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Module 05

Spin and the Bloch sphere

Turn a spin state from a pile of complex numbers into a geometric object you can drag, watch precess, and watch collapse.

What you will see

  • A sphere on which every point is a pure state; drag it and the |ψ⟩ in the readout follows
  • A spin in a magnetic field precesses about the field, with the angle between them never changing
  • Measurement snaps the vector to a pole; single outcomes are random, hundreds of repeats converge on (1 ± r⃗·n̂)/2

Assumed background

  • Complex numbers and Euler's formula
  • Two-dimensional complex vectors, inner products, modulus squared
  • The Born rule: modulus squared gives probability

Spin 1/2 is the smallest non-trivial system in quantum mechanics: its state space is two-dimensional.

How small is that? Small enough that we can draw every state at once, leaving nothing out. No other system allows this — hydrogen’s state space is infinite-dimensional and you can only look at one orbital at a time. The pure states of a spin-1/2 form exactly a sphere, and suddenly the abstract notion of “quantum state” is something you can push around with a finger.

Look first, calculate second.

The Bloch sphere: a spin state you can grab with your hand

Every point on the sphere is one pure state. Drag the arrowhead to change it, switch on a magnetic field to watch it precess, pick an axis and measure to watch it collapse.

Loading 3D scene…

Prepare a state
60°
120°
Magnetic field and precession
Field direction
1.20
Speed

A magnetic field never "pulls" the spin towards itself. It makes the spin circle around the field direction, and the angle between them never changes.

Measurement
Measurement axis
Theory P₊0.750
Measured frequency (0 runs)—
∣ψ⟩=\ket{\psi} =∣0⟩+\ket{0} +∣1⟩\ket{1}
r⃗ = (-0.433, 0.750, 0.500)(x, y, z)|r⃗| = 1.0000⟨Z（σ_z）⟩ = 0.500
  • State vector r⃗
  • Precession cone
  • Projection onto the measurement axis
  • Measurement axis n̂

What to look for

  • The formula in the readout is clickable. Click the cos(θ/2) term and θ appears on the sphere; click e^{iφ} and φ appears in the equatorial plane. Every symbol in the formula is some geometric quantity on the sphere.
  • Start by dragging θ to 90°. The state lands on the equator, and measuring along z gives 0 and 1 with equal probability. On the sphere, "equal superposition" simply means "standing on the equator".
  • Now hold θ fixed and drag only φ: the probabilities along z do not budge. φ is the relative phase — invisible in the z basis, yet it fixes the outcome along x and y. Switch to σx and drag φ again to see the readout move.
  • Switch on the field and press play. The angle between the vector and the field never changes; only φ increases steadily. That is Larmor precession.
  • Hit "Measure once" a dozen times and watch how random single outcomes are. Then hit "Re-prepare ×200" and watch the measured frequency close in on the theoretical P₊.

✕ "A point on the Bloch sphere is the direction the particle points in space"

It is not. The sphere is state space, not real space. θ = 90° does not mean the spin "lies flat"; it means that measuring along z gives up and down half the time each.

✕ "Measuring again gives a different answer"

It does not. Measurement projects the state onto an eigenstate, so an immediate second measurement along the same axis repeats the first result. To see the statistics you must re-prepare every time — which is exactly what "Re-prepare ×200" does.

Think it through

  1. Are |ψ⟩ and e^{iα}|ψ⟩ the same point on the sphere? If so, why is the relative phase φ observable while the global phase is not?
  2. With the field along z and the initial state exactly |0⟩, pressing play changes nothing on screen. Explain that using "an eigenstate only picks up a global phase".
  3. Set the measurement axis to σx with the state at |0⟩, so P₊ = 0.5. After one measurement the state sits on ±x. Switch back to σz — what are the probabilities now? What does that say about measuring different directions in sequence?

Why one point on a sphere is enough

A two-dimensional complex vector has four real parameters:

∣ψ⟩=(ab),a,b∈C.\ket{\psi} = \begin{pmatrix} a \\ b \end{pmatrix},\qquad a, b \in \mathbb{C}.

