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5) quantum waves / SurprizingFacts

Reminder: a quantum sphere on a spring

In the first article of the series we studied the ball of mass M on the spring of rigidity K, and found that its oscillations:

• There will be a formula

 $ z (t) = z_0 + A cos [ 2 π ν t ] $ "data -tex = "inline

.
• Energy

 $ E = 2 π ^ 2 ν ^ 2 A ^ 2 M $ "data-tex =" inline

.
• Equation of motion

 $ d ^ 2z / dt ^ 2 = - K / M (z - z_0) $ "data-tex =" inline

Where the equation of motion Leads to ν = √ K / M / 2π, but allows the amplitude A to be of any positive value. Then in the second article we saw that quantum mechanics, applicable to oscillations, limits their amplitude – it can not be any more. Instead, it is quantized, it must take one of an infinite number of discrete quantities.

 $ A = (1/2 π)  sqrt {2 nh } / Ν M $ "data-tex =" display

Where n = 0, 1, 2, 3, or 44, or generally any integer greater than or equal to zero. In particular, A can be equal to

 $ (1/2 π)  sqrt {2 h} / ν M $ "data-tex =" inline

but it can not be less – Only zero. We say that n is the number of quanta of oscillations of the ball's motion. The energy of the ball is now quantized also:

where n = 0, or 1, or 2, or 3, or 44, or any integer greater than or equal to zero. In particular, A can be as small as (1/2 π) √ 2 h / ν M, but it can not be anything smaller, except zero. We say that n is the number of quanta of oscillation in the ball's motion. The energy of the ball is also now quantized:

 $ E = (n + 1/2) h ν $ "data-tex =" display

]
The most important thing here is that to add one quantum of ball vibrations, we need the energy of the quantity hν – we can say that each quantum carries energy 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 + A Cos (2π [ν t – x/λ]).
• Energy per one wavelength:

$ inline $ 2 π ^ 2 ν ^ 2 A ^ 2 J_λ $ inline $

(where J λ is a constant depending on From, say, the rope, if we are talking about waves on a rope), several possible equations of motion, of which we select two for study:

 $ Class 0: d ^ 2Z / dt ^ 2-cw ^ 2 d ^ 2Z / dx ^ 2 = 0 $ "data-tex =" display

 $ Class 1: d ^ 2Z / dt ^ 2 - cw ^ 2 d ^ 2Z / dx ^ 2 = - (2 π μ) ^ 2 (Z-Z_0) $ "data-tex =" display

Again, quantum mechanics limits the amplitude A to discrete values. Just as for oscillations on a spring,

• One simple wave of a certain frequency and length consists of n quanta,
• The allowed magnitude of the amplitude A is proportional to √n,
• The allowed energy values ​​of E are proportional to (n + 1/2).

More precisely, as for a sphere on a spring,

• Permitted energy values ​​E = (n + 1/2) h ν
• Each wave quantum transfers the energy of the quantity 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's just write a formula that conveys the correct Idea. Most of the formulas are obtained by studying infinite waves, but for any real wave in nature, the duration is finite. If the wavelength is approximately L and its L / λ crests, then the amplitude is approximately

$$ display $$ A = (1/2 π) sqrt { frac {2 nh λ } {Ν L J_λ}} $$ display $$

What is proportional to

 $  sqrt {nh / ν} $ "data-tex =" inline

as in the case of a spring, but depends on L. The longer the wave, the smaller its amplitude – so that for each quantum of the wave energy is always equal to hν.

That's all – it's shown in the picture below.

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

Corollary

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

Since waves of class 0 can be of any frequency, they can have any energy. Even for a tiny ε value, 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 large wavelength, but it can exist.

Waves satisfying the equation of class 1 are not the same. Since there is a minimum frequency ν for them min = μ, there is also a quantum of minimal energy for them:

 $ E_ {min} = h ν_ {min} = h Μ $ "data-tex =" display

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

The result

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 oscillations of a sphere on a spring. And usually the amplitude can only take discrete values. The admissible amplitudes are such that both for oscillations of the ball on the spring and for waves of any class with a certain frequency ν

• To add one vibration quantum requires an energy h ν
• For oscillations of n quanta, the oscillation energy will be (n + 1/2) h ν

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

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