Atomic orbits. Part 2.1

Introduction.

There is no model or even suggestion about the actual shape of an electron orbit of a Hydrogen or Helium atom. Experiments have established that an electron’s orbit lays within the sphere of Bohr’s radius.

So Hydrogen and Helium orbits are spherical. But they are not circular.

Things are not helped by Quantum Mechanics and its claims that it isn’t possible to understand the exact shape of the orbit due to the Uncertainty Principle.

But we shall ignore the Uncertainty Principle to find out for certain whether it is or isn’t possible.

We are going to start by calculating the shape of electron orbits in atoms. We shall start with Hydrogen and Helium atoms.

1. Quantum Mechanics orbitals do not show atomic orbits. Orbitals indicate the approximate volume where electrons can be found, and the probability that an electron is actually on its orbit. According to Quantum Mechanics, the actual probability to find an electron in the volume of a spherical shell with the Bohr’s radius is less than 50%. Basically if you want to guess whether there is an electron on the orbit of a Hydrogen atom it will be a 50:50 percent chance, according to Quantum Mechanics. An experiment gives 100% assurance that there is an electron on the orbit. If there is no electron then it is an ion and not an atom.
2. When a quantum operator of Coulomb repulsion between electrons is expressed in the same form as in the formula for two point charges with the distance $r$ between them, it creates confusion. In Quantum Mechanics orbitals of electrons are supposed to have the shape of a spherical cloud. These clouds result from a rigid rotator solution, which considers nucleus and electrons as parts of the same rotator. There is no specific distance between electron clouds in a Helium atom. It is a contradiction, which needs to be resolved.
3. Expressions for Hamiltonians for both Helium and Hydrogen neglect to include Maxwell Electrodynamics. Magnetic fields, induced by rotating electrons are simply ignored.
4. In the case of Helium atoms, the difference between ortho-Helium and para-Helium is not restricted to having opposite spin. It is different atomic orbits configurations with different sets of energy levels. Moreover, the ground state is not the lowest energy state in a Helium atom. The nature of that problem as well as the nature of the difference in energy levels are not discussed. The solution to this problem is impossible according to the Uncertainty Principle. 
5. Rutherford’s planetary model was substituted by Bohr’s quantum planetary model, and then the Quantum Mechanical model, but the result is still the circular trajectory of electrons in an atomic orbit, as was in the original model, suggested by Rutherford. 
6. Circular model contradicts the orbital moment postulate of Quantum Mechanics:

$L = \lambda = 2 \pi r$       (1).

$M_n = \hbar n$       (2).

these two equations contradict one another. Out of these two equations, Bohr took the second, because it accurately described the experimental results.

That is the reason for why the second equation is called “hypothesis” of Bohr and one of the postulates of Quantum Mechanics. Equation (1) is usually ignored.

 

History.

Translated from Greek the word atom means that it cannot be divided. As soon as it became clear that atoms consist of electrons and nuclei, scientists attempted to determine the structure of atoms.

Based on experimental results, Rutherford suggested the Planetary model with a positively charged nucleus at the center and electrons rotating around.

 

Figure 1. Rutherford model of Hydrogen atom.

Rutherford’s model suggested that an electron’s orbit represents a circle with the center occupied by a positive nucleus. 

A contradiction with Electrodynamics was pseudo resolved with the postulate – electron in the lowest orbit does not emit radiation.

Bohr model was created in 1913, before Schrodinger and Heisenberg introduced Quantum Mechanics.

Niels Bohr based his theory on several postulates.

1. Atoms can occupy discreet orbitals and accordingly they can exist in discreet stationary energy states.
2. Atoms in stationary states do not emit electromagnetic radiation.
3. Atomic transition between stationary states is accompanied by emission or absorption of photons.
4. Electron moves in a circular orbit and its total energy is equal to the sum of kinetic energy of the electron and the potential energy of the electron-nucleus interaction.
5. Only the orbits for which electron angular momentum $m \cdot v \cdot r=\hbar \cdot n$ are allowed as electron stationary orbits.

Using these postulates, Bohr derived his formula for Hydrogen atom energy levels:

$E= \frac{-m{\ }e^4}{8{\ }{\epsilon_0 }^2 {\ }h^2 {\ }n^2}$       (3).

