The Effects of Dimensionality on a Ground State Hydrogen Atom Electron’s Position Probability

G.R.Dixon, 12/04/2005

The data points plotted in this article were computed by ground state modeling software. The motion is that of the electron in a ground state Hydrogen atom, as suggested in a previous article.

1. The Probability of Finding the Electron in a Given Volume Element.

Let K_{A} be a rectangular coordinate system. And let the ground state electron’s maximum distance from the Origin during time interval t be D=.529E-10 meters. Since the electron’s motion is periodic, a reasonable choice for t is a cycle time. Let us assume for openers that, when at a distance D from the Origin, the electron could be anywhere on a spherical surface of radius D. Call the volume enclosed by this surface ‘V’. Divide V into N equal increments in such a manner that, regardless of where on the spherical surface the electron is when it is D-distant from the Origin, it spends some nonzero amount of time in each volume element over the course of a cycle time. (This requirement can be met by defining dV_{i} to be the volume between spherical surfaces of radii r_{i-1} and r_{i}.)

Define dt_{i} to be the time the electron spends in volume element dV_{i}. Then p_{i}, the probability of the electron being in volume element dV_{i} is defined to be the fraction of the whole time t that the electron spends in dV_{i}:

. (1_1)

It is a virtual certainty that the electron will always be somewhere within the volume:

. (1_2)

2. dV and the Economy of Dimensions.

If it is known that the electron invariably moves along a __particular line__ (say the x-axis), then a reasonable choice for "dV" is an increment of the x-axis. If it is known only that the electron moves in a particular __plane__ (say the xy-plane), then a reasonable choice for "dV" might be an annular increment of area in said plane. And if
it is known only that the electron moves in a 3-dimensional __volume__, then dV would be the volume between two concentric spherical surfaces. In all cases "dV" must, by definition, be single-valued.

3. Why Deal with Probabilities?

In cases where a particle’s (a) initial position, (b) velocity, and (c) the force acting on it at every possible position are known, then there is little point in considering probabilities. Under these circumstances one need only apply Newton’s laws to predict what the particle’s behavior will be.

Probabilities are useful when only partial information about a particle’s motion is known. For example, in the 1-dimensional case we might not know __where__, on the x-axis between x=__+__D, our electron is at time t=0. In the 2-dimensional case, even if we know that the electron is initially on the circle of radius D, we may not know precisely __where__. Similar remarks apply to the 3-dimensional case. In all such cases the mathematics of probability may provide the most useful information.

Of course if we __observe__ a particle and determine (more or less precisely) where it is at any moment, then we can fall back to the more deterministic physics of Newtonian mechanics. That is, depending upon on how __precise__ our knowledge is when we observe the particle, we can apply the more deterministic mathematics for a period of time. But generally our observation can never be absolutely precise (particularly with "microscopic" or very small-mass particles whose motion we may perturb in the very act of observing). In due course, barring follow-on observations, our knowledge __invariably__ becomes less certain as time passes, and we must again view the system in probabilistic terms.

4. A 1-Dimensional Example.

Let our ground state electron’s motion be known to be solely on the x-axis and its speed, v(x), be as depicted in Fig. 4_1. Since this motion is mirrored every quarter cycle, we shall set the time interval t equal to the first quarter cycle of an oscillation. The oscillation amplitude is the radius of a Bohr Hydrogen atom (.529E-10 meters). v is plotted as a function of x, and the average v in each interval, of magnitude dx = .529E-10/1000 meters, is plotted.

Figure 4_1

v_{i}, 1-Dimensional Oscillation

"Economy of Dimensions" suggests in this 1-dimensional case that the "volume" element be dx. The time spent in interval dx_{i} would then be

, (4_1)

where v_{i} is the average speed in dx_{i}. Since dx is single-valued, dt_{i} depends only on the variable v_{i}. Thus p_{i} is a function of v_{i}:

. (4_2)

Fig. 4_2 plots p_{i} over the range 0<x__<__D. (The last 10 values, 991 through 1000, are not plotted in order to accentuate the smaller values of p_{i}.)

