm and moves in free space. How far does its centre move in 2.0 fs?Show your working
Working
Answer:
11.
A free electron plane wave has wavenumber k = 1.0×10¹⁰ m⁻¹. Use the Schrödinger equation's result E = ħ²k²/2m to find its energy in eV and its group velocity.Show your working
Working
Answer:
12.
A hydrogen atom is placed in a superposition of its n = 2 and n = 3 levels (E₂ = −3.40 eV, E₃ = −1.51 eV). At what frequency does its probability cloud oscillate? Compare with the Balmer line from n = 3 to n = 2.Show your working
Working
Answer:
13.
An electron is localized in a Gaussian packet of width σ₀ = 1.0 nm. Find the spreading time τ = 2mσ₀²/ħ and the packet's width after 1.0 ps.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The sloshing electron. An electron is trapped in a box 1.0 nm wide whose two lowest stationary states have real spatial shapes φ₁(x) and φ₂(x) and energies E₁ = 0.376 eV and E₂ = 1.504 eV (you will derive these in Chapter 11). At t = 0 it is prepared in ψ = (φ₁ + φ₂)/√2. (a) Write ψ(x, t). (b) Show that |ψ|² = ½φ₁² + ½φ₂² + φ₁φ₂ cos(ΔE·t/ħ), where ΔE = E₂ − E₁. (c) Is this a stationary state? Describe what the probability does, given that φ₁ is symmetric about the box's centre and φ₂ is antisymmetric. (d) Find the period of the motion. (e) What photon wavelength carries the same energy difference?

Working

Memory aid

Mnemonic The Engine Rhyme. Recite it with the equation in front of you: 'Bend means kinetic, V is the land; H-hat turns the clock with a steady hand. Add two solutions and the sum still flies; packets cruise at p over m and grow in size. Fix the energy and the cloud stands still, yet the particle moves, and always will.' Line one is the two halves of the Hamiltonian, curvature for kinetic energy and the potential as landscape. Line two is the time-dependent equation, deterministic as clockwork. Line three is linearity, line four is group velocity and spreading, and the last couplet is stationary states plus the myth they kill.

In one line Schrödinger's equation deterministically evolves ψ: curvature is kinetic energy, V is the landscape, and definite-energy states keep |ψ|² frozen.