Spinning Spirals and the Quantum Wave Function
1. PlaneY 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
Where possible it is usually convenient to locate the tail ofYís "vector" at the origin.
Complex Plane Representation ofY
In quantum wave mechanics the symbolY 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
In this case the vectorY in Fig. 1.1 would rotate clockwise around an axis through the origin and perpendicular to the figureís plane. For example, assuming ReY(0) is positive and ImY(0)=0,
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 planeís 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 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 depictingY as a function of x as well as t, say
In Eq. 1.4 at any given instant t (and for constant k), the tail ofY 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
In quantum theory,Y(x,t) in Eq. 1.4 is the wave function for an ensemble of identical particles with a common, positive mvx (momentum). DeBroglie postulated that the wavelengths and frequencies of such quantum waves are functions of particle momentum and kinetic energy:
Born then proposed that |Y|2dx (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 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 vx particles ostensibly have the same, clockwise spin, since time always flows "forward.") The equation for such a right handed spiral would then be
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.
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.
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
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 reflectedY(+) 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 toY(+) 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
And more generally (sinceY(-) is a right handed spiral), we have for x<0:
Invoking superposition, the wave function for all the particles is then
The Probability Density (|Y|2) is thus
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 sin2(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|2dx 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 wellís width is L.
An Infinite, Square Potential Energy Well
The wave function for particles traveling to the right, inside the well, is reflected at x=L. AndY(-) is reflected at x=0. Since Y=0 for all x<0 and x>L, we again require that
Since the twoYís 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
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 thatY(+) 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:
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 |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.
VisualizingY 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 isnít difficult to visualize how right and left handed spirals might add at any given moment, etc.