Simple Nature - Light and Matter

(Martin Jones) #1
two states, one in which the atom has decayed and your brain has
observed that fact, and one in which the atom has not yet decayed
and that fact is instead recorded in your brain.
To get more of a feeling for what is meant by unitarity, it may be
helpful to consider some examples of how it could be violated. One is
the mythical “collapse” of the wavefunction in naive interpretations
of the Copenhagen approximation (p. 887). Another example of
nonunitarity is given in example 14 on p. 989.
A more exotic example is the disappearance of matter into a
black hole. If I throw my secret teenage diary into a black hole,
then it contributes a little bit to the black hole’s mass, but the
embarrassing information on the pages is lost forever. This loss
of information seems to imply nonunitarity. This is one of several
arguments suggesting that quantum mechanics cannot fully handle
the gravitational force. Thus although physicists currently seem to
possess a completely successful theory of gravity (Einstein’s theory
of general relativity) and a completely successful theory of the mi-
croscopic world (quantum mechanics), the two theories are irrecon-
cilable, and we can only make educated guesses, for example, about
the behavior of a hypothetical microscopic black hole.

14.5 Methods for solving the Schrodinger equa- ̈
tion

14.5.1 Cut-and-paste solutions
Quite a few of the interesting phenomena of quantum mechanics
can be demonstrated by finding solutions to the one-dimensional
Schr ̈odinger equation using the following “cut and paste” method.
We break up thexaxis into pieces, where the potentialU(x) does
different things, and such that we already know the solutions of the
Schr ̈odinger equation for each piece. We then splice together the
different parts of the solution, requiring that no discontinuities occur
in the wavefunction Ψ or its derivative∂Ψ/∂x. (If the momentum
and kinetic energy are to be finite, andUis finite, then we need all
derivatives up to the second to be defined.)

Partial reflection at a step
The simplest example of this kind is a potential step,

U(x) =

{


U 1 , x < 0
U 2 , x >0,

whereU 1 andU 2 are constants, and the energy of the particle is
such that both sides are classically allowed. We have discussed this
example 18 on p. 907, where we cheated by drawing real-valued
wavefunctions, and simply assumed that we could still use our pre-
vious results for classical wave reflection (p. 381). It is not actually

970 Chapter 14 Additional Topics in Quantum Physics

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