7.2 The spin 1/2 particle in a magnetic field

7.2 The spin 1/2 particle in a magnetic field

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · eigenvalue · Lie algebra · Lie algebra representation · irreducible representation

In chapter 3 we saw that a general quantum system with $\mathcal{H}={\mathbf{C}}^2$ could be understood in terms of the action of $U(2)$ on ${\mathbf{C}}^2$ . The self-adjoint observables correspond (up to a factor of i) to the corresponding Lie algebra representation. The $U(1)\subset U(2)$ subgroup commutes with everything else and can be analyzed separately, here we will consider only the $SU(2)$ subgroup. For an arbitrary such system, the group $SU(2)$ has no particular geometric significance. When it occurs in its role as double cover of the rotational group, the quantum system is said to carry ${}^6{\mathrm{spin}}^{†}$ , in particular “spin $\frac{1}{2}d{s}^{,}$ for the two dimensional irreducible representation (in chapter 8 we will discuss state spaces of higher spin values).

As before, we take as a standard basis for the Lie algebra su(2) the operators $X_j,j=1,2,3$ , where

$$X_j=-i\frac{{\sigma }_j}{2}$$

which satisfy the commutation relations

$$[X_1,X_2]=X_3,[X_2,X_3]=X_1,[X_3,X_1]=X_2$$

To make contact with the physics formalism, we’ll define self-adjoint operators

$$S_j=i{X}_j=\frac{{\sigma }_j}{2}\text{\hspace{1em}(7.1)}$$

In general, to a skew-adjoint operator (which is what one gets from a unitary Lie algebra representation and what exponentiates to unitary operators) we will associate a self-adjoint operator by multiplying by $i$. These self-adjoint operators have real eigenvalues (in this case $\pm \frac{1}{2})$ , so are favored by physicists as observables since experimental results are given by real numbers. In the other direction, given a physicist’s observable self-adjoint operator, we will multiply by −$i$ to get a skew-adjoint operator (which may be an operator for a unitary Lie algebra representation).

Note that the conventional definition of these operators in physics texts includes a factor of ℏ:

$$S_j^{phys}=i\hslash X_j=\frac{\hslash {\sigma }_j}{2}$$

A compensating factor of $1/\hslash$ is then introduced when exponentiating to get group elements

$$\Omega (\theta ,\mathbf{w})=e^{-i\frac{\theta }{\hslash }\mathbf{w}\cdot {\mathbf{S}}^{phys}}\in SU(2)$$

which do not depend on ℏ. The reason for this convention has to do with the action of rotations on functions on ${\mathbf{R}}^3$ (see chapter 19) and the appearance of ℏ in the definition of the momentum operator. Our definitions of $S_j$ and of rotations using (see equation 6.3)

$$\Omega (\theta ,\mathbf{w})=e^{-i\theta \mathbf{w}\cdot \mathbf{S}}=e^{\theta \mathbf{w}\cdot \mathbf{X}}$$

will not include these factors of $\hslash ,$ but in any case they will be equivalent to the usual physics definitions when we make our standard choice of working with units such that $\hslash =1$

States in $\mathcal{H}={\mathbf{C}}^2$ that have a well-defined value of the observable $S_j$ will be the eigenvectors of $S_j$ , with value for the observable the corresponding eigenvalue, which will be $\pm \frac{1}{2}$ . Measurement theory postulates that if we perform the measurement corresponding to $S_j$ on an arbitrary state $|\psi ⟩$ , then we will

• with probability $c_{+}$ get a value of $+\frac{1}{2}$ and leave the state in an eigenvector $|j,+\frac{1}{2}⟩$ of $S_j$ with eigenvalue $+\frac{1}{2}$

• with probability c₋ get a value of $:-\frac{1}{2}$ and leave the state in an eigenvector $|j,-\frac{1}{2}⟩$ of $S_j$ with eigenvalue $-\frac{1}{2}$

where if

$$|\psi ⟩=\alpha |j,+\frac{1}{2}⟩+\beta |j,-\frac{1}{2}⟩$$

we have

$$c_{+}=\frac{|\alpha {|}^2}{|\alpha {|}^2+|\beta {|}^2},c_{-}=\frac{|\beta {|}^2}{|\alpha {|}^2+|\beta {|}^2}$$

After such a measurement, any attempt to measure another $S_k,k\ne j$ will give $\pm \frac{1}{2}$ with equal probability (since the inner products of $\mid j,\pm \frac{1}{2}⟩$ and $|k,\pm \frac{1}{2}⟩$ are equal up to a phase) and put the system in a corresponding eigenvector of $S_k$

If a quantum system is in an arbitrary state $|\psi ⟩$ it may not have a well-defined value for some observable $A,$ but the “expected value” of A can be calculated. This is the sum over a basis of H consisting of eigenvectors (which will all be orthogonal) of the corresponding eigenvalues, weighted by the probability of their occurrence. The calculation of this sum in this case $(A=S_j)$ using expansion in eigenvectors of $S_j$ gives

$$\begin{array}{rl}\frac{⟨\psi |A|\psi ⟩}{⟨\psi |\psi ⟩}& =\frac{(\overline{\alpha }⟨j,+\frac{1}{2}|+\overline{\beta }⟨j,-\frac{1}{2}|)A(\alpha |j,+\frac{1}{2}⟩+\beta |j,-\frac{1}{2}⟩)}{(\overline{\alpha }⟨j,+\frac{1}{2}|+\overline{\beta }⟨j,-\frac{1}{2}|)(\alpha |j,+\frac{1}{2}⟩+\beta |j,-\frac{1}{2}⟩)}\\ & =\frac{|\alpha {|}^2(+\frac{1}{2})+|\beta {|}^2(-\frac{1}{2})}{|\alpha {|}^2+|\beta {|}^2}\\ & =c_{+}(+\frac{1}{2})+c_{-}(-\frac{1}{2})\end{array}$$

One often chooses to simplify such calculations by normalizing states so that the denominator $⟨\psi |\psi ⟩$ is 1. Note that the same calculation works in general for the probability of measuring the various eigenvalues of an observable $A,$ , as long as one has orthogonality and completeness of eigenvectors.