Two of them are fake. Normalisation ∣a∣2+∣b∣2=1|a|^2 + |b|^2 = 1 removes one; the global phase — ∣ψ⟩\ket{\psi} and eiα∣ψ⟩\ee^{\ii\alpha}\ket{\psi} give identical results for every possible measurement — removes another. Two real parameters remain, and they are precisely the two angles of a sphere:

∣ψ⟩=cos⁡θ2∣0⟩+eiφsin⁡θ2∣1⟩.\ket{\psi} = \cos\frac{\theta}{2}\ket{0} + \ee^{\ii\varphi}\sin\frac{\theta}{2}\ket{1}.

The matching geometric object is the Bloch vector

r⃗=⟨ψ|σ⃗ ψ⟩=(sin⁡θcos⁡φ, sin⁡θsin⁡φ, cos⁡θ),\vec{r} = \braket{\psi}{\vec{\sigma}\,\psi} = (\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta),

which is the arrow in the scene. Pure states always satisfy ∣r⃗∣=1|\vec r| = 1 and lie on the surface; points inside the ball mean something too — they are mixed states, and they wait until module 12.

Precession: the field does not do what you expect

Put the spin in a field along zz, so H=−ℏω2σzH = -\tfrac{\hbar\omega}{2}\sigma_z. The resulting equation of motion is remarkably clean:

dr⃗dt=ω⃗×r⃗.\frac{\dd \vec r}{\dd t} = \vec\omega \times \vec r .

A cross product means the rate of change is always perpendicular to both r⃗\vec r and ω⃗\vec\omega. Hence:

  • ∣r⃗∣|\vec r| is constant (the state stays pure);
  • r⃗⋅ω^\vec r \cdot \hat\omega is constant (the angle to the field is locked);
  • only the azimuth around the field increases, at a constant rate.

That is the cone in the scene. Classical intuition usually expects the field to bend the spin round until it points along the field — but that is what a dissipative system does (a compass needle in air with friction). Unitary evolution of an isolated quantum system does no such thing; it only turns. Laying the spin down along the field requires coupling to an environment, which is module 12.

Precession from the Schrödinger equation

Take H=−ℏω2σzH = -\tfrac{\hbar\omega}{2}\sigma_z. In the Heisenberg picture operators obey

dσ^idt=iℏ[H,σ^i].\frac{\dd \hat\sigma_i}{\dd t} = \frac{\ii}{\hbar}[H, \hat\sigma_i].

Using the Pauli commutators [σi,σj]=2i ϵijkσk[\sigma_i, \sigma_j] = 2\ii\,\epsilon_{ijk}\sigma_k:

[σz,σx]=2iσy,[σz,σy]=−2iσx,[σz,σz]=0.[\sigma_z, \sigma_x] = 2\ii\sigma_y, \qquad [\sigma_z, \sigma_y] = -2\ii\sigma_x, \qquad [\sigma_z,\sigma_z]=0 .

Substituting gives

σ˙x=−ω σy,σ˙y=ω σx,σ˙z=0,\dot\sigma_x = -\omega\,\sigma_y,\qquad \dot\sigma_y = \omega\,\sigma_x,\qquad \dot\sigma_z = 0,

and taking expectation values yields r⃗˙=ω⃗×r⃗\dot{\vec r} = \vec\omega\times\vec r with ω⃗=ωz^\vec\omega = \omega\hat z. The third equation, σ˙z=0\dot\sigma_z = 0, is exactly the statement that the angle to the field never changes.

Measurement: projection, not inspection

Measuring the spin along a unit axis n^\hat n can only return ±1\pm 1, with probabilities

P±=1±r⃗⋅n^2.P_\pm = \frac{1 \pm \vec r\cdot\hat n}{2}.

The dashed line dropping from the arrowhead onto the measurement axis in the scene draws exactly the projection r⃗⋅n^\vec r \cdot \hat n. After the measurement the state collapses onto the corresponding eigenstate — the arrow jumps to one end of the axis.

There is a trap here that catches nearly everyone: measuring again along the same axis necessarily repeats the first result. To see the statistics of P±P_\pm you must re-prepare the same ∣ψ⟩\ket{\psi} every time. That is what the “Re-prepare ×200” button does, and the label spells it out on purpose.

