so is a rotation aboutwtaking place with angular velocityge 2 mc|B|.
The amount of non-trivial physics that is described by this simple system is
impressive, including:
- The Zeeman effect: 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 inH=C^2 have the
same energy, the term in the Hamiltonian given above adds
±
ge|B|
4 mc
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^12 quantum
systems through an inhomogeneous magnetic field. We have not yet dis-
cussed 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 directionw, and split the beam into two components,
of eigenvalue +^12 and−^12 for the operatorw·S. - Nuclear magnetic resonance spectroscopy: a spin^12 can be subjected to
a time-varying magnetic fieldB(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^12
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 try-
ing to put together multiple systems of this kind (qubits), keeping them
isolated from perturbations by the environment, but still allowing inter-
action with the system in a way that preserves its quantum behavior.
7.3 The Heisenberg picture
The treatment of time-dependence so far has used what physicists call the
“Schr ̈odinger picture” of quantum mechanics. States inHare functions of time,
obeying the Schr ̈odinger equation determined by a Hamiltonian observableH,
while observable self-adjoint operatorsOare time-independent. Time evolution
is given by a unitary transformation
U(t) =e−itH, |ψ(t)〉=U(t)|ψ(0)〉U(t) can instead be used to make a unitary transformation that puts the
time-dependence in the observables, removing it from the states, giving some-
thing called the “Heisenberg picture.” This is done as follows:
|ψ(t)〉→|ψ(t)〉H=U−^1 (t)|ψ(t)〉=|ψ(0)〉, O →OH(t) =U−^1 (t)OU(t)