# Quantum mechanics

Corresponding Wikipedia article: Quantum mechanics

## Statistics

The Gibbs ("Particle system", 8) and Boltzmann ("Particle system", 15) distributions quite accurately simulate any system of the real material particles.

The photonic statistics should take into account the fact, that the quantization does not generally apply to an absorption, and the harmonics (multiple frequencies) make up a single entity with the fundamental frequency of a photon, that is the photonic wave may be non-sinusoidal. The average energy of this photon is expressed in terms of probability $$p_n$$ for each harmonic quant: $\epsilon(\omega)=\sum_{n=0}^\infty{p_n\epsilon_n}\;\;\;\;\;\epsilon_n=n\hbar\omega\tag{1}$ Since the quanta $$\epsilon_n$$ are produced by the real material particles, they also are subject to the Gibbs distribution: $p_n=A\exp\left(-\frac{\epsilon_n}{kT}\right)\tag{2}$ The obvious condition $$\sum{p_n}=1$$, and substituting $$x=\hbar\omega⁄kT$$, gives: $\epsilon(\omega)=A\hbar\omega\sum_{n=0}^\infty{ne^{-nx}}=\hbar\omega\frac{\sum_{n=0}^{\infty}ne^{-nx}}{\sum_{n=0}^{\infty}e^{-nx}}\tag{3}$ Application of the geometric progression formula gives: $S=\sum_{n=0}^{\infty}e^{-nx}=\frac{1}{1-e^{-x}}\tag{4}$ $-\frac{\mathrm{d}S}{\mathrm{d}x}=\sum_{n=0}^{\infty}ne^{-nx}=\frac{e^{-x}}{(1-e^{-x})^2}\tag{5}$ As a result, the average energy of a photon is: $\epsilon(\omega)=\frac{\hbar\omega}{\exp(\hbar\omega/kT)-1}\tag{6}$ Hence the photons distribution, which is assumed in the Planck's law, has the form: $n=n_0\frac{1}{\exp(\epsilon/kT)-1}\tag{7}$ This distribution is close to the Gibbs distribution when $$\epsilon\gg kT$$. It is also called the Bose-Einstein distribution and describes the so-called bosons. The bosons are the photons and also the unrealistic hypothetical or artificial unstable superheavy particles, which, of course have $$\epsilon\gg kT$$. This distribution itself is paradoxical, because $$n\to\infty$$ at $$\epsilon\to 0$$.

The Bose-Einstein condensate is formed of the paired low energy particles.

The properties of the electron gas are understandable without the Fermi-Dirac distribution (see "Metals").

## Uncertainty principle

The uncertainty principle relates rather to the metrology than to the fundamental laws of nature.

The product of uncertainties (measurement errors) of a position and a momentum can be written using ("Mass and momentum", 8): $\Delta x\Delta p=h\frac{\Delta x}{\Delta\lambda}\tag{8}$ The wavelength $$\lambda$$ characterizes a particle size, and obviously the coordinate $$x$$ resolution (uncertainty) is directly proportional to the particle size: $\frac{\Delta x}{\lambda}\geq const\tag{9}$ The wavelength measurement is the oscillation phase measurement, so it has a resolution or uncertainty $$\Delta\lambda⁄\lambda=const$$, thus: $\frac{\Delta x}{\Delta\lambda}\geq const\tag{10}$ The measurements of a wavelength (momentum) and a coordinate are really interrelated, because both measurements are reducible to the measurement of an oscillation phase, the uncertainty of which is expressed as: $\Delta\varphi=2\pi\frac{\Delta x}{\Delta\lambda}\tag{11}$ Heisenberg had arbitrarily defined the minimum value of $$\Delta\varphi$$ as $$1/2$$ radians, which is much smaller than the resolution $$\Delta\varphi=\pi/2$$, corresponding to distinguishing of the maximum and minimum magnitudes.

## Schrödinger equation

The presented aether theory eliminates the concept of the particle wave function. It corresponds to the model described in Ch. "Matter". The probability density corresponds to the mass density.

Due to the denial of the wave function, this theory denies the Schrödinger equation as a law of nature. And, therefore, it denies the assignment of discrete quantum numbers to the particles.

It is stated that the quantum numbers have a meaning only for certain particle interactions:

Number Physical meaning
Principal Quantized energy of an excited electron within the atom.
Azimuthal (orbital) Degree of the electron interaction with other atomic electrons.
Magnetic Does not exist in nature.
Spin Magnetic moment of the particle with respect to its paired particle.

Consequently, the so-called sublevels S, P, D and F in the atomic electron shells are denied. Importantly, the Schrödinger equation does not define the number of electrons on the filled atomic shell, without introducing the “Aufbau principle”, which is known as the empirical Madelung rule.

The Pauli exclusion principle is a consequence of the electromagnetic particle properties. A particle pair has a weak total magnetic field and does not attract a third particle.

The state of a nucleus is an arrangement of its nucleons. The radiation and absorption of energy occurs due to the nucleons motion in an electromagnetic field. Since the number of stable configurations is limited, the energy is quantized.

The Stern–Gerlach experiment is considered as a proof, that each particle has its own spin independently of other particles. All the particles are oriented by the magnet field, i.e. the beam is polarized. Since the field is non-uniform, the magnetic flux density is increased at the poles, and a particle, which moves between the poles, is deflected by the magnetic forces towards one of them. Since the particles, which are focused and attracted to each other by the magnetic forces, move with the same speed, they are divided into two distinct groups, passing into one of two paths.

The Sokolov–Ternov effect is the orientation of particles in the external magnetic field direction, i.e. the polarization.

The quantum teleportation or the linked (entangled) particles is a way to justify the spending on the quantum mechanics research.