34.2 Solutions of the Pauli equation and representations of E(3)

34.2 Solutions of the Pauli equation and representations of E(3)

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

Since

$$\frac{\mathbf{\sigma }\cdot \mathbf{p}}{|\mathbf{p}|}$$

is an invertible operator with eigenvalues ±1, solutions to 34.4 will be given by solutions to

$$\frac{\mathbf{\sigma }\cdot \mathbf{p}}{|\mathbf{p}|}\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_{E,1}(\mathbf{p})}{{\tilde{\psi }}_{E,2}(\mathbf{p})}\right)=\pm \left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_{E,1}(\mathbf{p})}{{\tilde{\psi }}_{E,2}(\mathbf{p})}\right)\text{\hspace{1em}(34.6)}$$

where $|\mathbf{p}|=\sqrt{2mE}$ . We will write solutions to this equation with the + sign as ${\psi }_{E,+}(\mathbf{p})$ , those for the − sign as ${\tilde{\psi }}_{E,-}(\mathbf{p})$ . Note that ${\tilde{\psi }}_{E,+}(\mathbf{p})$ and ${\tilde{\psi }}_{E,-}(\mathbf{p})$ are each two-component complex functions on the sphere $\sqrt{2mE}=\left|\mathbf{p}\right|$ (or, more generally distributions on the sphere). Our goal in the rest of this section will be to show

Theorem

The spaces ofsolutions ${\tilde{\psi }}_{E,\pm }(\mathbf{p})$ to equations $34.6$ provide irreducible representations of $\stackrel{\sim }{E}(3)$ , the double cover of $E(3)$ , with eigenvalue 2mE for the first Casimir operator

$${\left|\mathbf{P}\right|}^2=(\mathbf{\sigma }\cdot \mathbf{P}{)}^2$$

and eigenvalues $\pm \frac{1}{2}\sqrt{2mE}$ for the second Casimir operator $J$ · P.

We will not try to prove irreducibili $\mathrm{J},$ but just show that these solution spaces give representations with the claimed eigenvalues of the Casimir operators (see sections 19.2 and 19.3 for more about the Casimir operators and general theory of representations of $E(3))$ . We will write the representation operators as $u(\mathbf{a},\Omega )$ in position space and $\tilde{u}(\mathbf{a},\Omega )$ in momentum space, with a a translation, $\Omega \in$ $SU(2)$ and $R=\Phi (\Omega )\in SO(3)$

The translation part of the group acts as in the one-component case of chapter 19, by the multiplication operator

$$\tilde{u}(\mathbf{a},1){\tilde{\psi }}_{E,\pm }(\mathbf{p})=e^{-i\mathbf{a}\cdot \mathbf{p}}{\tilde{\psi }}_{E,\pm }(\mathbf{p})$$

and

$$\tilde{u}(\mathbf{a},1)=e^{-i\mathbf{a}\cdot \mathbf{P}}$$

so the Lie algebra representation is given by the usual $\mathbf{P}$ operator. This action of the translations is easily seen to commute with $\sigma$ · P and thus act on the solutions to 34.6. It is the action of rotations that requires a more complicated discussion than in the single-component case.

In chapter 19 we saw that $R\in SO(3)$ acts on single-component momentum space solutions of the Schr¨odinger equation by

$${\tilde{\psi }}_E(\mathbf{p})\to \tilde{u}(0,R){\tilde{\psi }}_E(\mathbf{p})={\tilde{\psi }}_E(R^{-1}\mathbf{p})$$

This takes solutions to solutions since the operator $\tilde{u}(0,R)$ commutes with the Casimir operator $|\mathbf{P}{|}^2$

$$\tilde{u}(0,R)|\mathbf{P}{|}^2=|\mathbf{P}{|}^2\tilde{u}(0,R)⟺\tilde{u}(0,R)|\mathbf{P}{|}^2\tilde{u}(0,R{)}^{-1}=|\mathbf{P}{|}^2$$

This is true since

$$\begin{array}{rl}\tilde{u}(0,R)|\mathbf{P}{|}^2\tilde{u}(0,R{)}^{-1}\tilde{\psi }(\mathbf{p})& =\tilde{u}(0,R)|\mathbf{P}{|}^2\tilde{\psi }(R\mathbf{p})\\ & =|R^{-1}\mathbf{P}{|}^2\tilde{\psi }(R^{-1}R\mathbf{p})=|\mathbf{P}{|}^2\tilde{\psi }(\mathbf{p})\end{array}$$

