The Effects of Dimensionality on a Ground State Hydrogen Atom Electron’s Position Probability
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 KA 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 dVi to be the volume between spherical surfaces of radii ri-1 and ri.)
Define dti to be the time the electron spends in volume element dVi. Then pi, the probability of the electron being in volume element dVi is defined to be the fraction of the whole time t that the electron spends in dVi:
It is a virtual certainty that the electron will always be somewhere within the volume:
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.
vi, 1-Dimensional Oscillation
"Economy of Dimensions" suggests in this 1-dimensional case that the "volume" element be dx. The time spent in interval dxi would then be
where vi is the average speed in dxi. Since dx is single-valued, dti depends only on the variable vi. Thus pi is a function of vi:
Fig. 4_2 plots pi over the range 0<x<D. (The last 10 values, 991 through 1000, are not plotted in order to accentuate the smaller values of pi.)
pi, 1-Dimensional Oscillation
As expected, pi 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 =pD2/N.
Since (with a central force) the electron must oscillate along a radial line, it is reasonable to define area increment dAi to be the area between circles of radii ri-1 and ri, with rN = D. Here again the requirement that every pi be greater than zero will be met. But now, since all the dAi are equal, dri and the ri must both be variables. For example, starting with the outermost dAi = dAN,
For adequately small drN,
Having determined rN-1, rN-2 can be found from
etc. A key difference from the 1-dimensional case is that the size of a given dti now depends on both the variables vi and dri. Consequently pi is a function of these two variables:
Fig. 5_1 plots the dri in this 2-dimensional case. Note how, unlike the 1-dimensional case, each dri has a unique value.
dri, 2-dimensional Range of Motion
Fig. 5_2 plots pi . 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 dri dominate close to the origin, and pi actually grows in the vicinity of the origin!
pi, 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 pi is even more pronounced. Here dVi would be defined to be the volume between spherical surfaces of radii ri-1 and ri. And the requirement that dV be single-valued again dictates that ri and dri vary with distance from the Origin. Fig. 6_1 plots pi for this 3-dimensional case. (Note the lower bound of i=2, implemented in order to suppress the very large value of p1 at i=1.)
pi, 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 on 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.)