Geekly Articles each Day

1. A ball on a spring, the Newtonian version

2. Quantum ball on spring

3. Waves, classic look

4. Waves, the classical equation of motion

5. Quantum waves

6. Fields

7. Particles are quanta

8. How particles interact with fields

## Reminder: quantum ball on spring

In the first article of the series, we studied the ball of mass M on a spring of rigidity K, and found that it had vibrations:

• There will be a formula $$$z(t)={z}_{0}+Acos[2\text{}pi\text{}nut]$.

• Energy $$$E=2\text{}p{i}^{2}\text{}n{u}^{2}{A}^{2}M$.

• Equation of motion $$${d}^{2}z/d{t}^{2}=-K/M(z-{z}_{0})$

Where the equation of motion forces ν = √ K / M / 2π, but allows the amplitude A to be any positive value. Then in the second article we saw that quantum mechanics, applicable to oscillations, limits their amplitude - it can no longer be any. Instead, it is quantized, it must take one of an infinite number of discrete values.

Where n = 0, 1, 2, 3, or 44, or generally any integer greater than or equal to zero. In particular, A can equal $$$(1/2\text{}pi)\text{}sqrt2h/\text{}nuM$, but it cannot be less - only zero. We say that n is the number of quanta of oscillations of the ball. The energy of the ball is now also quantized:

The most important thing is that to add one quantum of oscillations of the ball, the energy of the magnitude of hν is needed - it can be said that each quantum carries energy of hν.

## Quantum wave

With the waves, everything is essentially the same. For a wave with frequency ν and wavelength λ, oscillating with amplitude A around the equilibrium position Z_{0} ,

• Equation of motion: $$$Z(x,t)={Z}_{0}+Acos(2\text{}pi[\text{}nut-x/\text{}lambda])$.

• Energy per wavelength: $$$2\text{}p{i}^{2}\text{}n{u}^{2}{A}^{2}{J}_{\text{}}lambda$.

(where J_{λ} is a constant depending on, say, a rope, if we are talking about waves on a rope), several possible equations of motion, from which we choose two for study:

And again, quantum mechanics limits amplitude A to discrete values. Just like for spring oscillations,

• One simple wave of a certain frequency and length consists of n quanta,

• The allowed magnitudes of amplitude A are proportional to √n,

• The allowed energy values of E are proportional to (n + 1/2).

More precisely, as for the ball on the spring,

• Permitted energy values E = (n + 1/2) h ν

• Each quantum of wave carries energy of magnitude h ν

The formula for the expression A is quite complicated, because we need to know how long the wave is, and the exact formula will be too confusing - so let me just write a formula that conveys the right idea. We received most of the formulas by studying endless waves, but for any real wave in nature, the duration is finite. If the wave duration is approximately equal to L, and it has L / λ ridges, then the amplitude is approximately equal to

Which is proportional $$$\text{}sqrtnh/\text{}nu$as in the case of a spring, but it depends on L. The longer the wave, the smaller its amplitude - so that for each quantum of the wave the energy is always hν.

That's all - it is shown in the figure below.

On the left - a naive image of waves, where the amplitude is proportional to the square root of the number of quanta, and other amplitudes cannot exist. On the right, a slightly less naive image that takes into account the quantum oscillations inherent in the quantum world. Even in the case of n = 0, some oscillations exist.

## The investigation

What does this mean for our class 0 and class 1 waves?

Since waves of class 0 can have any frequency, they can have any energy. Even for a tiny value of ε, one can always make one quantum of a class 0 wave with a frequency ν = ε / h. For such a small energy, this quantum wave will have a very small frequency and a very long wavelength, but it can exist.

Waves satisfying a class 1 equation are not. Since for them there is a minimum frequency ν_{min} = μ, for them there is also a quantum of minimum energy:

If your tiny energy value ε is less than this, the quantum of such a wave will not work. For all quanta of class 1 waves with a finite wavelength and a higher frequency, E ≥ h μ.

## Total

Before we begin to take into account quantum mechanics, the amplitude of the waves, like the amplitude of the ball on the spring, can change continuously; they can be made arbitrarily large or small. But quantum mechanics implies the existence of a minimal nonzero wave amplitude, as in the case of a ball oscillation on a spring. And usually the amplitude can take only discrete values. The allowable amplitudes are such that both for the oscillations of a ball on a spring, and for a wave of any class with a certain frequency ν

• To add one quantum of oscillations, energy h ν is required

• For oscillations of n quanta, the oscillation energy will be (n + 1/2) h ν

Now it's time to apply the knowledge gained to the fields and see when and how the quanta of the waves in these fields can be interpreted as what we call the "particles" of nature.

2. Quantum ball on spring

3. Waves, classic look

4. Waves, the classical equation of motion

5. Quantum waves

6. Fields

7. Particles are quanta

8. How particles interact with fields

In the first article of the series, we studied the ball of mass M on a spring of rigidity K, and found that it had vibrations:

• There will be a formula $$$z(t)={z}_{0}+Acos[2\text{}pi\text{}nut]$.

