At a glance
A spin-½ sits in a magnetic field along z, and the sim asks, at every instant, how likely a measurement of Sx is to return +ħ/2. The same picture, a spin that precesses and is read out along a different axis, underlies magnetic resonance.
What you'll learn
- What a measurement of Sx can return, and what P+ is the probability of.
- How P+ is tied to the arrow on the Bloch sphere.
- How the size of the P+ oscillation depends on α, and how its rate depends on μB.
- Which quantities change in time and which stay fixed.
The state is an arrow on the sphere. The field turns it about z; Sx is measured along x.
What's on screen
- Bloch sphere (top): the state as a blue arrow with a fading trail, and the blue circle it follows. Axes: x pink, y amber, z light blue.
- |→⟩, |←⟩ (pink) on the ±x ends, |↑⟩, |↓⟩ (blue) on the poles.
- P+(t) (bottom): the blue curve over two periods, with a cursor at the current time.
- Output and state panels: the clock t, the current P+, the starting state |ψ(0)⟩ and the P+ equation.
See itRun the opening settings. Watch for: the blue arrow turning about the vertical z axis while P+ in the card keeps changing.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| α | The |↑⟩ amplitude of the starting state α|↑⟩ + β|↓⟩. It changes the preparation, not the field. | 0–1 opens at 0.707 | |
| β | The |↓⟩ amplitude. It is tied to α by β = √(1 − α²): moving either slider moves the other. | 0–1 opens at 0.707 | |
| μB | The field strength, which is the precession rate ω₀ = μB (radians per unit of the sim's time). | 0.1–10 opens at 1.0 | |
| Speed | How fast the animation clock runs. The physics and the plotted curve stay the same. | 0.1×–5× opens at 1.0× | |
| Bloch sphere | Drag it to turn the view. Only the camera changes; the state does not. | any azimuth, elevation limited | |
| Maximize (plot corners) | Expands the sphere or the P+ plot to fill the stage; press again to restore. | per plot | |
| Output | The clock t and the current P+, the value the cursor on the curve shows. | t wraps every 4π/μB | |
| Quantum state and P+ equation | The starting state |ψ(0)⟩, and the closed form the curve follows. | panel | |
| Pause / Play | Freezes the clock so you can read P+ at one instant, or lets it run. | top bar | |
| Reset | Sets the clock back to t = 0 and returns α, μB and Speed to their opening values. | top bar | |
| Hide / Open (edge tab) | Collapses the sidebar so the plots fill the window, then brings it back. | tab |
Fixed: ħ = 1, the field points along +z, and α, β are real and non-negative with α² + β² = 1.
What an Sx measurement can return
A measurement of Sx returns one of two values, +ħ/2 or −ħ/2. P+ is the probability of +ħ/2. It lies between 0 and 1 and is not a component of the spin. In the sim the field is along z:
The sim's Hamiltonian and sign convention.
The two Sx outcomes belong to the states |→⟩ and |←⟩, which the sim labels on the sphere. In the z basis:
The x basis.
P+ is the weight of |→⟩ in the state. A spin prepared exactly along z, |↑⟩, is an equal superposition of |→⟩ and |←⟩, so for it both outcomes have probability ½.
Key idea. P+ = |⟨→|ψ⟩|², the probability of one outcome. It is not the x-component of the spin.
See itPrepare the spin along +z (α = 1). Watch for: the arrow sitting on the north pole while P+ stays at one value.
How the field moves the odds
The field gives the two z-amplitudes opposite phases. Only their relative phase, μBt, matters for P+.
The starting state and its evolution.
Projecting onto |→⟩ gives the curve on screen:
The sim's P+ equation.
- The ½ is the average value. The cosine is the interference between the |↑⟩ and |↓⟩ amplitudes.
- On the sphere, the arrow turns about z at the Larmor rate μB, and P+ = ½(1 + nx), where nx is the arrow's x-coordinate.
See itStart on the equator, with the arrow along +x (α = β). Watch for: P+ reading 1 when the arrow points at |→⟩ and 0 when it points at |←⟩.