Figure 4_2

p_{i}, 1-Dimensional Oscillation

As expected, p_{i} is maximum close to x=D (where v is minimum) and minimum near the Origin (where v has its maximum values).

5. A 2-Dimensional Example.

In Sect. 4 the implication was that the electron was __known__ to oscillate along the x-axis. But what if we only know that the electron is initially somewhere on the circle of radius D, lying in the xy-plane, with the electron subject to the same __spherically symmetric__ central force? In this case the "volume" element might more reasonably be defined to be an __area__ increment of size dA = pD^{2}/N.

Since (with a central force) the electron must oscillate along a __radial__ line, it is reasonable to define area increment dA_{i} to be the area between circles of radii r_{i-1} and r_{i}, with r_{N} = D. Here again the requirement that every p_{i} be greater than zero will be met. But now, since all the dA_{i} are equal, dr_{i} and the r_{i} __must both be variables__. For example, starting with the outermost dA_{i} = dA_{N},

, (5_1)

and

. (5_2)

For adequately small dr_{N},

, (5_3)

and

. (5_4)

Having determined r_{N-1}, r_{N-2} can be found from

, (5_5)

etc. A key difference from the 1-dimensional case is that the size of a given dt_{i} now depends on both the variables v_{i} __and__ dr_{i}. Consequently p_{i} is a function of these two variables:

. (5_6)

Fig. 5_1 plots the dr_{i} in this 2-dimensional case. Note how, unlike the 1-dimensional case, each dr_{i} has a unique value.

Figure 5_1

dr_{i}, 2-dimensional Range of Motion

Fig. 5_2 plots p_{i} . Note how the plot differs from that depicted in Fig. 4_2, notwithstanding the fact that the same central force drives the electron in both cases. In this 2-dimensional case, even though v steadily increases as the electron moves toward the origin, the increasingly larger values of dr_{i} dominate close to the origin, and p_{i} actually __grows__ in the vicinity of the origin!

Figure 5_2

p_{i}, 2-Dimensional Range of Motion

6. 3 Dimensions.

In the 3-dimensional case we know only that the electron is initially somewhere on the spherical __surface__ of radius D. The modulating effect on p_{i} is even more pronounced. Here dV_{i} would be defined to be the volume between spherical surfaces of radii r_{i-1} and r_{i}. And the requirement that dV be single-valued again dictates that r_{i} and dr_{i} vary with distance from the Origin. Fig. 6_1 plots p_{i} for this 3-dimensional case. (Note the lower bound of i=2, implemented in order to suppress the very large value of p_{1} at i=1.)

Figure 6_1

p_{i}, 3-Dimensional Range of Motion

7. Uncertainties in D.

In a previous article it was found (according to Maxwell) that electrons, subject to a force that results in Fig. 1_1, will radiate zero net energy per cycle __only__ for D=.529E-10 meters. Other values of D will result in net energy either being emitted or absorbed during a cycle time.

This sharply defined "stable state" value of D is "smeared out" by a remarkable conclusion drawn by Einstein and Planck. They deduced that, in the real world, any emitted or absorbed radiant energy can be no smaller than a finite quantum. It is found that this quantum always equals the product of Planck’s constant and some state-transition frequency. Such a finite quantization of emitted/absorbed radiant energy suggests that there may be a __range__ of D’s for a stable "ground" state. In the model of such a state the electron speed (Fig. 1_1) would not be so sharply defined. A bit of excess energy might be created in the field during one short interval of time, but a bit of excess energy might be deleted from the field during a subsequent interval of time. So long as these excess energies do not rise to the amount of energy in a quantum, then o__n average__ the net created/deleted field energy per several cycle times is practically zero. The curve in Fig. 1_1 would then represent the electron’s __average__ speed.

The "smeared out" nature of v indicates that the probability function will also be smeared out. In particular, a plot of p might not drop vertically to zero at r=__+__D, and it might not have a cusp at r=0. The matter will not be further investigated here. Suffice it to say that it is one of many interesting possibilities warranting further consideration. (Another would be the motions of stable states whose energies are greater than that of the ground state system.)