Quantum Mechanics 101 · Unit 3: The Mathematical Machinery

Quantum Academy · Chapter 9: The Wavefunction & Born's Rule

A wave of possibilities, and the recipe for turning it into probabilities

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Instructions. Work these on paper before checking the answers. Unless told otherwise, ψ describes one particle moving along a line. Tools: probability density |ψ(x)|²; probability of a region = area under |ψ|² across it; narrow-window shortcut P ≈ |ψ(x)|² × Δx; normalization means the total area is 1; ⟨x⟩ = area under x·|ψ|²; σ = √(⟨x²⟩ − ⟨x⟩²); a measured mean has typical error σ/√N; a count of N trials with probability p scatters by √(N·p·(1 − p)). Box states: ψₙ = √(2/L)·sin(nπx/L). 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.Wavefunction
2.Probability amplitude
3.Complex conjugate
4.Probability density
5.Born rule
6.Normalization
7.Expectation value
8.Node
9.Global phase
10.Relative phase
11.Configuration space

A.The space of all possible arrangements of a system's particles, with 3N dimensions for N particles in three-dimensional space; many-particle wavefunctions live there.
B.The function ψ that assigns a complex probability amplitude to every possible position, or for many particles to every arrangement; the complete quantum description of a state.
C.A phase factor multiplying the whole wavefunction; it changes no measurable prediction.
D.A complex number whose squared magnitude gives a probability or a probability density.
E.A complex number with the sign of its imaginary part reversed, the mirror-image arrow; a number times its conjugate is its squared magnitude.
F.The 1926 postulate that probabilities are squared magnitudes of amplitudes; for position, the probability of a region is the area under |ψ|² across it.
G.A point where ψ, and therefore |ψ|², is zero, so the particle is never found there.
H.The probability-weighted average of a quantity, written ⟨x⟩ for position: the long-run mean of many measurements on identically prepared systems.
I.The phase difference between the parts of a superposition; it controls their interference and so moves probability.
J.Probability per unit length, area or volume; for position it is |ψ(x)|², and only its areas are probabilities.
K.Scaling a wavefunction so that the total area under |ψ|² is exactly 1, because the particle must be found somewhere.

