Spin one-half states · Quantum Mechanics

Intuition

The state of a spin one-half is a vector in two complex dimensions: an amplitude for up and an amplitude for down along $z$ . Take away the overall phase and the length, and two real numbers remain, which can be read as the angles of a direction in space. Every spin state is spin up along some direction — there is no state that is up along no axis — and the set of all states is a sphere, the Bloch sphere, with up and down along $z$ at its poles and up and down along $x$ and $y$ around its equator.

A compass needle can point anywhere on a sphere of directions. A spin one-half has exactly as many states as a needle has directions — but two opposite directions are orthogonal states, not two different amounts of the same thing.

A cut through the Bloch sphere in the $x$ – $z$ plane. $\ket{\uparrow}$ and $\ket{\downarrow}$ sit at the poles, $\ket{\pm x}$ on the equator, and the state is the point in the direction at angle from , drawn for . Opposite points are orthogonal states.

The state space of spin one-half

$\ket{\uparrow}$ and $\ket{\downarrow}$ , the eigenstates of $\hat{S}_{z}$ , form a basis. Every normalised state is, up to an overall phase, spin up along the direction with polar angles and .

\ket{\chi}=\cos\frac{\theta}{2}\ket{\uparrow}+e^{i\varphi}\sin\frac{\theta}{2}\ket{\downarrow},\qquad 0\le\theta\le\pi

Properties

$\ket{\pm x}=\tfrac{1}{\sqrt2}\bigl(\ket{\uparrow}\pm\ket{\downarrow}\bigr)$ and .

Every spin state points somewhere

\begin{aligned}&\ket{\chi}=a\ket{\uparrow}+b\ket{\downarrow},\ \ |a|^{2}+|b|^{2}=1\\ &\Longrightarrow\ \ket{\chi}\simeq\cos\tfrac{\theta}{2}\ket{\uparrow}+e^{i\varphi}\sin\tfrac{\theta}{2}\ket{\downarrow}\end{aligned}

An overall phase changes nothing, so choose it to make $a$ real and not negative. Then $a$ and $|b|$ are two non-negative numbers whose squares add to one: the cosine and sine of one angle, which is called $\theta/2$ . What is left of $b$ is its phase, .

Proof steps

e^{-i\arg a}\ket{\chi}=|a|\ket{\uparrow}+b'\ket{\downarrow}

Multiply by a phase that makes the first amplitude real and not negative; the state is physically the same.

|a|^{2}+|b'|^{2}=1,\qquad |a|,|b'|\ge0

Normalisation.

|a|=\cos\tfrac{\theta}{2},\ \ |b'|=\sin\tfrac{\theta}{2},\qquad 0\le\theta\le\pi

Two non-negative numbers with squares adding to one.

b'=e^{i\varphi}\sin\tfrac{\theta}{2}

The remaining freedom is the phase of $b'$ .

(\theta,\varphi)\ \text{is a point on a sphere}

Two angles label every state: the Bloch sphere.

Applications

Practice

Up Along x

The state that is spin up along $x$ is an equal superposition of up and down along $z$ .

\ket{+x}=\tfrac{1}{\sqrt2}\bigl(\ket{\uparrow}+\ket{\downarrow}\bigr)

Try it

Which state is spin up along $y$ ?