Unitary Evolution Operators

Suppose I change the state of a quantum system ``smoothly''. For example, I could move the system through space, or I could ``move it through time'' (i.e., just wait - hence the term, ``evolution operator''), or I could (surprise!) rotate it.

It happens quite generally in quantum mechanics that all such state changes are induced by unitary operators. Proving this clearly would require some physical assumptions, and I won't go into this at all. For our spinning electron, it turns out you can even assume a little more: the change of state caused by rotating the electron is induced by an operator in $SU(2)$. The operator in question is called a rotation operator.

How should we visualize the action of a rotation operator $R$ on a state vector $v$? We saw how to picture $R$ as a rotation in 3-space by looking at its effects on traceless Hermitian matrices: $A\mapsto RAR^*$, where $A=x\sigma_x+y\sigma_y+z\sigma_z$. How can we hook up the action of $R$ on state vectors with the action of $R$ on traceless Hermitian matrices? It seems we need a correspondence between states (say $(a:b)$) and matrices of the form $x\sigma_x+y\sigma_y+z\sigma_z$. We won't get quite this, but we'll get something just as good.

The trick is to set up a correspondence between states and yet another kind of matrix: a projection matrix. You probably noticed that the matrix $\sigma_z$ does a pretty good job specifying the state ``spin up along the z-axis''. As it turns out, $\sigma_z$ is not a projection matrix, but it corresponds in a natural fashion to $\frac{1}{2}{\bf 1}+\frac{1}{2}\sigma_z)$, which is.

Here's how it goes for an arbitrary state vector $v=a\vert{\rm up}\rangle +b\vert{\rm down}\rangle$. Suppose $v$ is normalized, so $\vert a\vert^2+\vert b\vert^2=1$. The projection matrix for $v$ is given by taking the product of the column vector $v$ with the row vector $v^*$:

$$vv^*= \left[\begin{array}{cc}a\\ b\end{array}\right][a^*,b... ...\left[\begin{array}{cc}aa^*&ab^*\\ a^*b&bb^*\end{array}\right]$$

(Standard physicists' notation is $\vert v\rangle$ for $v$ and $\langle v\vert$ for $v^*$. The product is $\vert v\rangle\langle v\vert$. The norm is $\langle v\vert v\rangle$. Mathematicians prefer to talk about a vector space and its dual instead of column vectors and row vectors, but these notes prefer concreteness to elegance.)

Now, $vv^*$ is a Hermitian matrix with determinant 0 (check!). It must therefore take the form:

$$\left[ \begin{array}{cc} t+z & x-iy \\ x+iy & t-z \end... ...{\bf 1}+x\sigma_x+y\sigma_y+z\sigma_z, \quad t^2-x^2-y^2-z^2=0$$

(If the appearence of $t^2-x^2-y^2-z^2$ makes you think ``Special Relativity!'', you're on the right track, but I won't get into that.) However, the trace is not 0, but $aa^*+bb^*=2t$. Since I took $v$ to be normalized ($\vert a\vert^2+\vert b\vert^2=1$), it follows that $t=\frac{1}{2}$.

So we have a mapping from state vectors to Hermitian matrices of the form $\frac{1}{2}{\bf 1}+x\sigma_x+y\sigma_y+z\sigma_z$ with $x^2+y^2+z^2=\frac{1}{4}$. And the latter are in an obvious one-one correspondence with points on a sphere of radius one-half.

The mapping $v\mapsto vv^*$ (restricted to normalized vectors) actually establishes a one-one correspondence between states and our special class of Hermitian matrices. For let $c$ be a complex number of norm 1; then $(cv)(cv)^*=vv^*$.

Why do I call $vv^*$ a projection matrix? Answer: by analogy with projections in ordinary real vector spaces, say ${\bf R}^3$. If $v$ is a vector of norm 1, and $w$ is an arbitrary vector, then the projection of $w$ ``along the vector $v$'' (i.e., in the subspace spanned by $v$) is $(v\cdot w)v$. Analogous to this, we define ${\rm proj}_v(w)=\langle v,w\rangle v$, using the notation $\langle v,w\rangle$ for the inner product. In the ``row vector, column vector'' notation, this is $(v^*w)v=v(v^*w)=vv^*w$. In physicists' notation, this is $\vert v\rangle\langle v\vert w\rangle$.

We have acquired a new way of picturing the action of $SU(2)$ on 3-space. The formula $v\mapsto Avv^*A^*$ captures it succinctly. The mapping $v\mapsto vv^*$ sets up a one-one correspondence between the states $(a:b)$ (i.e., the complex projective line) and points on a sphere in 3-space. In fact this is just the Riemann sphere mapping!

So the quantum states for the spin of an electron can be pictured as points on a sphere. Elements of $SU(2)$ correspond to the change in state induced by rotating the electron, and this action of $SU(2)$ can be pictured as a rotation of the sphere. The naive pictures match up with the $SU(2)$ formalism flawlessly. The element $-{\bf 1}$ of $SU(2)$ induces the identity mapping on the space of states, since $v$ and $-v$ represent the same quantum state.

A simple computation illustrates how everything meshes. The rotation operator for a clockwise $90^\circ$ rotation about the y-axis is $\frac{1}{\sqrt{2}}({\bf 1}+i\sigma_y)$. Indeed, if you work out $({\bf 1}+i\sigma_y)\sigma_x({\bf 1}-i\sigma_y)$, you get $2\sigma_z$, and likewise $({\bf 1}+i\sigma_y)\sigma_z({\bf 1}-i\sigma_y)=-2\sigma_x$. The x-axis maps to the z-axis, and the z-axis maps to minus the x-axis.

The example of electron spin illustrates two features of quantum mechanics very clearly.

• Probabilistic character: The quantum state does not uniquely determine the result of experiment. Hence Einstein's famous complaint, ``I shall never believe that God plays dice with the universe.'' (Perhaps he plays cards with the physicists?)
• Correspondence with classical physics: Classical physics emerges from quantum mechanics by taking averages.

Some more technical features, also embodied in this example, and typical of quantum mechanics:

• Need for complex numbers: The neat correspondence with the classical spinning ball picture wouldn't work if we did everything over R.
• Non-commuting observables: You cannot simultaneously measure the x and z components of spin (for example), because $\sigma_x$ and $\sigma_z$ do not commute.
• Symmetry groups and observables: The rotation symmetry group gives rise indirectly to the $\sigma$ matrices, and ultimately to the notion of angular momentum. The mathematical basis is the Lie groupLie algebra correspondence.

Had I started with the first historical example, the single spinless particle coasting in space, I would be illustrating the same morals with different actors:

• Need for complex numbers: The appropriate Hilbert space is the space of complex-valued $L^2$ functions on 3-space.
• Non-commuting observables: The momentum and position operators do not commute, and you cannot simultaneously measure position and momentum.
• Symmetry groups and observables: The group of translations in 3-space gives rise to Lie group acting on the $L^2$ Hilbert space; the momentum operator emerges from the corresponding Lie algebra.

© 2001 Michael Weiss

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