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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 . The operator in question is called a rotation operator.

How should we visualize the action of a rotation operator on a state vector ? We saw how to picture as a rotation in 3-space by looking at its effects on traceless Hermitian matrices: , where . How can we hook up the action of on state vectors with the action of on traceless Hermitian matrices? It seems we need a correspondence between states (say ) and matrices of the form . 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 does a pretty good job specifying the state spin up along the z-axis''. As it turns out, is not a projection matrix, but it corresponds in a natural fashion to , which is.

Here's how it goes for an arbitrary state vector . Suppose is normalized, so . The projection matrix for is given by taking the product of the column vector with the row vector :

(Standard physicists' notation is for and for . The product is . The norm is . 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, is a Hermitian matrix with determinant 0 (check!). It must therefore take the form:

(If the appearence of 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 . Since I took to be normalized (), it follows that .

So we have a mapping from state vectors to Hermitian matrices of the form with . And the latter are in an obvious one-one correspondence with points on a sphere of radius one-half.

The mapping (restricted to normalized vectors) actually establishes a one-one correspondence between states and our special class of Hermitian matrices. For let be a complex number of norm 1; then .

Why do I call a projection matrix? Answer: by analogy with projections in ordinary real vector spaces, say . If is a vector of norm 1, and is an arbitrary vector, then the projection of along the vector '' (i.e., in the subspace spanned by ) is . Analogous to this, we define , using the notation for the inner product. In the row vector, column vector'' notation, this is . In physicists' notation, this is .

We have acquired a new way of picturing the action of on 3-space. The formula captures it succinctly. The mapping sets up a one-one correspondence between the states (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 correspond to the change in state induced by rotating the electron, and this action of can be pictured as a rotation of the sphere. The naive pictures match up with the formalism flawlessly. The element of induces the identity mapping on the space of states, since and represent the same quantum state.

A simple computation illustrates how everything meshes. The rotation operator for a clockwise rotation about the y-axis is . Indeed, if you work out , you get , and likewise . 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 and do not commute.
• Symmetry groups and observables: The rotation symmetry group gives rise indirectly to the 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 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 Hilbert space; the momentum operator emerges from the corresponding Lie algebra.

Next: Loose Ends Up: Quantum Mechanics: Two-state Systems Previous: Hermitian Observables