Simple Nature - Light and Matter

are cases where it is not the same, e.g., when we adopt a rotating
frame of reference. In both classical and quantum mechanics, the
Hamiltonian is what determines the time-evolution of a system; in
quantum mechanics, this is because it is the Hamiltonian that oc-
curs in the Schr ̈odinger equation. Because the Hamiltonian occurs
so frequently, we will notate it asHˆ rather than the more cumber-
someOE, where the hat is to remind us that it is an operator. A
similar notation can be used for other operators when it is easier to
write, e.g., ˆszrather than the clumsyOsz. In the hat notation, the
time-dependent Schr ̈odinger equation looks like this:

i~

∂Ψ

∂t

=HˆΨ.

An illegal energy operator example 14

We have pointed out on p. 976 some reasons to think that it would

be bad to have a quantum-mechanical observable whose eigen-

values were not real, i.e., one represented by a non-hermitian

operator (p. 982). Even worse things happen if we try to use a

non-hermitian operator for our energy operator, the Hamiltonian.

As the simplest possible example, consider a system consisting

of a particle at rest, and the Hamiltonian defined by

HˆΨ=i kΨ,

wherekis a nonzero real constant with units of energy. That is,

the energy of the system is a constant value, which is the imagi-

nary numberi k. This operator has a single eigenvalue,i k, which

is not real. The fact that it has a non-real eigenvalue is equivalent

to a statement that it is non-hermitian (problem 8, p. 1009). If we

plug this in to the Schrodinger equation, we get ̈ i~∂Ψ/∂t=i kΨ,

or

∂Ψ

∂t

=

k

~

Ψ.

This differential equation is not hard to solve by the guess-and-

check method. A function whose derivative is itself (except for a

multiplicative constant) is an exponential. The solution is

Ψ=Ae(k/~)t,

whereAis a constant. This is bad. Very bad. IfΨis properly nor-

malized att= 0, then it will not be normalized at other times. Ifk

is positive, then the total probability will become greater than 1 for

t>0, which we could perhaps interpret as meaning that the par-

ticle is spawning more copies of itself. Almost as bad is the case

ofk<0, for which the particle exponentially vanishes into noth-

ingness like the Cheshire cat. Either behavior would violate the

principle of the unitary evolution of the wavefunction (p. 969).

Section 14.6 The underlying structure of quantum mechanics, part 2 989