To get a representation on two-component wavefunctions that commutes with the operator $\mathbf{\sigma }\cdot \mathbf{P}$ we need to change the action of rotations to

$${\tilde{\psi }}_{E,\pm }(\mathbf{p})\to \tilde{u}(0,\Omega ){\tilde{\psi }}_{E,\pm }(\mathbf{p})=\Omega {\tilde{\psi }}_{E,\pm }(R^{-1}\mathbf{p})$$

With this action on solutions we have

$$\begin{array}{rl}\tilde{u}(0,\Omega )(\sigma \cdot \mathbf{P})\tilde{u}(0,\Omega {)}^{-1}{\tilde{\psi }}_{E,\pm }(\mathbf{p})& =\tilde{u}(0,\Omega )(\sigma \cdot \mathbf{P}){\Omega }^{-1}{\tilde{\psi }}_{E,\pm }(R\mathbf{p})\\ & =\Omega (\sigma \cdot R^{-1}\mathbf{P}){\Omega }^{-1}{\tilde{\psi }}_{E,\pm }(R^{-1}R\mathbf{p})\\ & =(\sigma \cdot \mathbf{P}){\tilde{\psi }}_{E,\pm }(\mathbf{p})\end{array}$$

where we have used equation $6.5$ to show

$$\Omega (\mathbf{\sigma }\cdot R^{-1}\mathbf{P}){\Omega }^{-1}=\mathbf{\sigma }\cdot R{R}^{-1}\mathbf{P}=\mathbf{\sigma }\cdot \mathbf{P}$$

The $SU(2)$ part of the group acts by a product of two commuting diferent actions on the two factors of the tensor product 34.2. These are:

1. The same action on the momentum coordinates as in the one-component case, just using $R=\Phi (\Omega )$ , the $SO(3)$ rotation corresponding to the $SU(2)$ group element Ω. For example, for a rotation about the x-axis by angle $\varphi$ we have

$${\tilde{\psi }}_{E,\pm }(\mathbf{p})\to {\tilde{\psi }}_{E,\pm }(R(\varphi ,{\mathbf{e}}_1{)}^{-1}\mathbf{p})$$

Recall that the operator that does this is $e^{-i\varphi L_1}$ where

$$-i{L}_1=-i(Q_2{P}_3-Q_3{P}_2)=-\left(q_2\frac{\partial }{\partial q_3}-q_3\frac{\partial }{\partial q_2}\right)$$

and in general we have operators

$$-i\mathbf{L}=-i\mathbf{Q}\times \mathbf{P}$$

that provide the Lie algebra version of the representation (recall that at the Lie algebra level, $SO(3)$ and $Spin(3)$ are isomorphic).

2. The action of the matrix $\Omega \in SU(2)$ on the two-component wavefunction by

$${\tilde{\psi }}_{E,\pm }(\mathbf{p})\to \Omega {\tilde{\psi }}_{E,\pm }(\mathbf{p})$$

For R a rotation by angle $\varphi$ about the x-axis one choice of Ω is

$$\Omega =e^{-i\varphi \frac{{\sigma }_1}{2}}$$

and the operators that provide the Lie algebra version of the representation are the 1

$$-i\mathbf{S}=-i\frac{1}{2}\sigma$$

The Lie algebra representation corresponding to the action of these transformations on the two factors of the tensor product is given as usual (see chapter 9) by a sum of operators that act on each factor

$$-i\mathbf{J}=-i(\mathbf{L}+\mathbf{S})$$

The standard terminology is to call $L$ the “orbital” angular momentum, $S$ the “spin” angular momentum, and J the “total” angular momentum.

The second Casimir operator for this case is

J · P

and as in the one-component case (see section 19.3) a straightforward calculation shows that the L · P part of this acts trivially on our solutions ${\tilde{\psi }}_{E,\pm }(\mathbf{p})$ . The spin component acts non-trivially and we have

$$(\mathbf{J}\cdot \mathbf{P}){\tilde{\psi }}_{E,\pm }(\mathbf{p})=(\frac{1}{2}\sigma \cdot \mathbf{p}){\tilde{\psi }}_{E,\pm }(\mathbf{p})=\pm \frac{1}{2}|\mathbf{p}|{\tilde{\psi }}_{E,\pm }(\mathbf{p})$$

so we see that our solutions have helicity (eigenvalue of J · P divided by the square root of the eigenvalue of $|\mathbf{P}{|}^2)$ values $\pm \frac{1}{2}.$ , as opposed to the integral helicity values discussed in chapter 19, where $E(3\overline{)}$ appeared and not its double cover. These two representations on the spaces of solutions ${\tilde{\psi }}_{E,\pm }(\mathbf{p})$ are thus the $E(3)$ representations described in section 19.3, the ones labeled by the helicity $\pm \frac{1}{2}$ representations of the stabilizer group SO(2).

