ing states according to their`z. You can simulate experiments like
this using an app at physics.weber.edu/schroeder/software/
Spins.html. We have already found that the wavefunction of the
intermediate beam is equal to the sum|`z=− 1 〉/√
2 +|`z= 1〉/√
2,
so interpreting squares of amplitudes as probabilities we predict a
probability (1/√
2)^2 = 1/2 that each particle will be measured to
have`z =−1, the same probability for−1, and zero probability
for 0. As explored in discussion question C on p. 959, this doesnot
mean that the two beams that emerge from the second spectrometer
have definite values of both`xand`z; those two observables are not
compatible.
In most of the examples we’ve encountered so far, it has been
possible to think of the “wavefunction” as exactly what the word
implies: a mathematical function ofx(and possibly also ofy, and
z), whose shape we visualize as a wave. The completeness principle,
however, does not assign any special role to the position operator,
nor does quantum mechanics in general. And there are cases where
we do not even have the option of resorting to the picture of a wave
that exists in space. For example, the intrinsic angular momentum
~/2 of an electron is not a possible amount of angular momentum for
a particle to generate by moving through space. In section 14.7.1,
p. 990, we will discuss a very simple quantum-mechanical system
consisting of an electron, at rest, surrounded by a uniform magnetic
field. In this example, the motion of the electron through space
is not even of interest, and a complete set of observables simply
consists ofLandLz(orsandsz, in notation that emphasizes that
we’re talking about intrinsic spin).14.6.4 The Schrodinger equation in general ̈
This raises the question of what we mean by “the Schr ̈odinger
equation” in cases where nothing is being expressed as a function
ofx. The basic idea of the Schr ̈odinger equation is that a parti-
cle’s energy is related to its frequency byE=hf, orE=~ω. In
the form of the time-dependent Schr ̈odinger equation that we have
discussed on p. 967,i~∂Ψ/∂t=−(~^2 / 2 m)∇^2 Ψ +UΨ, the quantity
on the right-hand side of the equation is just the energy operator
acting on the wavefunction. So to generalize this to cases where
the wavefunction isn’t expressed in terms ofx, we just make that
substitution:
i~∂Ψ
∂t=OEΨ.
This is as good a point as any to introduce a not-very-memorable
piece of terminology, which is that the energy operator in quantum
mechanics is called theHamiltonian, after W.R. Hamilton. There
is a classical version of the Hamiltonian, which is usually a syn-
onym for the energy of a system, although it turns out that there
comes more plausible if you consider the randomness of the unpolarized beam
as being defined by its having maximum entropy.988 Chapter 14 Additional Topics in Quantum Physics