Simple Nature - Light and Matter

the linearity of the Schr ̈odinger equation, the result is that the mea-

suring apparatus or observer ends up in a Schr ̈odinger’s-cat state

that looks like

α|observedv〉+β|observedw〉.

We interpret squares of amplitudes as probabilities, so

P=|α|^2 =|〈u|v〉|^2

gives us the probability that we will have observed the state to be

v. This final leap in the logic, to a probability interpretation, has

felt mysterious to several generations of physicists, but recent work

has clarified the situation somewhat.

On p. 938 we stated the Pauli exclusion principle by saying that

two particles with half-integer spins could never occupy the same

state. This was not a completely rigorous definition of the principle,

since we didn’t really define “same state.” A more mathematically

precise statement is that if one electron’s wavefunction isu and

another’s isv, then〈u|v〉= 0. In other words, we are ruling out not

just the case whereuandvare the same wavefunction,〈u|v〉= 1,

but also the intermediate case where〈u|v〉is greater than 0 but less

than 1.

A unitary transformation is one that preserves inner products.

That is,〈Ou|Ov〉=〈u|v〉. This is similar to the way in which rota-

tions preserve dot products in Euclidean geometry. This provides a

more rigorous definition of what we meant by postulating the uni-

tary evolution of the wavefunction (p. 969). It can be shown that if

the Hamiltonian is hermitian, then the evolution of the wavefunc-

tion over time is a unitary operation. This protects us from bad

scenarios like the one described in example 14, p. 989.

Traveling waves in the quantum moat example 12

On p. 920 we discussed the “quantum moat,” in which a parti-

cle is constrained to a circle like the moat around a castle. For

the`= 1 state, the two degenerate traveling wave solutions to

the Schrodinger equation are (ignoring normalization) the coun- ̈

terclockwise|ccw〉=eiθand the clockwise|cw〉=e−iθ. These

states are distinguishable by their angular momenta`z=±1, so

we expect them to be orthogonal. Let’s check that directly.

〈ccw|cw〉=

∫ 2 π

0

[

(eiθ)∗

]

e−iθdθ

=

∫ 2 π

0

e−iθe−iθdθ

=

∫ 2 π

0

e−^2 iθdθ

984 Chapter 14 Additional Topics in Quantum Physics