Spin Measurements

A spin measurement is a measurement of a chosen spin component. For spin-$1/2$, measuring along a unit axis $\hat{\mathbf n}$ means measuring

$$S_{\hat n} = \hat{\mathbf n}\cdot\mathbf S = \frac{\hbar}{2}\hat{\mathbf n}\cdot\boldsymbol\sigma.$$

The possible outcomes are

$$\pm\frac{\hbar}{2}.$$

The axis is part of the measurement specification. There is no axis-independent measurement of “the spin value” for spin-$1/2$; there are measurements of $S_z$, $S_x$, $S_{\hat n}$, and so on.

This page applies the projective measurement formalism to spin. The physical Stern–Gerlach analyzer model is treated in Stern–Gerlach Revisited, and the general measurement rules are treated in Projective Measurement.

Measurement Along z

In the standard $S_z$ basis,

$$\lvert\uparrow\rangle=\lvert+z\rangle, \qquad \lvert\downarrow\rangle=\lvert-z\rangle,$$

with

$$S_z\lvert\uparrow\rangle = \frac{\hbar}{2}\lvert\uparrow\rangle, \qquad S_z\lvert\downarrow\rangle = -\frac{\hbar}{2}\lvert\downarrow\rangle.$$

The corresponding projectors are

$$P_{+z} = \lvert\uparrow\rangle\langle\uparrow\rvert = \frac12(I+\sigma_z),$$

and

$$P_{-z} = \lvert\downarrow\rangle\langle\downarrow\rvert = \frac12(I-\sigma_z).$$

For a normalized state

$$\lvert\psi\rangle = \alpha\lvert\uparrow\rangle +\beta\lvert\downarrow\rangle,$$

the Born probabilities are

$$p(+z)=\lvert\alpha\rvert^2, \qquad p(-z)=\lvert\beta\rvert^2.$$

These are not probabilities for the particle to be “really up” or “really down” before measurement in a classical hidden-variable sense. They are the probabilities assigned by the state and the specified projective measurement.

Arbitrary-Axis Projectors

Let

$$\hat{\mathbf n} = (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$$

be a unit vector. The spin component along $\hat{\mathbf n}$ is

$$S_{\hat n} = \frac{\hbar}{2}\hat{\mathbf n}\cdot\boldsymbol\sigma.$$

Because

$$(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=I,$$

the eigenvalues of $\hat{\mathbf n}\cdot\boldsymbol\sigma$ are $\pm1$, and the eigenvalues of $S_{\hat n}$ are $\pm\hbar/2$.

The projectors are

$$P_\pm^{(\hat n)} = \frac12 \left( I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

They obey the projective-measurement identities

$$P_+^{(\hat n)}+P_-^{(\hat n)}=I,$$ $$P_\pm^{(\hat n)}P_\pm^{(\hat n)} = P_\pm^{(\hat n)},$$

and

$$P_+^{(\hat n)}P_-^{(\hat n)}=0.$$

Thus $\{P_+^{(\hat n)},P_-^{(\hat n)}\}$ is the two-outcome projective measurement associated with the spin component $S_{\hat n}$.

Probabilities from Spinors

For a pure state $\lvert\psi\rangle$, the probability of outcome $\pm$ along $\hat{\mathbf n}$ is

$$p_\pm = \langle\psi|P_\pm^{(\hat n)}|\psi\rangle.$$

If the state is the $+\hat{\mathbf a}$ eigenstate, where $\gamma$ is the angle between $\hat{\mathbf a}$ and $\hat{\mathbf n}$, then

$$p_+ = \frac12 \left( 1+\hat{\mathbf a}\cdot\hat{\mathbf n} \right) = \cos^2\frac{\gamma}{2},$$

and

$$p_- = \sin^2\frac{\gamma}{2}.$$

For example, $|+z\rangle$ measured along $x$ has $\gamma=\pi/2$, so both outcomes have probability $1/2$.

Probabilities from the Bloch Vector

For a spin-$1/2$ density operator,

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right),$$

where $\lvert\mathbf r\rvert\leq1$, the measurement probabilities are

$$p_\pm = \operatorname{Tr} \left( \rho P_\pm^{(\hat n)} \right) = \frac12 \left( 1\pm\mathbf r\cdot\hat{\mathbf n} \right).$$

This is the compact Bloch-sphere form of the Born rule. The component of the Bloch vector along the analyzer axis controls the bias; transverse components affect other axes but not this outcome probability directly.

The expectation value is

$$\langle S_{\hat n}\rangle = \frac{\hbar}{2}\mathbf r\cdot\hat{\mathbf n}.$$

Thus a state can have zero expectation value along an axis while still giving two possible outcomes $\pm\hbar/2$.

State Update

If the outcome $+$ along $\hat{\mathbf n}$ occurs in an ideal selective projective measurement, a pure state updates as

$$\lvert\psi\rangle \longrightarrow \frac{ P_+^{(\hat n)}\lvert\psi\rangle }{ \sqrt{p_+} }.$$

Similarly, outcome $-$ gives

$$\lvert\psi\rangle \longrightarrow \frac{ P_-^{(\hat n)}\lvert\psi\rangle }{ \sqrt{p_-} }.$$

For density operators, the update is

$$\rho \longrightarrow \frac{ P_\pm^{(\hat n)}\rho P_\pm^{(\hat n)} }{ \operatorname{Tr} \left( \rho P_\pm^{(\hat n)} \right) }.$$

This is a conditional state assignment after a specified outcome. It is not a full microscopic model of a detector.

