XVI Spin

The spin was discovered in the Stern-Gerlach experiment in 1922. In this experiment, silver atoms are heated in an oven, from which they escape through a tiny hole. They travel into $x$ direction and enter an apparatus where they pass through an inhomogeneous magnetic field $𝐁=bz𝐤$. This field points into $z$ direction and has a finite gradient $\partial B_z/\partial z=b$. If the atoms carry a magnetic moment $𝐦$ this gives rise to a force

$$F_z=-\frac{\partial V_B}{\partial z}=m_{z}b$$

(279)

which deflects the atoms into $z$ direction.

In the experiment it is observed that the atoms arrive on the screen at two spots, only. This is not what is classically expected, but also incompatible with quantisation of the orbital angular momentum $𝐋$, which always yields an odd number $2l+1$ of possible values of $F_z$ since $l$ is an integer. However, the experiment can be fully explained by introducing an intrinsic magnetic moment

$$\widehat{𝐦}=-\frac{e}{m_e}\widehat{𝐒},$$

(280)

where $\widehat{𝐒}$ has the properties of angular momentum, but with $l=l_s=1/2$ not an integer. Since the spin magnetic quantum number is restricted to $m_s=-l_s,-l_s+1,\mathrm{\dots },l_s$ this allows only two values $m_s=\pm \frac{1}{2}$, hence

$$S_z=\frac{\mathrm{\hslash }}{2}\text{ or }{S}_z=-\frac{\mathrm{\hslash }}{2}.$$

(281)

Given the force from Eq. (279) it can be deduced that the observed locations of the arriving atoms indeed correspond to these two values.

The spin state of the electron is described by a two-component vector (called spinor)

$$\psi =\left(\genfrac{}{}{0pt}{}{{\psi }_{\uparrow }}{{\psi }_{\downarrow }}\right)$$

(282)

where

	${|{\psi }_{\uparrow }|}^2$	$=$	$\text{probability that }{S}_z=\mathrm{\hslash }/2\text{ (‘spin up’)},$		(283)
	${|{\psi }_{\downarrow }|}^2$	$=$	$\text{probability that }{S}_z=-\mathrm{\hslash }/2\text{ (‘spin down’)}.$		(284)

Hence $\psi =\left(\genfrac{}{}{0pt}{}{1}{0}\right)$ describes an electron with spin up and $\psi =\left(\genfrac{}{}{0pt}{}{0}{1}\right)$ describes an electron with spin down.

The scalar product of spinors is given by (see the mathematical appendix)

$$⟨\psi |\phi ⟩={\psi }_{\uparrow }^{*}{\phi }_{\uparrow }+{\psi }_{\downarrow }^{*}{\phi }_{\downarrow }.$$

(285)

Normalisation of the probability requires ${|{\psi }_{\uparrow }|}^2+{|{\psi }_{\downarrow }|}^2=1$, hence $⟨\psi |\psi ⟩=1$.

The spin operator $\widehat{𝐒}={\widehat{S}}_x𝐢+{\widehat{S}}_y𝐣+{\widehat{S}}_z𝐤$ consists of the three spin matrices

			(286)
			(287)
			(288)

The three matrices

(289)

are called Pauli matrices.

Matrix multiplication (see the mathematical appendix) gives

(290)

The commutation relations also follow from matrix multiplication,

	$[{\widehat{S}}_x,{\widehat{S}}_y]=i\mathrm{\hslash }{\widehat{S}}_z,[{\widehat{S}}_y,{\widehat{S}}_z]=i\mathrm{\hslash }{\widehat{S}}_x,[{\widehat{S}}_z,{\widehat{S}}_x]=i\mathrm{\hslash }{\widehat{S}}_y,$
			(291)
	$[{\widehat{S}}_x,{\widehat{𝐒}}^2]=0,[{\widehat{S}}_y,{\widehat{𝐒}}^2]=0,[{\widehat{S}}_z,{\widehat{𝐒}}^2]=0.$		(292)

This is identical to the commutation relations of angular momentum, see Eqs. (264) and (265).

Since

(293)

all vectors are eigenvectors ${\widehat{𝐒}}^2$. The eigenvalue $\frac{3}{4}{\mathrm{\hslash }}^2$ corresponds to angular momentum with $l=l_s=1/2$, see Eq. (266).

Since

(294)

the normalised eigenvectors of ${\widehat{S}}_z$ are $\psi =\left(\genfrac{}{}{0pt}{}{1}{0}\right)$ (with eigenvalue $\mathrm{\hslash }/2$) and $\psi =\left(\genfrac{}{}{0pt}{}{0}{1}\right)$ (with eigenvalue $-\mathrm{\hslash }/2$). This again corresponds to angular momentum with $l=l_s=1/2$, see Eq. (267).