Rydberg formula for spectral lines of a Hydrogen atom is the same as Bohr’s formula (3), but recalculated in the units of reciprocal centimeters:

$\frac {1} {\lambda} = \frac{E}{h{}c} = R \cdot ( \frac {1}{n{}_b^2}$ -$ \frac{1}{n{}_a^2})$       (4).

The letters $n_a$ and $n_b$ mark two different lines in the Hydrogen spectrum.

In spite of the errors in some of Bohr’s postulates indicated in Quantum Mechanics, Bohr’s Theory compliments the experimental data well. Measurements of the Hydrogen spectra confirmed the validity of Bohr’s model.

Thirteen years later in 1926 Schrodinger and Quantum Mechanics substituted Bohr model and reformulated his postulates.

Quantum Mechanics is considered as a significant step forward in understanding the atomic structure.

Quantum Mechanics apparatus is quite different from any other science. It operates in a multitude of imaginary dimensions.  Quantum Mechanics uses a statistical approach. It calculates the probability of events happening.

It means that by definition, Quantum Mechanics cannot calculate the behavior of a single electron, because Statistics deals with multitude of events, probabilities, uncertainties and expectations. No amount of accumulated statistical data can accurately predict a single toss of a coin, unless the toss is rigged.

Formally, Quantum Mechanics is based on the wave properties of moving particles. Both diffraction and interference experiments leave no doubts in a waves characteristics of behavior of electrons and other elementary particles.

Schrodinger named the wave function as a state function in order to underline that this function was introduced in order to define the state of the whole system.

Schrodinger’s equation:

$H \Psi = -i{} \hbar \frac{\partial \Psi}{\partial t}$       (5)

describes the state of a particle as a sum of kinetic and potential energy. In this expression, Hamilton operator $H$ is usually expressed as the sum of kinetic and potential $V(x,y,z)$ energies:

$H = \frac{1}{2m} (p{}_x^2 + p{}_y^2 + p{}_z^2) + V(x,y,z)$       (6).

In a Hydrogen atom, an electron in a circular orbit is usually represented by a stationary time independent wave. The circular orbit is postulated and it is approximated by a resonator in the shape of a rectangular box with specific conditions at each end of the box, such that Hamilton’s equation describes stationary energy levels:

$H \psi = E \psi$       (7).

With capital letter $\Psi$ we marked the time dependent wave function, while regular $\psi$ represents the time independent function. Obviously these types of state functions are defined in an imaginary space and cannot have any physical meaning. According to Quantum Mechanics, a wave function should not be confused as a physical wave. It is just a mathematical abstraction, which can be used in some calculations.

The initial struggle with the interpretation of the physical meaning of a wave function was resolved by Max Born, who postulated that $|\psi |^2$ gives probability density for finding particles at a given location in space.

Normalization requirement below means at any moment of time, that particle exists somewhere in space:

$\int {|\Psi|^2 d \tau} $       (8).

For a time independent function, in order to find a particle at a specific location in space, the normalization condition means that for any specific time, a particle exists somewhere in space

$\int\int\int{\ } {|\psi|^2} dx{\ }dy{\ }dz = 1$       (9).

For one particle and a one-dimensional system with potential energy $V$ Schrodinger’s equation becomes:

$- \frac{h^2}{2m} \frac{\partial ^2 \Psi}{\partial t^2} + V(x) {\ } \Psi = \frac{- h}{2 \pi i} \frac{\partial \Psi}{\partial t}$       (10)

An electrons orbit can be considered as ring resonator and can be modeled as a one-dimensional box. For a particle in the box, the wave function can be found as a sum of two waves. Two waves do not propagate and they represent a standing wave:

$\psi = A {\ } (e^{ikx} – e^{-ikx})$       (11)

This expression for a wave function is chosen for two main reasons. First, it helps to get rid of the imaginary part in $|\psi|^2$ . Second, it makes Schrodinger’s equation time-independent and opens the path to the only analytical solution of such an equation, which currently exists in Quantum Mechanics. But this form of wave function also brings up a question about kinetic energy. If the wave is not propagating then what is the meaning of the impulse and kinetic energy? Also it is strange that an electron is moving, but its wave is standing. In a resonator or rectangular box, the standing wave is the result of a reflection from the wall. There is no wall in a circular electron orbit and one electron cannot produce two waves which move in opposite directions. But these don’t comprise the biggest problems.