If the measurement is performed but the outcome is not recorded, the nonselective update is

$$\rho \longrightarrow P_+^{(\hat n)}\rho P_+^{(\hat n)} + P_-^{(\hat n)}\rho P_-^{(\hat n)}.$$

In Bloch-vector language, this keeps only the component of $\mathbf r$ along $\hat{\mathbf n}$:

$$\mathbf r \longrightarrow (\mathbf r\cdot\hat{\mathbf n})\hat{\mathbf n}.$$

The measurement removes coherence between the two eigenspaces of $S_{\hat n}$.

Noncommuting Axes

Spin measurements along different axes are generally incompatible. For two unit vectors $\hat{\mathbf a}$ and $\hat{\mathbf b}$,

$$[S_{\hat a},S_{\hat b}] = i\hbar S_{\hat{\mathbf a}\times\hat{\mathbf b}}.$$

If the axes are not parallel, the commutator is generally nonzero. A state prepared with definite $S_z$ is therefore not generally definite in $S_x$.

For spin-$1/2$,

$$\lvert+z\rangle = \frac{1}{\sqrt2} \left( \lvert+x\rangle+\lvert-x\rangle \right).$$

An $x$ measurement of a $+z$ state gives equal probabilities. If one outcome is selected, the post-measurement state is an $x$ eigenstate, and a later $z$ measurement again has equal probabilities. This is the spin version of the general Sequential Measurements rule.

Repeated and Sequential Measurements

Repeated measurement along the same axis is deterministic in the ideal projective model. If a measurement of $S_{\hat n}$ gives $+$ and is immediately repeated, the second measurement gives $+$ with probability $1$ because the state has been updated into the $+\hat{\mathbf n}$ eigenspace.

Changing the axis changes the measurement. A useful three-step example is:

1. prepare $|+z\rangle$;
2. measure $S_x$ and select $+x$;
3. measure $S_z$ again.

The final $S_z$ measurement gives

$$p(+z)=p(-z)=\frac12.$$

The intermediate $S_x$ selection prepares a new state. It does not merely reveal a value while leaving the original $S_z$ certainty intact.

Connection to Stern–Gerlach

An ideal Stern–Gerlach analyzer oriented along $\hat{\mathbf n}$ is the standard physical picture for the measurement $\{P_+^{(\hat n)},P_-^{(\hat n)}\}$. The apparatus correlates the two spin projectors with two spatial output paths.

This page owns the compact measurement formulas. Stern–Gerlach Revisited owns the analyzer model, path separation, sequential-analyzer interpretation, and historical caveats.

Common Mistakes

• Saying “measure spin” without specifying the axis.
• Using $|\alpha|^2$ and $|\beta|^2$ for every axis instead of only the chosen basis.
• Forgetting that $P_\pm^{(\hat n)}$ changes when the analyzer axis changes.
• Treating the post-measurement state as the same as the pre-measurement state after a selective outcome.
• Assuming that definite $S_z$ implies definite $S_x$.
• Confusing expectation value $\langle S_{\hat n}\rangle$ with a single-shot outcome.
• Treating the projective update rule as a complete physical detector model.

Cross-Links

• Spin-1/2 Hilbert Space
• Pauli Matrices
• Bloch Sphere
• Stern–Gerlach Revisited
• Spin Rotations
• Spin Problems
• Born Rule
• Projective Measurement
• State Update Rule
• Sequential Measurements
• Projectors
• Spin-1/2 as a Canonical System

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. A state is $\lvert\psi\rangle=(\lvert\uparrow\rangle+i\lvert\downarrow\rangle)/\sqrt2$. Find the probabilities for an $S_z$ measurement.

Solution

In the $S_z$ basis,

$$\alpha=\frac{1}{\sqrt2}, \qquad \beta=\frac{i}{\sqrt2}.$$

Therefore

$$p(+z)=\lvert\alpha\rvert^2=\frac12, \qquad p(-z)=\lvert\beta\rvert^2=\frac12.$$

2. Use the Bloch-vector formula to find the probabilities for measuring along $\hat z$ when $\mathbf r=(1,0,0)$.

Solution

For measurement along $\hat z$,

$$p_\pm = \frac12 \left( 1\pm\mathbf r\cdot\hat z \right).$$

Here $\mathbf r\cdot\hat z=0$, so

$$p_+=p_-=\frac12.$$

3. Verify that $P_\pm^{(\hat n)}=(I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma)/2$ are orthogonal projectors.

Solution

Use

$$(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=I.$$

Then

$$\left( P_\pm^{(\hat n)} \right)^2 = \frac14 \left( I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma \right)^2 = \frac12 \left( I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma \right) = P_\pm^{(\hat n)}.$$

Also,

$$P_+^{(\hat n)}P_-^{(\hat n)} = \frac14 \left[ I- (\hat{\mathbf n}\cdot\boldsymbol\sigma)^2 \right] = 0.$$

4. A spin is prepared in $|+z\rangle$, then $S_x$ is measured and the $+x$ outcome is selected. What are the probabilities for a later $S_z$ measurement?

Solution

After selecting $+x$, the state is $|+x\rangle$. In the $S_z$ basis,

$$|+x\rangle = \frac{1}{\sqrt2} \left( |+z\rangle+|-z\rangle \right).$$

Therefore the later $S_z$ probabilities are

$$p(+z)=\frac12, \qquad p(-z)=\frac12.$$