(295)

Hence the normalised eigenvectors of ${\widehat{S}}_x$ are $\psi =\frac{1}{\sqrt{2}}\left(\genfrac{}{}{0pt}{}{1}{1}\right)$ (with eigenvalue $\mathrm{\hslash }/2$) and $\psi =\frac{1}{\sqrt{2}}\left(\genfrac{}{}{0pt}{}{1}{-1}\right)$ (with eigenvalue $-\mathrm{\hslash }/2$).

(296)

Hence the normalised eigenvectors of ${\widehat{S}}_x$ are $\psi =\frac{1}{\sqrt{2}}\left(\genfrac{}{}{0pt}{}{1}{i}\right)$ (with eigenvalue $\mathrm{\hslash }/2$) and $\psi =\frac{1}{\sqrt{2}}\left(\genfrac{}{}{0pt}{}{1}{-i}\right)$ (with eigenvalue $-\mathrm{\hslash }/2$).

The expectation value of $S_z$ is given by

	$⟨{\widehat{S}}_z⟩$	$=$	$⟨\psi |{\widehat{S}}_z\psi ⟩$		(297)
		$=$
		$=$	${\displaystyle \frac{\mathrm{\hslash }}{2}}({\psi }_{\uparrow }^{*},{\psi }_{\downarrow }^{*})\left({\displaystyle \genfrac{}{}{0pt}{}{{\psi }_{\uparrow }}{-{\psi }_{\downarrow }}}\right)$
		$=$	${\displaystyle \frac{\mathrm{\hslash }}{2}}({|{\psi }_{\uparrow }|}^2-{|{\psi }_{\downarrow }|}^2),$

Note that

	$P(S_z=\mathrm{\hslash }/2)$	$=$	${|{\psi }_{\uparrow }|}^2,$		(298)
	$P(S_z=-\mathrm{\hslash }/2)$	$=$	${|{\psi }_{\downarrow }|}^2.$		(299)

are the probabilities for spin up and down, respectively.

By similar calculations,

	$⟨{\widehat{S}}_x⟩$	$=$	$\mathrm{\hslash }\mathrm{Re}{\psi }_{\uparrow }^{*}{\psi }_{\downarrow },$		(300)
	$⟨{\widehat{S}}_y⟩$	$=$	$\mathrm{\hslash }\mathrm{Im}{\psi }_{\uparrow }^{*}{\psi }_{\downarrow },$		(301)

where Re denotes the real part and Im denotes the imaginary part of the complex number ${\psi }_{\uparrow }^{*}{\psi }_{\downarrow }$.

The probabilities for spin measurements in $x$ or $y$ directions are given by

	$P(S_x=\mathrm{\hslash }/2)$	$=$	${\displaystyle \frac{1}{2}}{|{\psi }_{\uparrow }+{\psi }_{\downarrow }|}^2,$		(302)
	$P(S_x=-\mathrm{\hslash }/2)$	$=$	${\displaystyle \frac{1}{2}}{|{\psi }_{\uparrow }-{\psi }_{\downarrow }|}^2,$		(303)

and

	$P(S_y=\mathrm{\hslash }/2)$	$=$	${\displaystyle \frac{1}{2}}{|{\psi }_{\uparrow }-i{\psi }_{\downarrow }|}^2,$		(304)
	$P(S_y=-\mathrm{\hslash }/2)$	$=$	${\displaystyle \frac{1}{2}}{|{\psi }_{\downarrow }-i{\psi }_{\uparrow }|}^2,$		(305)

respectively.

A convenient alternative way to represent a spinor state is given by the 3-dimensional real polarisation vector

	$\overrightarrow{P}$	$=$	$(⟨{\sigma }_x⟩,⟨{\sigma }_y⟩,⟨{\sigma }_z⟩)$		(306)
		$=$	$(2\mathrm{Re}{\psi }_{\uparrow }^{*}{\psi }_{\downarrow },2\mathrm{Im}{\psi }_{\uparrow }^{*}{\psi }_{\downarrow },{|{\psi }_{\uparrow }|}^2-{|{\psi }_{\downarrow }|}^2).$		(307)

If $\psi$ is normalised then the polarisation vector is of unit length, i.e., it is restricted to the surface of a sphere, the so-called Bloch sphere.

