Quantum Mechanics 101 · Unit 4: Atoms, Spin & Identity

Quantum Academy · Chapter 17: Spin & the Stern–Gerlach Experiment

A tiny magnet that refuses to point anywhere in between

Name Date Score / 26

Instructions. Work through every question with the lesson closed, then check yourself. Useful facts: a spin-½ component along any axis is +ħ/2 or −ħ/2; |+x⟩ = (1/√2)(|↑⟩ + |↓⟩); a spin prepared up along one axis gives up along an axis at angle θ with probability cos²(θ/2); the binomial scatter of a count is √(N·p·(1 − p)); h = 6.626×10⁻³⁴ J·s, c = 2.998×10⁸ m/s, 1 eV = 1.602×10⁻¹⁹ joules, μB = 9.274×10⁻²⁴ joules per tesla, the electron g-factor is 2.0023, and protons resonate at 42.58 MHz per tesla. Show every step in the calculation questions. 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.Spin
2.Spin quantum number
3.Magnetic moment
4.Bohr magneton
5.Field gradient
6.Stern-Gerlach apparatus
7.Space quantization
8.g-factor
9.Spinor
10.Larmor precession
11.Magnetic resonance
12.Electron spin resonance

A.The number s fixing a particle's spin; its component along any axis takes the 2s + 1 values mₛħ, from −sħ to +sħ in whole steps.
B.A magnet with a strong field gradient that sorts particles into separate beams according to their spin component along its axis.
C.A two-component quantum state, such as a spin-½ state, that changes sign under a 360-degree rotation and returns to itself only after 720 degrees.
D.The ratio of a particle's magnetic moment to its angular momentum, in natural units; about 2.0023 for electron spin, 1 for orbital motion.
E.The restriction of an angular momentum's component along a chosen axis to discrete values.
F.The rate at which a magnetic field changes with position; only a gradient exerts a net force on a magnetic moment.
G.Magnetic resonance of unpaired electron spins, near 28.0 GHz per tesla; discovered by Yevgeny Zavoisky in 1944.
H.The absorption of radiation whose photon energy matches the energy gap between spin states in a magnetic field, flipping spins; the basis of NMR and MRI.
I.The wheeling of a magnetic moment's direction around a magnetic field, at a rate proportional to the field.
J.The natural unit of electron magnetism, μB = eħ/2mₑ ≈ 9.274×10⁻²⁴ joules per tesla.
K.A measure of how strongly something acts as a magnet, setting both the torque in a uniform field and the force in a non-uniform one; unit joules per tesla.
L.Intrinsic angular momentum carried by a particle itself rather than by its motion through space; for an electron, s = ½.

Part B Questions

1.
Why did Stern and Gerlach need a strongly non-uniform magnetic field rather than simply a very strong uniform one?Circle one
1. AA uniform field would demagnetize the silver atoms before they reached the plate
2. BA uniform field only twists a magnet; a net sideways push needs a field that changes with position
3. CA uniform field would deflect the atoms too far for them to land on the collector plate
4. DOnly a non-uniform field could be produced by the electromagnets available in 1922
2.
Which feature of the result shows most directly that spin cannot be ordinary orbital angular momentum?Circle one
1. AThe beam was deflected at all, so the atoms must be magnetic
2. BThe two stripes were equally bright, so up and down are equally likely
3. CThere were exactly two stripes, an even number
4. DThe size of the deflection depended on the speed of the atoms
3.
Atoms that left the up exit of a Z magnet pass a second magnet tilted 90 degrees (an X magnet). Its −x exit is blocked, and the survivors meet a third magnet, also an X magnet. What fraction of the atoms entering the third magnet leave its +x exit?Circle one
1. AOne quarter
2. BOne half
3. CThree-quarters
4. DAll of them
4.
A spin prepared up along z and measured along an axis 45 degrees away can return the result (ħ/2)cos 45°.Circle one
TrueFalse
5.
After a Z magnet and then an X magnet, an atom leaving the +x exit has no definite z-spin.Circle one
TrueFalse
6.
Rotating an electron's spin state through a full 360 degrees leaves the state exactly unchanged.Circle one
TrueFalse
7.
A particle of spin s is split by a Stern-Gerlach magnet into beams.Fill in the blank
8.
The natural unit of electron magnetism, eħ/2mₑ ≈ 9.274×10⁻²⁴ joules per tesla, is called the .Fill in the blank
9.
Explain how the Z then X then Z experiment rules out the idea that each atom carries fixed, hidden values of both its z-spin and its x-spin.Short answer
10.
In what sense was the Stern-Gerlach experiment a case of 'right experiment, wrong theory'?Short answer
11.
4,000 atoms, all up along z, enter a magnet tilted 45 degrees from z. How many leave its up exit, and what is the typical scatter? What is the average spin component along the tilted axis, as a multiple of ħ?Show your working
Working
Answer:
12.
8,000 atoms, all up along z, enter a magnet tilted 60 degrees from z; its down exit is blocked. The survivors then pass a Z magnet. How many atoms leave each exit of the Z magnet?Show your working
Working
Answer:
13.
A hospital MRI scanner has a 1.5-tesla magnet. Find the proton resonance frequency, the wavelength of radio waves at that frequency, and the photon energy in electron-volts.Show your working
Working
Answer:
14.
An electron spin resonance spectrometer works at 0.35 tesla. Find the resonance frequency of a free electron spin, and compare it with the proton resonance frequency in the same field.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The 720-degree neutron. In a neutron interferometer, one path passes through a magnetic field that makes the neutron spin precess by an angle α, while the other path is left alone. When the paths recombine, the intensity in one output is I(α) = (I₀/2)(1 + cos(α/2)). (a) Find I(α) for α = 0, 180, 360, 540 and 720 degrees. (b) What would the same formula give at 360 degrees if spin states behaved like ordinary arrows, with cos α in place of cos(α/2)? (c) Thermal neutrons at 2,200 m/s cross a field region 2.0 centimetres long. The precession angle is α = γ·B·t, where t is the time spent in the field and γ = 1.832×10⁸ radians per second per tesla. What field gives α = 720 degrees? (d) A minus sign in front of a whole state is unobservable, yet this experiment sees it. Explain why there is no contradiction.

Working