Boundary conditions at the ends of the box limit possible values of energy to discreet series of values:

$E_n = \frac{n^2{\ } h^2}{8 m L^2}$, {n = 1,2,3…}       (12)

In this equation $L$ – is the size of the box, $m$ – is the mass of the electron, $h$ – is Planck’s constant and $n$ – can accept only integer values. It would be better to call this box as a resonator instead, similar to a laser resonator. In that situation, the  solutions would be identical to the modes of laser waves.

Quantum Mechanical analysis is quite similar to the analysis, made by Bohr. The resulting energy levels of standing waves in a resonator or in a harmonic oscillator are the same as in the Bohr Theory.

The solution for one particle is represented by an exponential function with the maximum density of the probability for the wave function at the center of that particle. This is the same result as the solution for the standing wave in a laser resonator in the case of $TEM_{000}$ mode of an electromagnetic wave.

This result means that the maximum density of the state of the wave function is located at the center of the particle. The probability density of the wave function decreases exponentially and decreases to an infinitely small value at an infinite distance away from the particle. This result, applied to a single electron or proton would make quite a lot of sense, because the probability to find a particle should have a maximum at the center of the particle and decrease to zero outside of the particle.

Unfortunately, Quantum Mechanics made the next step and attempted to extend that approach to the Hydrogen atom. Quantum Mechanics introduced the “rigid rotator” in which the Hydrogen atom is described as a proton and electron bound together with a strong stick or bond. That rigid bond is assumed to be the Coulomb force between the proton and electron. 

Such an approach represents several big assumptions, which are clearly wrong. Instead of calculating the wave functions of an electron and proton separately, and then analyzing the interference of these two waves, it was instead assumed that one particle named “rigid rotator”, with the sum of the masses of the proton and electron, rotates around the common center of mass and creates a common wave.

It means the wave interference is substituted with the sum of these waves.

For an illustration, observe and compare the waves created by a Battleship and a tiny speed boat with the distance between them 10 times bigger than the size of the big battleship. They will not create a common wave. Their individual waves will have a different size and frequency.

But in Quantum Mechanics, the result of the calculations for the combined electron-nucleus particle is applied to the wave function of an electron. There are other mistakes, but they are less significant.

For the ground state 1s the wave function looks like:

$\psi = \frac{1}{(\pi a^3)^{0.5}} \cdot e^{-r/a_0}$       (13).

Here $r$ – is the polar coordinates radius and $a_0$ – is Bohr radius.

(Gilbert W. Castellan, Physical Chemistry, Second Edition, Addison-Wesley Publishing Company, 1971)

The result is still a similar equation and the same types of waves as was obtained for a single electron. Harmonic oscillator or a standing wave in a quantum resonator cannot be described with any other function. The second power of the wave function, which describes probability density:

$P(r)=|\psi|^2=\frac{e^{-2r/a_0}}{\pi a^3}$       (14).

This function has its maximum at the center of a nucleus.

Figure 1 illustrates the result of rigid rotator calculations.

The left side of Figure 1 shows a normalized “Probability Density” function for an electron of the Hydrogen atom according to Quantum Mechanics. If we take a sphere with the radius 2 times bigger than the Bohr radius for the electron in a 1s orbit, then we shall find an electron inside that sphere with the probability of about 90%.

Such a result is trivial and does not add any new knowledge about the behavior of electrons. If an electron is not orbiting the nucleus, then the atom becomes an ion. There is no new knowledge about an electrons orbit, shape, or radius.

In order to justify all of its efforts, Quantum Mechanics suggests a procedure to find the correlation between the actual orbit of a 1s electron in a Hydrogen atom and of the state function. The obvious step is taking the volume integral.

When “Probability Density” is multiplied by volume, the result should be “Probability”. It would be reasonable to choose the volume to be about 10 times bigger than the volume of an electron and integrate or multiply the Probability Density function by the same volume.

Unfortunately, such an integration would not produce the desired result. The result would be the same shape of a Probability function as the shape of Density of a Probability function. Such a result would contradict experimental data, so this approach is ignored.

The trick is to choose changing and increasing integration volume.

Quantum Mechanics suggests integration by the volumes of spherical shells with each shell having a different radius. In that case, the convolution of the exponential Density Probability function and the parabolic function of the Shells Volume would produce the result on Figure 1b, on the right side.