Key formulas

  • Sphere parameterisation

    ∣ψ⟩=cos⁡θ2∣0⟩+eiφsin⁡θ2∣1⟩\ket{\psi} = \cos\frac{\theta}{2}\ket{0} + \ee^{\ii\varphi}\sin\frac{\theta}{2}\ket{1}

    Two real parameters: normalisation and global phase eat one each

  • Bloch vector

    r⃗=(sin⁡θcos⁡φ, sin⁡θsin⁡φ, cos⁡θ)\vec r = (\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta)

    Pure states have |r| = 1; mixed states live inside

  • Larmor precession

    r⃗˙=ω⃗×r⃗\dot{\vec r} = \vec\omega \times \vec r

    A cross product conserves both length and angle; only the azimuth moves

  • Measurement probability

    P±=12(1±r⃗⋅n^)P_\pm = \tfrac{1}{2}\left(1 \pm \vec r\cdot\hat n\right)

    The length of that dashed projection

Taking the sphere into the laboratory: Stern–Gerlach

The Bloch sphere is a mathematical picture. What Stern and Gerlach did in 1922 was the first time that picture hit people in a visible form: a beam of silver atoms crossing an inhomogeneous magnetic field did not smear into a continuous band. It split cleanly into two.

Stern-Gerlach in series: measurement is not "having a look"

Every dot that flies past is a real random draw. Set the second analyser to X, measure Z again with the third, and the beam that the first analyser had cleanly filtered out comes back.

Loading 3D scene…

Apparatus
Number of analysers

Analyser 1

Analyser 2

Analyser 3

Beam
0 so far
Speed
60
Counts at the screen
Upper / lower0 / 0

0 absorbed by the blockers. Orthogonal to the previous stage: fifty-fifty — the component filtered out earlier is back

∣⟨±x|±z⟩∣2=12|\braket{\pm_x}{\pm_z}|^2 = \tfrac12
Emitted 0Reached screen 0Upper fraction —
  • spin + (along that analyser's axis)
  • spin −
  • Magnets and blockers

What to look for

  • The default is the classic experiment: Z (block lower) → X (block lower) → Z. Note that the last analyser still splits the beam in two, even though the first one passed only "up".
  • Set the second analyser to Z as well: only one beam leaves the last one. Measurement is repeatable — along the same axis you can measure as often as you like without changing anything.
  • Switch the second analyser back to X and the lower beam returns at once. An X measurement does not read off a property; it genuinely changes the state.
  • Keep a single analyser and block nothing: unpolarised silver atoms split half and half. That is what Stern and Gerlach saw in 1922, and it was the first direct evidence that spin takes only two values.

✕ "The first analyser already removed the lower beam, so it should not return"

"Lower" is defined relative to Z. After the X analyser the state is an eigenstate of X, which is once again a superposition in the Z basis. Nothing leaked through the filter; the property itself was redefined.

✕ "The magnetic field bends the spin into a new direction"

The inhomogeneous field supplies a position-dependent force that separates the spin components in space. What actually changes the state is blocking one beam — that is the projection.

Think it through

  1. Suppose the X analyser blocks nothing and instead recombines both beams without looking at which is which. What does the third Z analyser give? (This is the germ of the quantum eraser.)
  2. Set all three analysers to Z and block nothing. How many beams reach the screen, and why?
  3. A silver atom has 47 electrons. Why does the whole thing behave like a single spin-1/2?

Chaining the apparatus makes it more interesting. Split along zz with the first magnet and block the lower beam; measure zz again with a second and everything still comes out upper — measurement repeats itself. But make the second one measure along xx and the output is fifty-fifty again; add a third along zz and the lower beam, supposedly filtered away, is back.

Nothing classical does this. It says that measuring along xx is not reading off a pre-existing property; it genuinely changes the state.

Three standard misreadings

Check yourself

  1. Set the state to ∣+⟩\ket{+} (θ = 90°, φ = 0) and the axis to σx. What will one measurement give, and with what probability? What about σz?
  2. With the field along z and the initial state ∣0⟩\ket{0}, pressing play changes nothing on screen. Explain it using “an eigenstate only accumulates a global phase”. What if the initial state is ∣+⟩\ket{+}?
  3. The states ∣ψ1⟩=∣+⟩\ket{\psi_1} = \ket{+} and ∣ψ2⟩=eiπ/3∣+⟩\ket{\psi_2} = \ee^{\ii\pi/3}\ket{+} are the same point on the sphere. How do “the phase is unobservable” and “the relative phase φ is observable” both hold at once?

Go deeper · matching textbook sections

The 3D scenes build the picture; the full derivations and exercises live in the textbook.

Having finished this module