At a glance
This sim draws the state of one spin-½ particle as an arrow on the Bloch sphere and lets you measure it along the Z, X or Y axis. With only two basis states the whole state fits in two angles, so spin is the cleanest place to study a quantum measurement.
What you'll learn
- How a spin state is written as two angles, θ and φ, or two amplitudes, α and β.
- What a measurement along an axis returns, and what it does to the state.
- How the probability bars and the sphere's colours follow from the state.
- How measurements along different axes relate to one another.
The sim's picture: a white arrow on a sphere, tilted by θ from the Z axis and turned by φ around it.
What's on screen
- White arrow: the spin state, tilted by θ from Z and turned by φ around it.
- Blue Z axis with the poles |↑⟩ and |↓⟩. Green X axis, |±x⟩. Amber Y axis, |±y⟩.
- Sphere colour and its P(n̂) legend, 0% to 100% (section 06).
- Right panel: |ψ⟩ with the P(↑) and P(↓) bars, the Bloch Coordinates, and the History.
See itTilt the spin to 60° from Z. Watch for: the white arrow leaning away from the blue Z axis, and the Bloch Coordinates line reading θ = 60.0°.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Measure Spin: Z, X, Y | Measures the spin along that axis. Pulse rings run for about 1 s, then the result is added to the History. | 3 buttons ignored while the arrow is moving | |
| Preset states | Sets the state to one end of an axis: |↑⟩, |↓⟩, |+x⟩, |−x⟩, |+y⟩ or |−y⟩. | 6 buttons | |
| α and β boxes | The amplitudes of |↑⟩ and |↓⟩, each as a real and an imaginary part. Nothing changes until you press Apply. | any numbers open at α = 1, β = 0 | |
| Apply Custom State | Divides all four numbers by √(sum of their squares), so |α|² + |β|² = 1, then swings the arrow to that state. All zeros are refused. | button | |
| Bloch Coordinates | Reads out the current θ and φ, in degrees. | θ 0° to 180° φ in degrees | |
| Wavefunction | Shows the current state as |ψ⟩, and two bars, P(↑) and P(↓), for the Z axis. | 0% to 100% read-only | |
| Measurement History | One line per measurement, newest first: its number, its axis and its result. | last 10 | |
| Drag on the sphere | Rotates the view only. The state does not change. | vertical tilt limited to ±90° | |
| ↻ Reset | Returns to |↑⟩, clears the History and stops any measurement in progress. | top bar |
Fixed: one spin-½ particle, no magnetic field and no time evolution. Only the controls above and your measurements change the state.
A state is two angles on a sphere
The two basis states |↑⟩ and |↓⟩ are the ends of the Z axis. Any state is a superposition of them, with complex amplitudes α and β:
One spin-½ state, written the way the sim's amplitude boxes hold it.
- |α|² + |β|² = 1 fixes the size, and an overall phase changes nothing observable. Two numbers remain: θ, the tilt from +Z (0° to 180°), and φ, the phase of β relative to α.
- The arrow points along ŝ = (sin θ cos φ, sin θ sin φ, cos θ).
- The X axis ends are |±x⟩ = (|↑⟩ ± |↓⟩)/√2, at θ = 90° and φ = 0° or 180°. The Y axis ends are |±y⟩ = (|↑⟩ ± i|↓⟩)/√2, at θ = 90° and φ = +90° or −90°.
- Opposite points on the sphere are orthogonal states. Only the ratio β/α matters: typing α = 1, β = 1 or α = 5, β = 5 gives the same state.
See itSet the state to |+y⟩. Watch for: the arrow lying flat on the equator along the amber Y axis, with θ = 90.0° and φ = 90.0°.
A measurement returns one of two values
Measuring along Z uses the operator Ŝz. It has two eigenvalues and no others:
The measured quantity has two eigenvalues; the classical projection is not one of them.
- A classical spin of length ħ/2 tilted by θ would have a Z component (ħ/2) cos θ, anywhere from −ħ/2 to +ħ/2. At θ = 60° that is ħ/4.
- The sim returns only +ħ/2 (↑ Spin Up in the History) or −ħ/2 (↓ Spin Down). ħ/4 never appears.
- X and Y measurements work the same way along their own axes: two results, Spin + or Spin −, which are +ħ/2 and −ħ/2 along that axis.
- The tilt is not lost. It sets how often each result turns up (section 06).
Left: the classical arrow, whose Z component varies continuously with the tilt. Right: the sim's Bloch sphere at θ = 60°, φ = 0.
See itTilt to 60°, then measure Z. Watch for: one new History line, either ↑ Spin Up or ↓ Spin Down, never anything in between.
The state lands on the result
After a measurement the sim sets the state to the eigenstate that was found. This is the projection rule of the textbook postulates:
The projection rule for a Z measurement. The sim then writes back the new θ and φ.