• Energy $$$E=2\text{}p{i}^{2}\text{}n{u}^{2}{A}^{2}M$.

• Equation of motion $$${d}^{2}z/d{t}^{2}=-K/M(z-{z}_{0})$

Where the equation of motion forces ν = √ K / M / 2π, but allows the amplitude A to be any positive value. Then in the second article we saw that quantum mechanics, applicable to oscillations, limits their amplitude - it can no longer be any. Instead, it is quantized, it must take one of an infinite number of discrete values.

$$

$$A=(1/2\text{}pi)\text{}sqrt2nh/\text{}nuM$$

Where n = 0, 1, 2, 3, or 44, or generally any integer greater than or equal to zero. In particular, A can equal $$$(1/2\text{}pi)\text{}sqrt2h/\text{}nuM$, but it cannot be less - only zero. We say that n is the number of quanta of oscillations of the ball. The energy of the ball is now also quantized:

$$

$$E=(n+1/2)h\text{}nu$$

The most important thing is that to add one quantum of oscillations of the ball, the energy of the magnitude of hν is needed - it can be said that each quantum carries energy of hν.

With the waves, everything is essentially the same. For a wave with frequency ν and wavelength λ, oscillating with amplitude A around the equilibrium position Z

• Equation of motion: $$$Z(x,t)={Z}_{0}+Acos(2\text{}pi[\text{}nut-x/\text{}lambda])$.

• Energy per wavelength: $$$2\text{}p{i}^{2}\text{}n{u}^{2}{A}^{2}{J}_{\text{}}lambda$.

(where J

$$

$$Class0:{d}^{2}Z/d{t}^{2}-c{w}^{2}{d}^{2}Z/d{x}^{2}=0$$

$$

$$Class1:{d}^{2}Z/d{t}^{2}-c{w}^{2}{d}^{2}Z/d{x}^{2}=-(2\text{}pi\text{}mu{)}^{2}(Z-{Z}_{0})$$

And again, quantum mechanics limits amplitude A to discrete values. Just like for spring oscillations,

• One simple wave of a certain frequency and length consists of n quanta,

• The allowed magnitudes of amplitude A are proportional to √n,

• The allowed energy values of E are proportional to (n + 1/2).

More precisely, as for the ball on the spring,

• Permitted energy values E = (n + 1/2) h ν

• Each quantum of wave carries energy of magnitude h ν

The formula for the expression A is quite complicated, because we need to know how long the wave is, and the exact formula will be too confusing - so let me just write a formula that conveys the right idea. We received most of the formulas by studying endless waves, but for any real wave in nature, the duration is finite. If the wave duration is approximately equal to L, and it has L / λ ridges, then the amplitude is approximately equal to

$$

$$A=(1/2\text{}pi)\text{}sqrt\text{}frac2nh\text{}lambda\text{}nuL{J}_{\text{}}lambda$$

Which is proportional $$$\text{}sqrtnh/\text{}nu$as in the case of a spring, but it depends on L. The longer the wave, the smaller its amplitude - so that for each quantum of the wave the energy is always hν.

That's all - it is shown in the figure below.

On the left - a naive image of waves, where the amplitude is proportional to the square root of the number of quanta, and other amplitudes cannot exist. On the right, a slightly less naive image that takes into account the quantum oscillations inherent in the quantum world. Even in the case of n = 0, some oscillations exist.

What does this mean for our class 0 and class 1 waves?

Since waves of class 0 can have any frequency, they can have any energy. Even for a tiny value of ε, one can always make one quantum of a class 0 wave with a frequency ν = ε / h. For such a small energy, this quantum wave will have a very small frequency and a very long wavelength, but it can exist.

Waves satisfying a class 1 equation are not. Since for them there is a minimum frequency ν

$$

$${E}_{min}=h\text{}n{u}_{min}=h\text{}mu$$

If your tiny energy value ε is less than this, the quantum of such a wave will not work. For all quanta of class 1 waves with a finite wavelength and a higher frequency, E ≥ h μ.

Before we begin to take into account quantum mechanics, the amplitude of the waves, like the amplitude of the ball on the spring, can change continuously; they can be made arbitrarily large or small. But quantum mechanics implies the existence of a minimal nonzero wave amplitude, as in the case of a ball oscillation on a spring. And usually the amplitude can take only discrete values. The allowable amplitudes are such that both for the oscillations of a ball on a spring, and for a wave of any class with a certain frequency ν

• To add one quantum of oscillations, energy h ν is required

• For oscillations of n quanta, the oscillation energy will be (n + 1/2) h ν

Now it's time to apply the knowledge gained to the fields and see when and how the quanta of the waves in these fields can be interpreted as what we call the "particles" of nature.

Source: https://habr.com/ru/post/405699/