Part B Questions

1.
Which quantity is the probability density for finding a particle near position x?Circle one
1. Aψ(x) itself
2. BThe phase of ψ(x)
3. C|ψ(x)|², the squared magnitude of ψ(x)
4. DThe area under ψ(x) from minus infinity to x
2.
A normalized wavefunction ψ(x) is given. Which expression gives the probability of finding the particle between x = a and x = b?Circle one
1. A|ψ(b)|² − |ψ(a)|²
2. BThe area under |ψ(x)|² between a and b
3. CThe area under ψ(x) between a and b
4. D|ψ(a)|² multiplied by (b − a), for any a and b
3.
An electron in a box is in the state ψ = 0.6ψ₁ + 0.8ψ₂. Which change alters the measured distribution of its positions?Circle one
1. AMultiplying the whole of ψ by −1
2. BMultiplying the whole of ψ by i
3. CChanging the sign of the ψ₂ part only
4. DNone of these, since phases can never be measured
4.
A probability density |ψ(x)|² can be greater than 1 per nanometre at some point without anything being wrong.Circle one
TrueFalse
5.
The expectation value ⟨x⟩ is always a position at which the particle can actually be found.Circle one
TrueFalse
6.
If two wavefunctions have exactly the same |ψ(x)|² at one instant, they must describe the same physical state.Circle one
TrueFalse
7.
Scaling a wavefunction so that the total area under |ψ|² equals 1 is called .Fill in the blank
8.
The wavefunction of N particles in three-dimensional space is a function on a configuration space with dimensions.Fill in the blank
9.
Give two pieces of evidence that the wavefunction is not the electron itself, smeared out like jam.Short answer
10.
A student measures the positions of 10,000 identically prepared electrons and finds that their average lies within 0.001 nm of the predicted ⟨x⟩. Yet she cannot predict where the 10,001st electron will land. Explain why both facts agree with the Born rule.Short answer
11.
A particle bound to a very narrow, deep well at x = 0 has the wavefunction ψ(x) = A·exp(−|x|/a), with a = 0.20 nm. (a) Find A. (b) Find the probability density at x = 0. (c) Find the probability that the particle is found within a distance a of the well. (d) Use the narrow-window shortcut to estimate the probability of finding it within 0.01 nm of the well, and compare with the exact value 1 − exp(−0.1).Show your working
Working
Answer:
12.
An electron is in the ground state of a 1.0 nm box, ψ₁ = √(2/L)·sin(πx/L). (a) What is the probability density at the centre? (b) Use the narrow-window shortcut to estimate the probability of finding it between 0.40 and 0.60 nm. (c) The exact probability is 0.2 + sin(0.2π)/π; evaluate it and explain why the shortcut comes out high. (d) The exact probability of finding it in the first 0.2 nm of the box is 0.2 − sin(0.4π)/(2π); evaluate it.Show your working
Working
Answer:
13.
In the lab's Gaussian preset (centre 0.40 nm, σ = 0.09 nm) you bet on the region from 0.22 to 0.58 nm. (a) What is the true probability? (b) In 500 detections, what count do you expect, and how much does it scatter, in detections and in percentage points? (c) By what factor does the scatter in percentage points shrink if you fire 2,000 detections instead? (d) How many detections bring the running mean to within about 0.003 nm of ⟨x⟩?Show your working
Working
Answer:
14.
An electron in a 1.0 nm box is in the state ψ = (ψ₁ + c·ψ₂)/√2, where ψ₁ and ψ₂ are the two lowest box states and c is a complex number of size 1. Over the left half of the box, ψ₁² and ψ₂² each have area 0.5 and ψ₁ψ₂ has area 4/(3π). Find the probability of detecting the electron in the left half for c = 1, c = i and c = −1.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Weigh hydrogen's electron cloud. In three dimensions Born's rule uses volume: the probability of finding the electron in a small volume is |ψ|² times that volume, and the total over all space must be 1. Hydrogen's ground state is ψ(r) = N·exp(−r/a₀), where r is the distance from the proton and a₀ = 0.0529 nm is the Bohr radius. (a) Using the volume of a thin spherical shell, 4πr²·dr, and the fact that the area under r²·exp(−2r/a₀) from 0 to infinity is a₀³/4, show that N = 1/√(πa₀³). (b) |ψ|² is greatest at r = 0, right at the proton, yet the most probable distance from the proton is a₀. Show this and resolve the apparent contradiction. (c) The probability of finding the electron within a distance R of the proton is 1 − exp(−2R/a₀)·(1 + 2R/a₀ + 2R²/a₀²). How often is it found within one Bohr radius? (d) The expectation value of the distance is ⟨r⟩ = 1.5a₀. Evaluate it and explain why it differs from the most probable distance. (e) The proton's radius is about 0.84 fm. Estimate the probability that the electron is found inside the proton.

Working

Memory aid

Mnemonic When you meet a wavefunction, obey its own command: SQUARE IT. S, Size and phase: ψ is a complex amplitude, an arrow at every point. Q, the Quantity you bet on is |ψ|², a probability per unit length, never ψ itself. U, Unity: normalize so the total area is exactly 1. A, Area: the probability of a region is the area under |ψ|² across it. R, Random clicks, Reliable averages: single outcomes are unpredictable, while ⟨x⟩ and σ emerge exactly from many runs. E, Extra particles, Extra dimensions: N particles live in a 3N-dimensional configuration space. I, Invisible overall phase. T, Telltale relative phase, which moves probability by interference. To use it, say 'square it' whenever you see a ψ, then walk the eight letters as a checklist before you trust any probability.

In one line ψ is a complex amplitude; square it for probability density, take areas for odds: single clicks are random, the statistics exact.