The author anj68 laughed at the nonsense of physicists and mathematicians. About the madness of phase velocities. Such speeds are a purely mathematical product. Since it was developed by mathematicians who absolutely do not care where it came from, that is, the physical essence. There are two waves in quantum physics and mathematics. One is Compton. It does not depend on the speed of movement of the particle, but is involved in the speed of light, and the internal energy of the particle. Strictly speaking, it is a constant for every particle. For example, for an electron
The electron frequency in the Compton construction is also constant
And the second mathematical wave construction is the de Broglie wave, which directly depends on the particle speed.
So, we get such a phase velocity if we multiply the frequency of the Compton wave, which is a constant, by the wavelength of another wave construction — the de Broglie wave.
It is clear that the lower the particle velocity, the longer the de Broglie wavelength will be. And at a constant frequency, the phase velocity will be the greater, the lower the particle velocity. But there is no physical sense in such a construction.
Now let’s try to figure out how this happened.
Let’s go back to the days of Huygens. He associated his light waves with ordinary water waves, which, roughly speaking, look like circles on the water surface. That is, circles. And the circle has a radius. And in general, convenient geometry. For example, if we take a conventional meter of radius, then the circumference will be 2πR, or 6.28 * 1m. On such a circumference, you can place a number of segments equal to the wavelength. For example, for a wavelength of 2.426 * 10 ^ -12m, the number of segments will be
2π / λ = 6.28 / 2.4264 * 10 ^ -12 = 2.588196505 * 10 ^ 12m. This is the wavenumber for the Compton wavelength of the electron. In fact, this is the number of circular frequencies that correspond to a water wave propagating a meter from the source.
And now you can make up the proportion:
If for a radius of 1m — segments k, then for 300000000m — segments ω
Since time is also involved here, the radius can also be represented in the form of speed. Well, we get our circular frequency. In the example, the frequency of an electron.
The geometric meaning is that with a radius of 4.5 * 10 ^ 11 m, the circumference will be 2.826 * 10 ^ 12m, and this circumference will include 7.76467 * 10 ^ 20 segments of 3.63956 * 10 ^ -9m each.
So far, this is exclusively the geometry of the water wave, which we transferred to the magnitudes of the Compton wave of the electron. And it must be said that the energy of a water wave depends on the magnitude of the amplitude, and the frequency is a conditional value. That is, the situation when 20 frequencies in a puddle from a sparrow are much less in energy than 1 frequency of a tsunami is normal. Therefore, the number of frequencies for a water-like wave did not play a special role, but it was possible to calculate the phase velocity and vice versa.
They used the analogy with the water wave for a very long time (and in some ways even now, for example, when explaining the Poisson spot), they got used to it, therefore, when switching to a flat sinusoidal e / m wave, they managed to adapt the wavenumber to it. True, the energy of a sinusoidal wave depends on frequency, so the angular frequency, which is 6.28 times higher than the usual one, to find the energy is multiplied not by Planck’s constant h, but by the reduced ħ, which is 6.28 times less.
And, besides, in the process of transition to a sine wave, a small postulate emerges unobtrusively
Trofimova T.I., «Course of Physics», «High School», 1985.
The point is that for the waves proper, their phase velocity does not depend on the frequency. Their speed depends on the characteristics of the medium in which these waves arise. For example, at the same frequency in a medium where the wave speed is higher, a longer wavelength will be obtained. The fact that, knowing the wavelength and frequency, we can calculate the phase velocity, does not mean that this velocity physically depends on them. And then suddenly it began to depend.
While this concerned only light, for which the phase velocity is always the same, and in general with = Vλ, there were no particular problems. For particles that move at different speeds, it turned out badly.
And for particles, as for any non-fictional object, several types of energy are distinguished. For example, internal, it is the rest energy (E = mc ^ 2), kinetic (Ek = mv ^ 2/2). We are not talking about the frequency yet. But it can go. Calculation frequency
They are based on energy (V = E / h), that is, the energy is divided by the minimum portion of energy h, and the number of such portions is obtained. By multiplying this number by 2π, you get the circular frequency. Moreover, they take internal energy as a basis. And they are not embarrassed when, having a completely constant circular frequency of the electron, the phase velocity, in contradiction with the newly appeared postulate, turns out to be different. And in fact, for particles, this mathematical postulate should sound completely different:
«The phase velocity of sine waves depends on the de Broglie wavelengths, at a constant circular frequency of the Compton wave.»
Well, it turns out that way if you analyze the course of calculations.
According to De Broglie, we may well consider ourselves a particle. Walking very slowly, at a speed of 0.8m / s, we will have a phase speed of 1.125 * 10 ^ 17m / s.
And we are frantically looking for phases.