The most general stationary Hamiltonian for a spin is a hermitian $2\times 2$-dimensional matrix. Such a matrix can always be written as

$$H=a_{0}I+a_x{\sigma }_x+a_y{\sigma }_y+a_z{\sigma }_z$$

(308)

where $a_0$, $a_x$, $a_y$, and $a_z$ are real constants, and

(309)

is the $2\times 2$-dimensional identity matrix. The stationary Schrödinger equation $E\psi =H\psi$ is solved by the eigenvectors of $H$. These can be found by writing the vector $𝐚=(a_x,a_y,a_z)=a𝐧$, where $𝐧$ is a unit vector and $a=|𝐚|$. The eigenvectors can then be written as

	${\psi }_{+}$	$=$	${\displaystyle \frac{1}{\sqrt{2(1+n_z)}}}\left({\displaystyle \genfrac{}{}{0pt}{}{1+n_z}{n_x+i{n}_y}}\right),$		(310)
	${\psi }_{-}$	$=$	${\displaystyle \frac{1}{\sqrt{2(1+n_z)}}}\left({\displaystyle \genfrac{}{}{0pt}{}{-n_x+i{n}_y}{1+n_z}}\right).$		(311)

On the Bloch sphere, these eigenvectors correspond to the polarisation vectors

$${\overrightarrow{P}}_{\pm }=\pm 𝐧.$$

(312)

The corresponding eigenvalues

$$E_{\pm }=a_0\pm a$$

(313)

determine the energy of these states.

In order to describe spin in the Dirac notation, we introduce the normalised and mutually orthogonal states $|\uparrow ⟩$ for “spin up” and $|\downarrow ⟩$ for “spin down”, with

$${\widehat{S}}_z|\uparrow ⟩=\frac{\mathrm{\hslash }}{2}|\uparrow ⟩,{\widehat{S}}_z|\downarrow ⟩=-\frac{\mathrm{\hslash }}{2}|\downarrow ⟩.$$

(314)

Accordingly, we can write

$${\widehat{S}}_z=\frac{\mathrm{\hslash }}{2}|\uparrow ⟩⟨\uparrow |-\frac{\mathrm{\hslash }}{2}|\downarrow ⟩⟨\downarrow |.$$

(315)

The other spin components are then given by the operators

$${\widehat{S}}_x=\frac{\mathrm{\hslash }}{2}|\uparrow ⟩⟨\downarrow |+\frac{\mathrm{\hslash }}{2}|\downarrow ⟩⟨\uparrow |,{\widehat{S}}_y=-i\frac{\mathrm{\hslash }}{2}|\uparrow ⟩⟨\downarrow |+i\frac{\mathrm{\hslash }}{2}|\downarrow ⟩⟨\uparrow |,$$

(316)

and the total spin is given as

$${\widehat{𝐒}}^2={\widehat{S}}_x^2+{\widehat{S}}_y^2+{\widehat{S}}_z^2=\frac{3{\mathrm{\hslash }}^2}{4}(|\uparrow ⟩⟨\uparrow |+|\downarrow ⟩⟨\downarrow |).$$

(317)

The eigenstates of ${\widehat{S}}_x$ are $|\pm x⟩=\sqrt{1/2}(|\uparrow ⟩\pm |\downarrow ⟩)$, while those of ${\widehat{S}}_y$ are $|\pm y⟩=\sqrt{1/2}(|\uparrow ⟩\pm i|\downarrow ⟩)$; the associated eigenvalues are $\pm \mathrm{\hslash }/2$. All the expressions for expectation values and probabilities given above can now be seen as special incarnations of our general rules for quantum mechanics, evaluated in the basis $|\uparrow ⟩$, $|\downarrow ⟩$.

In analogy to orbital angular momentum, we can also introduce the ladder operators

$${\widehat{S}}_{+}={\widehat{S}}_x+i{S}_y=|\uparrow ⟩⟨\downarrow |,{\widehat{S}}_{-}={\widehat{S}}_x-i{S}_y=|\downarrow ⟩⟨\uparrow |,$$

(318)

which are again non-hermitian and related by ${\widehat{S}}_{+}^{†}={\widehat{S}}_{-}$. These now convert or annihilate spin eigenstates, according to

$${\widehat{S}}_{+}|\downarrow ⟩=|\uparrow ⟩,{\widehat{S}}_{+}|\uparrow ⟩=|\mathbb{𝟘}⟩,{\widehat{S}}_{-}|\uparrow ⟩=|\downarrow ⟩,{\widehat{S}}_{-}|\downarrow ⟩=|\mathbb{𝟘}⟩.$$

(319)