- After Z: θ = 0° (↑) or 180° (↓), with φ = 0°.
- After X: θ = 90° with φ = 0° (+x) or 180° (−x). After Y: θ = 90° with φ = +90° (+y) or −90° (−y). The arrow lands on the axis you measured.
- Measure the same axis again and the state is already an eigenstate, so the same result comes back every time.
- The swing takes about 0.67 s in the sim. That is animation. The sim stores one state and shows the rule, and it does not decide what physically happens in the measurement.
See itStart from |+x⟩ and measure Z twice in a row. Watch for: the two Z lines in the History carrying the same label.
Probability is the squared amplitude
The Born rule, in the three forms this sim evaluates.
- For a Z measurement, P(↑) = |α|² = cos²(θ/2) and P(↓) = |β|² = sin²(θ/2). At θ = 60°, α = √3/2 ≈ 0.87 but P(↑) = 75%: the probability is the amplitude squared.
- The bars show only the Z numbers and ignore φ: |+x⟩ and |+y⟩ both read 50 / 50.
- For any direction n̂ the chance of "+" is P(n̂) = ½(1 + n̂·ŝ). That is the sphere's colour: the state is the white arrow, and the colour maps the answer for every axis.
- A probability shows up as a frequency over many identical preparations. The average ⟨Ŝz⟩ = (ħ/2)[P(↑) − P(↓)] = (ħ/2) cos θ is ħ/4 at 60°, a value no single run returns.
θ
φ
Drag θ and φ and the sim's α and β boxes follow. A new state is ignored while the arrow is moving, so a fast drag settles a moment later.
The two Z probabilities against θ, with ⟨Ŝz⟩/(ħ/2) dashed. Dots mark θ = 0°, 60° and 90°.
See itPrepare the 60° state afresh and measure Z eight times. Watch for: mostly ↑ Spin Up lines, about 6 of 8 expected, yet every single run still gives one of the two results.
Different axes cannot both be sharp
Z, X and Y measurements each have two results, but they do not share eigenstates. The operators do not commute, and an uncertainty relation follows:
Non-commuting observables, and the uncertainty relation that follows.
- |↑⟩ has a certain Z result. An X measurement on it returns + or − with probability ½ each, and the state lands on the equator, where P(↑) = ½. The earlier certainty about Z is gone.
- The sim implements one rule: a stored state, and results drawn at random with the Born probabilities. It has no hidden value behind a result.
- Whether a pre-existing value could still exist is decided by Bell-type experiments on entangled pairs. This one-spin sim cannot address that.
See itStart at |↑⟩ and measure X. Watch for: P(↑) falling from 100% to 50% once the X result arrives.
Where it matters
The Stern–Gerlach experiment: spin-½ atoms arrive as two spots, not a band.
- Stern–Gerlach, 1922. Silver atoms crossed a magnetic field that varies along Z. A classical spread of tilts predicts a smear; the plate showed two spots. A Z measurement in this sim is that experiment.
- NMR and MRI. Proton spin states split in a magnetic field, and the transition frequency is f = (γ/2π)B with γ/2π = 42.58 MHz/T, about 128 MHz at 3 T. This sim has no field or time evolution.
- Qubits. |ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩ is the state of section 03 with new labels, and reading out a qubit is a Z measurement.
Go deeper: any measurement axis
- Measuring along a unit vector n̂ uses Ŝn = n̂·Ŝ. Its results are ±ħ/2, with probabilities P(±) = ½(1 ± n̂·ŝ).
- The sim's three buttons are n̂ = ẑ, x̂ and ŷ. The sphere colour is P(+) for every n̂ on the sphere.
Summary
In one line. A spin state is a direction on the sphere. A measurement along n̂ returns +ħ/2 or −ħ/2 with probabilities ½(1 ± n̂·ŝ), and leaves the state on the result.
Key points
- A state is θ and φ, or α and β with |α|² + |β|² = 1.
- A Z measurement returns +ħ/2 or −ħ/2, with P(↑) = cos²(θ/2).
- The same axis again repeats the result; another axis re-randomises it.
Common mix-ups
- "Tilt it 60° and a Z measurement reads ħ/4." Every reading is +ħ/2 or −ħ/2. ħ/4 is only the average.
- "The amplitude is the probability." √3/2 ≈ 0.87 is the amplitude. The probability is its square, 0.75.
- "The coloured sphere shows where the spin is." The state is the white arrow. The colour is P(n̂) for each axis.
Things to try on your own
- Apply α = 1, β = 1, then α = 1, β = i. Compare the Z bars, then measure X on each.
- Use the sliders in section 06 to find the θ at which P(↑) = 25%, then confirm it with Z measurements.
- From |↑⟩ press X, then Z. Repeat a few times and count the ↑ results.