Solutions for either sign of equation 34.6 are given by a one dimensional subspace of ${\mathbf{C}}^2$ for each $\mathbf{p},$ and it is sometimes convenient to represent them as follows. Note that for each p one can decompose

$${\mathbf{C}}^2=\mathbf{C}\oplus \mathbf{C}$$

into ±-eigenspaces of the matrix $\frac{\sigma \cdot \mathbf{p}}{|\mathbf{p}|}$ . In our discussion of the Bloch sphere in section 7.5 we explicitly found that (see equation 7.6)

$$u_{+}(\mathbf{p})=\frac{1}{\sqrt{2(1+p_3)}}\left(\genfrac{}{}{0ex}{}{1+p_3}{p_1+i{p}_2}\right)\text{\hspace{1em}(34.7)}$$

provides a normalized element of the + eigenspace of $\frac{\sigma \cdot \mathbf{p}}{|\mathbf{p}|}$ that satisfies

$$\Omega u_{+}(\mathbf{p})=u_{+}(R\mathbf{p})$$

Similarly, we saw that

$$u_{-}(\mathbf{p})=\frac{1}{\sqrt{2(1+p_3)}}\left(\genfrac{}{}{0ex}{}{-(p_1-i{p}_2)}{1+p_3}\right)\text{\hspace{1em}(34.8)}$$

provides such an element for the − eigenspace.

Another way to construct such elements is to use projection operators. The operators

$$P_{\pm }(\mathbf{p})=\frac{1}{2}(1\pm \frac{\sigma \cdot \mathbf{p}}{|\mathbf{p}|})$$

provide projection operators onto these two spaces, since one can easily check that

$$P_{+}^2=P_{+},P_{-}^2=P_{-},P_{+}{P}_{-}=P_{-}{P}_{+}=0,P_{+}+P_{-}=1$$

Solutions can now be written as

$${\tilde{\psi }}_{E,\pm }(\mathbf{p})={\alpha }_{E,\pm }(\mathbf{p})u_{\pm }(\mathbf{p})\text{\hspace{1em}(34.9)}$$

for arbitrary functions ${\alpha }_{E,\pm }(\mathbf{p})$ on the sphere $|\mathbf{p}|=\sqrt{2mE}$ , where the $u_{\pm }(\mathbf{p})$ in this context are called “spin polarization vectors” . There is however a subtlety involved in representing solutions in this manner. Recall from section 7.5 that $u_{+}(\mathbf{p})$ is discontinuous at $p_3=-1$ (the same will be true for $u_{-}(\mathbf{p}))$ and any unit-length eigenvector of $\mathbf{\sigma }\cdot \mathbf{p}$ must have such a discontinuity somewhere. If ${\alpha }_{E,\pm }(\mathbf{p})$ has a zero at $p_3=-1$ the product ${\tilde{\psi }}_{E,\pm }(\mathbf{p})$ can be continuous. It remains a basic topological fact that the combination ${\tilde{\psi }}_{E,\pm }(\mathbf{p})$ must have a zero, or it will have to be discontinuous. Our choice of $u_{\pm }(\mathbf{p})$ works well if this zero is at $p_3=-1$ , but if it is elsewhere one might want to make a diferent choice. In the end one needs to check that computed physical quantities are independent of such choices.

Keeping in mind the above subtlety, the $u_{\pm }(\mathbf{p})$ can be used to write an arbitrary solution of the Pauli equation 34.3 of energy $E$ as

$$\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q},t)}{{\psi }_2(\mathbf{q},t)}\right)=e^{-iEt}\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)$$

where

$$\begin{array}{rl}\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)& =\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}\delta (|\mathbf{p}{|}^2-2mE)({\tilde{\psi }}_{E,+}(\mathbf{p})+{\tilde{\psi }}_{E,-}(\mathbf{p}))e^{i\mathbf{p}\cdot \mathbf{q}}{d}^3\mathbf{p}\\ & =\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}\delta (|\mathbf{p}{|}^2-2mE)({\alpha }_{E,+}(\mathbf{p})u_{+}(\mathbf{p})+{\alpha }_{E,-}(\mathbf{p})u_{-}(\mathbf{p}))e^{i\mathbf{p}\cdot \mathbf{q}}{d}^3\mathbf{p}\end{array}\text{\hspace{1em}(34.10)}$$

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