This approach cannot be justified by any science.

It would be the same as if a scientist compares the density of two cubes. One cube made of copper and another made of aluminum. The scientist would take the measurement of the weight of these two cubes and declare that the aluminum cube has a bigger density, because it has a larger weight. When other scientists would check his experiment, they would notice that the aluminum cube is much larger than the copper cube and would immediately realize that such an experiment is not correct.

But in Quantum Mechanics, nobody cares about any physical correctness.

Aside from being correct, the main question is: what new and useful information can such an approach produce?

What perspective for science development does Quantum Mechanics suggest?

1. The Uncertainty Principle denies the chance to study individual particles. It is pure agnosticism. As a result of the Statistical approach, Heisenberg stated that the uncertainty of an electrons position, multiplied by the uncertainty of an electrons speed must always be bigger than Planck’s constant

$\Delta x \cdot \Delta p \geq \hbar$       (15).

This equation could have physical meaning for photon-type waves. Assuming the photon will have constant speed, the longer the measurement distance used, the more accurate the result of the speed can be produced.

But a wave in Quantum Mechanics and an electromagnetic wave are quite different types of waves. Experiments prove that  electrons can be accelerated to any speed in an electric field. The question is: What happens to a Quantum Mechanical wave when an electron is stopped between two electrodes in a capacitor or trap? The answer is clear: the energy of the wave will become equal to zero and the wave will disappear:

$h \nu = \frac{mv^2}{2} $       (16).

Once the Quantum Mechanical wave will disappear, the Electron becomes a classic particle and no uncertainty could be observed.

2. Electron orbitals, introduced in Quantum Mechanics, have nothing to do with an electrons orbit. According to Quantum Mechanics, the probability to find an electron on its orbit is less than 50%. It is not only a contradiction to the law of Energy conservation. Quantum Mechanics contradicts itself.

First it postulates that the length of the electron’s orbit must be equal to the whole number of an electrons wavelength, without any uncertainty. But later Quantum Mechanics comes to the conclusion that an electrons orbit can be longer or shorter with almost the same probability. This conclusion also contradicts the linear character of experimental spectral data.

3. In the calculation of an electrons orbit, the function was integrated in spherical shells of different volume and the result is a spherical shell with the radius close to the Bohr radius of an electron orbit. If we would decide to integrate in rectangular coordinates, using shells of cubic boxes, the result would be cubic orbits of electrons. If we would choose rectangular or elliptical shells, the results would be ellipse, etc. It means that the shape of the orbital was not determined by Quantum Mechanics calculations, but was chosen arbitrarily by the person, who made the calculations.
4. Nobody tried to complete analytical calculations for other types of orbitals. It is impossible to obtain an 8-shaped electron p-orbit from an 8-shaped second harmonic orbital. It would be very difficult to justify the 8-shaped shells for integration of p-orbitals. Just try to integrate other orbitals with spherical shells. The result will be a sphere instead of a p-orbital. Spherical orbitals were chosen because an experiment indicated the spherical shape of an electron orbit in a Hydrogen atom. Quantum Mechanics did not add any reason for the shape of an electron orbit, besides getting the only possible analytical solution. 
5. The wave function consists of two waves. Although one of these waves is an imaginary wave and cannot be measured in real space. The result of the interaction of two waves is a standing wave in real space and time. These waves move in opposite directions. In a resonator of a rectangular box, these waves are original and reflected waves. But in an orbital electrons movement, there is no wall to reflect the original wave.
6. Electron is distributed in a spherical shell. That spherical cloud is not moving, it is a stationary time-independent function, but it still has both orbital moment and spin.
7. If an electron is transformed into a distributed wave, what is the mechanism of such a transformation? If two clouds from two electrons in a Helium atom are mixed within the same sphere, then what is the meaning of the distance between two charges in the Coulomb formula?

The list of funny questions can be continued, but the main conclusion is clear:

Quantum Mechanics produces no new information about atoms and molecules. But it imposes the Uncertainty Principle on all parameters in order to justify its lack of results in spite of all of these efforts.

Still there is one correct conclusion in Quantum Mechanics: An electron exhibits wave properties and they should be taken into account.

In the next page we shall calculate the Orbit of a Hydrogen Atom.