In the case of a spin $\begin{array}{l}\frac{1}{2}\end{array}$ particle, the group $Spin(3)=SU(2)$ acts on states by the spinor representation with the element $\Omega (\theta ,\mathbf{w})\in SU(2)$ acting as

$$|\psi ⟩\to \Omega (\theta ,\mathbf{w})|\psi ⟩$$

As we saw in chapter 6, the $\Omega (\theta ,\mathbf{w})$ also act on self-adjoint matrices by conjugation, an action that corresponds to rotation of vectors when one makes the identification

$$\mathbf{v}\leftrightarrow \mathbf{v}\cdot \mathbf{\sigma }$$

(see equation 6.5). Under this identification the $S_j$ correspond (up to a factor of 2) to the basis vectors ${\mathbf{e}}_j$ . Their transformation rule can be written as

$$S_j\to S_j^{\prime }=\Omega (\theta ,\mathbf{w})S_j\Omega (\theta ,\mathbf{w}{)}^{-1}$$

and

$$\left(\begin{array}{c}{S}_1^{\prime }\\ S_2^{\prime }\\ S_3^{\prime }\end{array}\right)=R(\theta ,\mathbf{w}{)}^T\left(\begin{array}{c}{S}_1\\ S_2\\ S_3\end{array}\right)$$

Note that, recalling the discussion in section 4.1, rotations on sets of basis vectors like this involve the transpose $R(\theta ,\mathbf{w}{)}^T$ of the matrix $R(\theta ,\mathbf{w})$ that acts on coordinates.

Recalling the discussion in section 3.3, the spin degree of freedom that we are describing by $\mathcal{H}={\mathbf{C}}^2$ has a dynamics described by the Hamiltonian

$$H=-\mathbf{\mu }\cdot \mathbf{B}\text{\hspace{1em}(7.2)}$$

Here $B$ is the vector describing the magnetic field, and

$$\mu =g\frac{-e}{2mc}\mathbf{S}$$

is an operator called the magnetic moment operator. The constants that appear are: −$e$ the electric charge, $c$ the speed of light, $m$ the mass of the particle, and ${}^{g,}$ a dimensionless number called the “gyromagnetic ratio”, which is approximately 2 for an electron, about 5.6 for a proton.

The Schr¨odinger equation is

$$\frac{d}{dt}|\psi (t)⟩=-i(-\mu \cdot \mathbf{B})|\psi (t)⟩$$

with solution

$$|\psi (t)⟩=U(t)|\psi (0)⟩$$

where

$$U(t)=e^{it\mathbf{\mu }\cdot \mathbf{B}}=e^{it\frac{-ge}{2mc}\mathbf{S}\cdot \mathbf{B}}=e^{t\frac{ge}{2mc}\mathbf{X}\cdot \mathbf{B}}=e^{t\frac{ge|\mathbf{B}|}{2mc}\mathbf{X}\cdot \frac{\mathbf{B}}{|\mathbf{B}|}}$$

The time evolution of a state is thus given at time t by the same $SU(2)$ element that, acting on vectors, gives a rotation about the axis $\begin{array}{r}\mathbf{w}=\frac{\mathbf{B}}{|\mathbf{B}|}\end{array}$ by an angle

$$\frac{ge|\mathbf{B}|t}{2mc}$$

so is a rotation about w taking place with angular velocity $\frac{ge|\mathbf{B}|}{2mc}$

The amount of non-trivial physics that is described by this simple system is impressive, including:

• The Zeeman efect: this is the splitting of atomic energy levels that occurs when an atom is put in a constant magnetic field. With respect to the energy levels for no magnetic field, where both states in $\mathcal{H}={\mathbf{C}}^2$ have the same energy, the term in the Hamiltonian given above adds

$$\pm \frac{ge|\mathbf{B}|}{4mc}$$

to the two energy levels, giving a splitting between them proportional to the size of the magnetic field.

• The Stern-Gerlach experiment: here one passes a beam of spin $\begin{array}{l}\frac{1}{2}\end{array}$ quantum systems through an inhomogeneous magnetic field. We have not yet discussed particle motion, so more is involved here than the simple two-state system. However, it turns out that one can arrange this in such a way as to pick out a specific direction $w$, and split the beam into two components, of eigenvalue $+\frac{1}{2}$ and $-\frac{1}{2}$ for the operator $\mathbf{w}\cdot \mathbf{s}$

• Nuclear magnetic resonance spectroscopy: a spin $\frac{1}{2}$ can be subjected to a time-varying magnetic field $\mathbf{B}(t)$ , and such a system will be described by the same Schr¨odinger equation (although now the solution cannot be found just by exponentiating a matrix). Nuclei of atoms provide spin $\begin{array}{l}\frac{1}{2}\end{array}$ systems that can be probed with time and space-varying magnetic fields, allowing imaging of the material that they make up.

• Quantum computing: attempts to build a quantum computer involve trying to put together multiple systems of this kind (qubits), keeping them isolated from perturbations by the environment, but still allowing interaction with the system in a way that preserves its quantum behavior.

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