Spin-1/2 Hilbert Space

A spin-$1/2$ Hilbert space is two-dimensional. A standard basis is the $S_z$ eigenbasis,

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

These states satisfy

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

Spin Basis and Components

In the $\{\lvert\uparrow\rangle,\lvert\downarrow\rangle\}$ basis,

$$\lvert\uparrow\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \qquad \lvert\downarrow\rangle = \begin{pmatrix}0\\1\end{pmatrix}.$$

This is a basis convention. A physical spin state is not literally a column of numbers; the column is a representation of a vector in a chosen basis.

General State

A normalized pure spin-$1/2$ state is

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

with

$$\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

The global phase is physically irrelevant, but the relative phase between $\alpha$ and $\beta$ affects measurements along axes other than $z$. A two-dimensional Hilbert space therefore has infinitely many pure states, not just “up” and “down.” A statistical mixture, or a spin entangled with unobserved motion, instead needs a density operator.

For example, $|+x\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt2$ gives

$$P(+x)=|\langle+x|\psi\rangle|^2 =\frac{|\alpha+\beta|^2}{2} =\frac12+\operatorname{Re}(\alpha^*\beta).$$

The interference term makes the relative phase observable. By comparison, an equal incoherent mixture of up and down gives $P(+x)=1/2$.

Measurement Along z

In the $S_z$ basis, the probabilities are

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

After an ideal projective measurement with outcome $+z$, the state updates to $\lvert\uparrow\rangle$ in the simplest projective model.

The Stern–Gerlach analyzer is the standard physical model for this kind of spin-component measurement; see Stern–Gerlach Revisited.

Spin Operators

For spin-$1/2$,

$$S_i=\frac{\hbar}{2}\sigma_i.$$

The Pauli matrices therefore provide the matrix representation of spin components in the chosen basis.

The same basis is commonly used to define spin-$1/2$ time reversal as $\Theta=-i\sigma_yK$; see Time Reversal for Spin-1/2 Particles.

Common Mistakes

• Treating $\lvert\uparrow\rangle$ and $\lvert\downarrow\rangle$ as spatial directions rather than basis states.
• Forgetting that $\alpha$ and $\beta$ are basis-dependent components.
• Discarding relative phase because global phase is unobservable.
• Assuming a density matrix diagonal in the $S_z$ basis must be a sharp state. It may describe a mixture; and even a sharp $S_z$ state is not sharp in $S_x$.

Exercises

1. For $\lvert\psi\rangle=(\lvert\uparrow\rangle+i\lvert\downarrow\rangle)/\sqrt2$, compute the probabilities for $S_z=\pm\hbar/2$ and for $S_x=+\hbar/2$.

Solution

Here $\alpha=1/\sqrt2$ and $\beta=i/\sqrt2$, so

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

Also $P(+x)=|1+i|^2/4=1/2$. Changing the relative phase from $i$ to $1$ leaves the $z$ probabilities unchanged but changes $P(+x)$ to one.

2. Explain why multiplying both $\alpha$ and $\beta$ by $e^{i\chi}$ does not change the physical state.

Solution

The whole state vector is multiplied by a global phase:

$$\lvert\psi\rangle\mapsto e^{i\chi}\lvert\psi\rangle.$$

Pure states are rays, so this represents the same physical state and gives the same measurement probabilities.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.