Quantum Mechanics 101 · Unit 2: Wave–Particle Duality

Quantum Academy · Chapter 8: Heisenberg's Uncertainty Principle

Not clumsy measurement — a built-in property of waves

Name Date Score / 26

Instructions. Work these on paper before checking the answers. Treat every Δ as a standard deviation. Use Δx·Δp ≥ ħ/2, Δv = Δp/m, ΔE·Δt ≳ ħ/2, Γ = ħ/τ, Δf = 1/(2πτ) and E(r) ≈ ħ²/(2·m·r²) − e²/(4πε₀r). Take ħ = 1.055×10⁻³⁴ J·s = 6.582×10⁻¹⁶ eV·s, electron mass 9.109×10⁻³¹ kg, proton mass 1.673×10⁻²⁷ kg, c = 2.998×10⁸ m/s, 1 eV = 1.602×10⁻¹⁹ J, e²/(4πε₀) = 1.440 eV·nm and hc = 1240 eV·nm. Show units at every step. 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.Uncertainty principle
2.Standard deviation
3.Ensemble
4.Wave packet
5.Wavenumber
6.Fourier transform
7.Heisenberg's microscope
8.Natural line width
9.Zero-point energy
10.Squeezed state
11.Incompatible observables
12.Commutator

A.The difference ÂB̂ − B̂Â between applying two operators in opposite orders; for position and momentum it equals iħ.
B.A localized lump of wave built by adding many waves of different wavelengths, which reinforce in one region and cancel elsewhere.
C.The number of radians of phase per metre, k = 2π/λ; a particle's momentum is p = ħk.
D.A state, often of light, in which the uncertainty of one property is reduced below the zero-point level at the cost of greater uncertainty in its partner.
E.The recipe listing how much of each wavelength or frequency is needed to build a given shape; narrow shapes need broad recipes.
F.Measurable quantities, such as position and momentum along the same axis, that no state can make definite at the same time.
G.The rule that for every quantum state the spreads of position and momentum along the same axis obey Δx·Δp ≥ ħ/2.
H.A large collection of systems prepared in exactly the same way, on which repeated measurements reveal a state's distributions.
I.A 1927 thought experiment in which a gamma-ray photon locates an electron but gives it an unknown kick; an illustration of uncertainty, not its cause.
J.The root-mean-square distance of a set of results from their average; the precise meaning of each Δ in an uncertainty relation.
K.The minimum spread in energy or frequency of a spectral line, Γ = ħ/τ, set by the lifetime τ of the emitting state.
L.The minimum energy that a confined quantum system retains even at absolute zero, because it cannot have sharp position and momentum together.

Part B Questions

1.
In the relation Δx·Δp ≥ ħ/2, what does Δx stand for?Circle one
1. AThe smallest distance that the best available ruler or detector can resolve
2. BThe error bar attached to one single measurement of the particle's position
3. CThe standard deviation of position results over many identically prepared runs
4. DThe distance the particle travels during the time it is being measured
2.
Your wave packet is a Gaussian sitting exactly on the bound, Δx·Δk = 1/2. You want a packet half as wide. What must you do to the range of wavelengths you add together?Circle one
1. AReplace them all with a single, shorter wavelength
2. BAt least double the spread of wavenumbers, and so of momenta, in the mix
3. CHalve the spread of wavenumbers, and so of momenta, in the mix
4. DNothing, because packet width and wavelength spread are unrelated
3.
Which pair of quantities can both have sharp values in the same quantum state?Circle one
1. APosition along x and momentum along x
2. BSpin along x and spin along z
3. CThe two complementary noise properties of a light beam that squeezing trades
4. DPosition along x and momentum along y
4.
With sufficiently advanced instruments, a laboratory could prepare electrons whose Δx·Δp is smaller than ħ/2.Circle one
TrueFalse
5.
A Gaussian wave packet can have Δx·Δp exactly equal to ħ/2.Circle one
TrueFalse
6.
An excited state with a shorter lifetime emits a spectral line with a narrower natural width.Circle one
TrueFalse
7.
A wave with a single, perfectly definite wavelength is spread uniformly over all of .Fill in the blank
8.
Squeezing an electron into a smaller region raises its energy, which is why atoms do not collapse.Fill in the blank
9.
Using the uncertainty principle, explain why the electron in a hydrogen atom cannot spiral into the nucleus.Short answer
10.
Heisenberg's microscope blames uncertainty on the kick from a photon. Explain why physicists today regard the uncertainty principle as intrinsic rather than as measurement disturbance.Short answer
11.
An electron in a quantum dot has a position spread Δx = 1.0 nm along one direction. Find (a) its minimum momentum spread, (b) its minimum velocity spread and (c) the confinement kinetic energy (Δp)²/(2m) for that direction, in eV.Show your working
Working
Answer:
12.
Hydrogen's 2p state lives τ = 1.60 ns before emitting a Lyman-alpha photon of wavelength 121.6 nm. Find the natural line width in eV and in MHz, and the width as a fraction of the photon energy.Show your working
Working
Answer:
13.
A muon is a heavier cousin of the electron, 206.8 times as massive. Using the uncertainty-principle estimate of Worked Example 2, and treating the proton as fixed, predict the radius and ground-state energy of muonic hydrogen, a muon bound to a proton.Show your working
Working
Answer:
14.
A titanium-sapphire laser emits pulses centred at 800 nm with a duration Δt = 5.0 fs (as a standard deviation). Find the minimum energy spread of the photons in a pulse, compare it with the photon energy, and estimate the spread of wavelengths.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The nucleus is no place for an electron. Before James Chadwick discovered the neutron in 1932, many physicists pictured nuclei as protons bound together with electrons. Suppose an electron were confined in a nucleus with position spread Δx = 5.0 fm, a generous value for a medium-sized nucleus. (a) Find its minimum momentum spread, and take that as its typical momentum. (b) Show that the non-relativistic formula v = p/m would give a speed faster than light. Then use E = √((pc)² + (mc²)²), with the electron's rest energy mc² = 0.511 MeV, to find its kinetic energy. (c) Electrons emitted in nuclear beta decay typically carry from a few keV to a few MeV. What does your answer suggest? (d) Repeat the estimate for a proton confined in the same nucleus, using (Δp)²/(2m) with the proton's mass, and explain why nucleons, unlike electrons, can comfortably live there.

Working

Memory aid

Mnemonic Remember the chapter as the lab's own command: SQUEEZE. S, Standard deviations: each Δ is the spread of many identically prepared runs, not one blurry reading. Q, Quarrelling partners: only incompatible pairs trade, such as position and momentum along the same axis, because their commutator is iħ. U, Unbeatable floor: Δx·Δp ≥ ħ/2 for every state, whatever the instruments. E, Every packet is a chord: a narrow lump of wave needs many wavelengths, and only a Gaussian touches the floor. E, Energy and time: short life, broad line, Γ = ħ/τ. Z, Zero-point jitter: nothing confined is ever perfectly still. E, Electrons cannot fall in: confinement costs kinetic energy, so hydrogen settles at a₀ and −13.6 eV. To use it, picture squeezing a water balloon: pinch it thin in position and it bulges in momentum. Then walk the seven letters whenever you meet a claim about uncertainty, and ask whether it concerns spreads, a true incompatible pair, and the floor of ħ/2.

In one line Squeeze a wave in position and it spreads in momentum; that trade, Δx·Δp ≥ ħ/2, is built into waves, not instruments, and holds atoms open.