What sets the swing
P+ runs between ½ − αβ and ½ + αβ. Its swing is 2αβ, which equals sin θ, where θ is the angle of the arrow from +z.
The tilt on the left fixes the swing on the right (μB = 1).
- The swing is largest, a full 0 to 1, for α = β = 1/√2, with the arrow on the equator.
- It is zero for α = 1 or β = 1, with the arrow on a pole.
- It depends on the product αβ, so α = 0.6 and α = 0.8 give the same swing.
α
μB
These sliders move the sim too. β follows α.
See itRun one cycle each at α = 1, 0.90 and 0.71. Watch for: the card listing the lowest and highest P+ of each run.
What sets the rate
The arrow makes one turn about z in the time T = 2π/μB, and P+ repeats with the same period. Doubling μB halves T.
- The time axis of the plot is ruled in units of π/μB and always spans 4π/μB, two periods. Changing μB therefore does not change the drawn curve; only the numbers on the t readout change.
- The two sliders do different jobs: α sets how far P+ swings, and μB sets how fast it does so.
See itTime how long P+ takes to fall from 1 to ½ at μB = 1, then at μB = 2. Watch for: the card showing the two times side by side.
What stays fixed, and what is measured
The Bloch vector the sim draws, and the quantities that do not move.
- P+ + P− = 1: these are the only two outcomes.
- The z odds never change. The field only turns the arrow about z, so its height nz = α² − β² and the probability α² of +ħ/2 along z stay constant.
- ⟨Sx⟩ = (ħ/2)(P+ − P−) is the average of the two outcomes. It lies between −ħ/2 and +ħ/2.
- |↑⟩ and |↓⟩ are the energy eigenstates here. A spin prepared in one of them only gains an overall phase, so its arrow never moves.
- Nothing is measured on screen. The sim evolves one state, with no collapse and no randomness, and plots the probability a measurement would have.
See itUse α = 0.8 and watch the arrow while P+ changes. Watch for: the arrow circling at a constant height above the equator while P+ rises and falls.
Where it matters
- The sim's μB is the Larmor frequency ω₀ = γB. Change the particle and only γ changes; the rest of this sheet carries over.
- MRI. A proton in a 1.5 T scanner precesses at about 63.9 MHz, the radio frequency the scanner transmits and listens at.
- Electron spin resonance. An electron's γ is about 658 times a proton's: at 0.35 T it precesses at about 9.81 GHz, in the microwave band.
- Ramsey-type measurements. A spin prepared on the equator is left to precess for a time t and then read out along x. The signal is the curve of this sim with α = β.
Go deeper: the sign of the field
- With H = −μB Sz the Bloch vector is n = (2αβ cos μBt, −2αβ sin μBt, α² − β²): it turns from +x towards −y.
- The opposite sign turns it the other way. P+ depends only on cos μBt, so it does not change.
Summary
In one line. A spin-½ in a field along z precesses at ω₀ = μB, and the probability that Sx returns +ħ/2 is P+ = ½(1 + 2αβ cos μBt).
Key points
- Sx returns only ±ħ/2. P+ is the probability of +ħ/2.
- P+ = ½(1 + nx): it follows the arrow's x-coordinate.
- The swing 2αβ is set by the preparation; the period 2π/μB by the field.
- The z odds are constant, and nothing is measured on screen.
Common mix-ups
- "The spin points along z, so P+ = 0." Sx never returns 0. |↑⟩ is an equal superposition of |→⟩ and |←⟩, so P+ = ½.
- "P+ is the x-component of the spin." The component is ⟨Sx⟩ = (ħ/2)(P+ − P−). On a pole P+ = ½ while ⟨Sx⟩ = 0.
- "A stronger field makes P+ swing wider." μB changes only the rate. The swing is 2αβ.
Things to try on your own
- Set α = 0.6, then α = 0.8, and compare the P+ range in each case.
- Set μB = 4 and use the t readout to find the time of the first minimum of P+. Compare it with π/μB.
- Drag the sphere to look down on it from above and follow the arrow's tip around its circle.