__Spinning Spirals and the
Quantum Wave Function__

** Abstract.**
The quantum wave function is represented as a spinning spiral around the x-axis.
Examples are then discussed.

__1.
Plane Y Waves.__

It is often advantageous to represent complex numbers as "vectors" in the complex plane. For example, Fig. 1.1 depicts the representation of the complex number

(1.1)

.

Where possible it is usually convenient to locate the tail of Y’s "vector" at the origin of such a plane.

Figure 1.1

Complex Plane Representation of Y

In quantum wave mechanics the symbol Y signifies the quantum wave function. If |Y| is constant in time but Y is a function of time, then this is tantamount to a time-dependent q in Fig. 1.1, say

(1.2)

In this case the vector Y in Fig. 1.1 would rotate clockwise around an axis through the origin and perpendicular to the figure’s plane. For example, assuming ReY(0) is positive and ImY(0)=0,

(1.3)

No loss of rigor is incurred by overlaying the complex plane in Fig. 1.1 on a Cartesian plane in 3-dimensional space, say one parallel to the yz-plane and with its origin on the x-axis. In such cases the complex plane’s Real (Re) axis can be oriented parallel to the z-axis, and its Imaginary (Im) axis can be oriented parallel to the y-axis. (The positive x-axis would then lie above and below the plane of Fig. 1.1.) A "stack" or set of such complex planes, one for each value of x, then provides a means for depicting Y as a function of x as well as t, say

(1.4)

In Eq. 1.4 at any given instant t (and for constant k), the tail of Y would lie on the x-axis, and the head of Y would lie on a left-handed spiral concentric with the x-axis. With the passage of time this spiral would spin clockwise around the x-axis, and its threads would appear to advance in the positive x-direction. Since a plane, circularly polarized electromagnetic wave can be similarly represented, it is natural to think of Y(x,t) as a wave with wavelength

(1.5)

and frequency

(1.6)

In
quantum theory, Y(x,t)
in Eq. 1.4 is the wave function for an ensemble of identical particles with a
common, positive mv_{x} (momentum).
DeBroglie postulated that the wavelengths and frequencies of such quantum waves
are functions of particle momentum and kinetic energy:

(1.7a)

(1.7b)

Born
then proposed that |Y|^{2}dx
(or YY*dx)
is the probability of finding a particle in the interval x to x+dx (or (x,dx)
for short). In the case of Eq. 1.4 |Y|
is constant, even though its Real and Imaginary parts vary in space and time.
Hence this probability is the same for all x and t.

If
a left handed spiral with clockwise spin can represent an ensemble of particles
traveling in the positive x-direction, then a right handed spiral (also with
clockwise spin) should be able to represent an ensemble of particles traveling
in the negative x-direction. That is, when such a right handed spiral spins
clockwise around the x-axis, its threads move in the negative x-direction. (The
spirals for both the positive and negative v_{x }particles ostensibly
have the same, clockwise spin, since time always flows "forward.") The
equation for such a right handed spiral would then be

(1.8)

Here again the probability of finding a particle moving in the negative x-direction and in the interval (x,dx) would be independent of x and t.

__2.
Reflections.__

Fig. 2.1 depicts an infinite potential energy (U) barrier at which all particles to the left of the origin and traveling in the positive x-direction are theoretically turned back. U is infinite at all x>0, and zero for x<0.

Figure 2.1

Infinite Potential Energy Barrier

As already discussed (Eq. 1.4), the wave function for particles traveling in the positive x-direction can be formulated as

(2.1)

What
should we postulate for the wave function of particles that have been turned
back at the barrier? Their wave function would correspond to the reflected Y_{(+) }wave.

In
the case of plane, circularly polarized electromagnetic waves, the electric
field vector is reversed upon reflection. It makes sense to assume that this
must happen to Y_{(+)} also,
particularly if we wish a plot of |Y|(x)
to be continuous (though not necessarily smooth) at the barrier. Since |Y|(x>0)
must be zero, imposing this requirement produces

(2.2)

And
more generally (since Y_{(-)} is
a right handed spiral), we have for x<0:

(2.3)

.

Invoking superposition, the wave function for all the particles is then

(2.4)

=

The
Probability Density (|Y|^{2})
is thus

(2.5)

For
negative values of x, the probability of finding a particle in (x,dx) is not
independent of x in the infinite barrier case! It goes as sin^{2}(kx)!
Furthermore, it is zero at x=0. The modulated form of |Y|^{2},
when the wave function is reflected, is the "quantum surprise."

Note
that here again, however, |Y|^{2} (in
Eq. 2.5) is independent of time (or is "stationary" in time); i.e., it
depends only upon x. Thus there are values of negative x where the probability
of finding a particle, traveling in either direction, is permanently zero! More
generally the particle will not interact with anything in its environment
(including us) at these locations. Does this mean that no particle is ever at
such a location? Or does it mean that no particle will interact when it is at
such a location? Logically the latter seems to make more sense. But there are
those who insist that, if a particle can never be observed at some location,
then by definition it never is at that location. Born (who perhaps grappled with
this conundrum) was careful to define |Y|^{2}dx
as the probability that a particle will be found (and thus interact with an
observer) in (x,dx).

__3.
The Infinite Square Well.__

Fig. 3.1 depicts two infinite potential energy barriers, collectively forming a "well" in which U=0. The well’s width is L.

Figure 3.1

An Infinite, Square Potential Energy Well

The
wave function for particles traveling to the right, inside the well, is
reflected at x=L. And Y_{(-) }is
reflected at x=0. Since Y=0
for all x<0 and x>L, we again require that

(3.1a)

(3.1b)

Since the two Y’s reverse phase at x=0 and x=L, and since Y is single-valued at any particular point in space and time, these conditions can only be satisfied if

(3.2)

Whereas any and all wavelengths are acceptable in the single barrier case (Sect. 2), only discrete wavelengths are acceptable in the square well case. Or, according to deBroglie, only discrete energies can persist in a square well.

We
incur no loss of rigor if we stipulate that Y_{(+) }is
real and positive at x=0 and t=0. Then Y_{(-) }is
real and negative. Keeping t constantly zero (i.e., taking a
"snapshot" of Y_{(+)} and Y_{(-) }at
t=0), the head of Y_{(+) }moves
along a left handed spiral as x increases, and the head of Y_{(-) }moves
along a right handed spiral. Y (defined
to be Y_{(+) }+ Y_{(-) })
is then purely imaginary (lying all in a plane) at time t=0. But the spirals for Y_{(+) }and
Y_{(-) }mutually
spin clockwise, and thus the plane of Y spins
clockwise in time. However, |Y|
is again independent of time:

(3.3)

Like a playground jump rope, Y whirls around the x-axis, with nodes at x=0 and x=L.

Again
there are "quantum surprises." Classically we might expect |Y|^{2} to
be the same everywhere inside the well. But in the lowest energy state |Y|^{2} is
zero at x=0 and x=L, and it is maximum at x=L/2. For the next higher permissible
energy (l=L) there are 3 zero points (at x=0, x=L/2, and x=L) and 2 maxima (at
x=L/4 and x=3L/4).

__4.
Concluding Remarks.__

Visualizing Y as a spinning spiral may serve to make this complex quantity less abstract (particularly to the beginning student of wave mechanics). Most of us are familiar with spiral springs or stretched "Slinky" toys, and it isn’t difficult to visualize how right and left handed spirals might add at any given moment, etc.