*Michael Fowler 10/1/08*

The momentum-position uncertainty principle _{} has an energy-time
analog, _{}. Evidently, though,
this must be a different kind of relationship to the momentum-position one,
because *t* is not a dynamical
variable, so this can’t have anything to do with non-commutation.

To illustrate the meaning of the equation _{}, let us reconsider _{}-decay, but with a slightly simplified potential to clarify
what’s going on. Specifically, we replace the combined nuclear
force/electrostatic repulsion barrier with a square barrier, high enough and
thick enough that there is a small probability per unit time of the particle
tunneling out of the well.

If the barrier thickness were increased to infinity (keeping
*r*_{0} fixed) there would be a
true bound state having energy *E*_{0},
and for *E*_{0} well below *V*_{0}, having approximately an
integral number of half wavelengths in the well. For a barrier of finite thickness, there is of
course some nonzero probability of the particle escaping—so no longer a true
bound state, but for a thick barrier the difference may be hard to detect.

As with the _{}-decay analysis, we’ll look at this semiclassically, picturing the particle as bouncing off the
walls backwards and forwards inside, time _{} between hits, and at
each hit probability of penetration some small number _{}. Therefore, the
probability that the particle is still in the well after a time _{} is _{}. Since _{} really is *very* small for _{}-decay (less than 10^{-12}), we can conveniently
write *P* as a function of time by using
the formula _{}.

From this, the probability of the particle being in the well
after *n* bounces, time _{},

_{}

The standard notation is to introduce a variable _{} having the dimensions
of *energy* (_{} being dimensionless), in terms of which

_{}

Evidently_{} measures the *decay rate*.
In other words, _{} is an inverse
lifetime.

The exponential decay law for radioactive elements is completely confirmed experimentally, it is the basis of the “half-life” rule: for any given amount of a radioactive nucleus, half of it will decay in a time period—the half-life—fixed for that nucleus.

Obviously, if the modulus of the wave function of the
particle in the well is decreasing with time, the time dependence of _{} can no longer be just the
_{} of the original “bound
state”. The wave function inside the
well will stay the same shape, but gradually decrease in amplitude:

_{}.

This is a far slower time dependence than that of the _{} term, so it is an
excellent approximation to put the time dependences together in one exponential
factor:

_{}

At this point, the analogy with _{} emerges. Recall we introduced the *p*, *x* uncertainty
principle by finding what spread _{} in the Fourier
components of a wave packet were necessary in order for the wave packet to die
away from its center over a distance of order _{} A true localized wave
packet has a continuum of *p* components,
but the right expression for the spread in momentum space _{} turns out to be given
by taking just two waves, _{}, and noticing that they fall out of phase in a distance _{}. The more
precise derivation based on Gaussian wave packets reaches essentially the same
conclusion.

Now, in the present situation the wave function decays in *time* rather than space, but the argument
is very similar. To construct the decaying
wave function we must add together “plane waves in time” _{} corresponding to
different energies. The required spread
in energy _{} follows from an
argument just like the one above for space: if the wave function is to be die
away in a time of order _{} in other words to be
“localized in time”, it must be constructed of waves having a range _{} in energies such that _{}and _{}get out of sync in a time _{} This gives
immediately _{}.

It’s worth looking a little further into just what superposition
of energy “plane waves” _{}gives the required exponential-in-time behavior of the wave
function for _{}-decay. Writing

_{}

the Fourier coefficients are given by

_{}.

This tells us that the energy with which the _{}-particle emerges has a probability distribution which is easily
normalized to give

_{}.

This distribution (called *Lorentzian*) has a narrow peak of width of order _{} and height of order _{} centered at *E*_{0}. (Strictly speaking, this is an approximation
in that *c*(*E*) must of course be zero for *E*
negative—we ignore that tiny correction here.)

In fact, for the _{}-decaying nucleus, this energy spread is undetectably small,
but that is certainly *not* the case
for other decaying states, where this same analysis applies. In particular, some of the resonant states
created in collisions of elementary particles have masses of order 1,000 Mev,
and lifetimes of order 10^{-23} seconds—corresponding to a width _{}of the order of 10% of *E*_{0}! Obviously, these transient bound states are
far from eigenstates of the Hamiltonian—but you will find them listed in
particle tables.

The *time-reversed*
wave function for _{}-decay is also a perfectly good solution to Schrödinger’s
equation. In principle, if we could
arrange for an _{}-particle to have a spherical ingoing wave function within
the narrow energy range corresponding to the quasi bound state, _{} would become very
large inside the nucleus, meaning that the _{}-particle would spend a very long time there compared with
the time spent at any other point on the way in: this is called *resonance*. This particular experiment is never likely to
take place, but precisely analogous experiments involving particles scattering
off each other, and electrons scattering resonantly from atoms, are common.

One can interpret the wave function for a *true* bound state as a standing wave,
radially containing a whole number of half wavelengths, so that when the wave
is reflected at the walls it has just the right phase to interfere
constructively with itself. A resonance
will have the same wavelength requirement (but the reflection at the walls is of
course no longer perfect).

We shall see later that in the case of nonzero angular momentum, the centrifugal barrier provides an effective repulsive term, which together with an attractive force can create a barrier configuration having similar effect to that in the toy model.

It is instructive to use the spreadsheet to
discover how the maximum value of _{} varies as the barrier
height is increased, and to vary the energy to explore how _{} varies with barrier
height.

The pictures below are for the energy right at resonance for
two different barrier heights.
Evidently, if the height of the barrier is increased, _{}inside at resonance increases as well. This is easy to
understand—the higher barrier is more difficult to penetrate, so the particle
spends even longer inside. The lifetime _{} is clearly
proportional to _{}inside.

*Important exercise*: Open the spreadsheet, find the resonant
energy for these cases, and then detune the energy away from resonance. As the energy moves away from resonance, which
_{} drops to half its
maximum value first?

This energy spread of the resonance is the _{} above, and is related
to the “lifetime” of the resonance by _{}. How does the ratio
of the two _{}’s for these two barriers relate to the ratio of the two
maximum values of _{}? Give a physical
explanation of your findings.