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Path Integrals in Quantum Mechanics
Michael Fowler, UVa
Huygen’s Picture of Wave Propagation
If a point source of light is switched on, the wavefront is
an expanding sphere centered at the source.
Huygens suggested that this could be understood if at any instant in
time each point on the wavefront was regarded as a source of secondary
wavelets, and the new wavefront a moment later was to be regarded as built up
from the sum of these wavelets. For a light shining continuously, this process
just keeps repeating.
What use is this idea? For one thing, it explains
refraction—the change in
direction of a wavefront on entering a different medium, such as a ray of light
going from air into glass.
If the light moves more slowly in the glass, velocity instead of with then Huygen’s picture explains Snell’s Law,
that the ratio of the sines of the angles to the normal of incident and
transmitted beams is constant, and in fact is the ratio This is evident from the diagram below: in
the time the wavelet centered at A has propagated to C,
that from B has reached D, the ratio of lengths AC/BD
being But the angles in Snell’s Law are in fact the
angles ABC, BCD, and those right-angled
triangles have a common hypotenuse BC, from which the Law follows.
Fermat’s Principle of Least Time
We will now temporarily forget about the wave nature of
light, and consider a narrow ray or beam of light shining from point A to
point B, where we suppose A to be in air, B in glass. Fermat showed that the path of such a beam is
given by the Principle of Least Time: a ray of light going from A to B
by any other path would take longer. How can we see that? It’s obvious that any
deviation from a straight line path in air or in the glass is going to add to
the time taken, but what about moving slightly the point at which the beam
enters the glass?
Where the air meets the glass, the two rays, separated by a
small distance CD = d along that interface, will look parallel:
(Feynman gives a nice illustration: a lifeguard on a beach
spots a swimmer in trouble some distance away, in a diagonal direction. He can
run three times faster than he can swim. What is the quickest path to the
swimmer?)
Moving the point of entry up a small distance the light has to travel an extra in air, but a distance less by in the glass, giving an extra travel time .
For the classical path, Snell’s
Law gives ,
so to first order. But if we look at a series of
possible paths, each a small distance away from the next at the point of crossing
from air into glass, becomes of order away from the classical path.
Suppose now we imagine that the light actually travels along
all these paths with about equal amplitude. What
will be the total contribution of all the paths at B? Since the times along the paths are
different, the signals along the different paths will arrive at B with
different phases, and to get the total wave amplitude we must add a series of
unit 2D vectors, one from each
path. (Representing the amplitude and
phase of the wave by a complex number for conveniencefor a real wave,
we can take the real part at the end.)
When we map out these unit 2D vectors, we find that in the neighborhood of the classical path,
the phase varies little, but as we go away from it the phase spirals more and
more rapidly, so those paths interfere amongst themselves destructively. To formulate this a little more precisely,
let us assume that some close by path has a phase difference from the least time path, and goes from air to
glass a distance away from the least time path: then for these
close by paths, where depends on the geometric arrangement and the
wavelength. From this, the sum over the
close by paths is an integral of the form . (We are assuming the wavelength of light is
far less than the size of the equipment.)
This is a standard integral, its value is all its weight is concentrated in a central
area of width exactly as for the real function
This is the explanation of Fermat’s Principleonly near the
path of least time do paths stay approximately in phase with each other and add
constructively. So this classical path rule has an underlying wave-phase
explanation. In fact, the central role
of phase in this analysis is sometimes emphasized by saying the light beam
follows the path of stationary phase.
Of course, we’re not summing over all paths herewe assume that
the path in air from the source to the point of entry into the glass is a
straight line, clearly the subpath of stationary phase.
Classical Mechanics: The Principle of Least Action
Confining our attention for the moment to the mechanics of a
single nonrelativistic particle in a potential, with Lagrangian the action is defined by
Newton’s
Laws of Motion can be shown to be equivalent to the statement that a particle
moving in the potential from A at to B at travels along the path that minimizes the
action. This is called the Principle
of Least Action: for example, the parabolic path followed by a ball thrown
through the air minimizes the integral along the path of the action where is the ball’s kinetic energy, its gravitational potential energy (neglecting
air resistance, of course). Note here
that the initial and final times are fixed, so since we’ll be summing over
paths with different lengths, necessarily the particles speed will be different
along the different paths. In other words, it will have different energies
along the different paths.
With the advent of quantum mechanics, and the realization
that any particle, including a thrown ball, has wave like properties, the
rather mysterious Principle of Least Action looks a lot like Fermat’s Principle
of Least Time. Recall that Fermat’s Principle
works because the total phase along a path is the integrated time elapsed along
the path, and for a path where that integral is stationary for small path
variations, neighboring paths add constructively, and no other sets of paths
do. If the Principle of Least Action has
a similar explanation, then the wave amplitude for a particle going along a
path from A to B must have a phase equal to some constant times
the action along that path. If this is the case, then the observed path
followed will be just that of least action, or, more generally, of stationary action, for only near that
path will the amplitudes add constructively, just as in Fermat’s analysis of
light rays.
Going from Classical Mechanics to Quantum Mechanics
Of course, if we write a phase factor for a path where is the action for the path and is some constant, must necessarily have the dimensions of
inverse action. Fortunately, there is a
natural candidate for the constant The wave nature of matter arises from quantum
mechanics, and the fundamental constant of quantum mechanics, Planck’s
constant, is in fact a unit of action. (Recall action has the same dimensions as and therefore the same as manifestly the same as angular momentum.) It turns out that the appropriate path phase
factor is