If Ψ_1 and Ψ_2 are arbitrary solutions of the equation, then Ψ_1 + Ψ_2 is, too.
Definition 2. Let
|Ψ_i(x)| < k^i for all x,
for some 0 < k < 1, and let all Ψ_i be solutions. We say that the solutions of the equation are closed under convergent sums if the above implies that the sum of all Ψ_i is a solution.
Linear wave equations have the property of Definition 2.
Definition 3. A wave equation is local if we can combine into a solution Ψ two solutions, Ψ_1 to the left from spatial point x_0, and Ψ_2 to the right of x_0, provided that the combined solution is continuous and has a continuous derivative at x_0, and the other parameters of the equation, the potential, the string weight, etc. stay constant in the vicinity of x_0. Note that far away from x_0, the parameters of the equation of the solutions Ψ_1 and Ψ_2 are allowed to be different.
Locality means that we can glue together two solutions if they match in the vicinity of x_0. The different parts of the solution can only "communicate" through the wave function in the vicinity of x_0.
In our previous post about the Klein paradox, we uncovered a fundamental question in the linear wave equation formulation of quantum mechanics.
Suppose that we have a smooth potential barrier, where the potential has a continuous derivative.
e- ------>
____
___/ \___ smooth potential
A B
<-- <-- reflections
A flux of particles meets the barrier from the left. Conventional wisdom says that if there is destructive interference of the reflections from each end of the barrier, then the transmitted flux is 100%.
There is ample evidence that the conventional wisdom is true. The antireflective coating in optics works like this. My eyeglasses seem to reflect mostly green light, because the coating causes destructive interference for yellow light.
The solutions of the Schrödinger wave equation are closed under convergent sums.
The question is what mechanism creates the extra flux from B to the right, if there happens to be a destructive interference of reflected fluxes at A?
If we just add an ad hoc flux from B to the right, that would make the wave function discontinuous at B.
Destructive interference in a linear wave equation
Ψ ---> Ψ_A ---> Ψ_B --> right-moving waves
------------------=========XXXXXXXXXX
A B
<--- Ψ_1 <--- Ψ_2 left-moving waves
Suppose that we have a tense string whose weight per meter differs at different sections. If we feed a wave Ψ from the left to the string, there will be reflections Ψ_1 from A and Ψ_2 from B to the left. We have marked in the diagram all waves in each section.
Theorem 4. Let us assume the solutions of a wave equation are closed under convergent sums and the equation is local. Then destructive interference in reflection can break conservation of energy.
Proof. We assume that reflected waves carry much less energy than transmitted waves.
We assume that there is a destructive interference, so that Ψ_1 contains much less energy than Ψ_2.
Is it possible that the energy is reflected to the right, so that energy is conserved?
We will prove that energy is not conserved.
From the uniform part of the string there is no reflection, because there is a symmetry by translation in the space coordinate x. There is no reason why a reflected wave of a specific phase φ should originate from a uniform section of the string.
Theorem 4. Let us assume the solutions of a wave equation are closed under convergent sums and the equation is local. Then destructive interference in reflection can break conservation of energy.
Proof. We assume that reflected waves carry much less energy than transmitted waves.
We assume that there is a destructive interference, so that Ψ_1 contains much less energy than Ψ_2.
Is it possible that the energy is reflected to the right, so that energy is conserved?
We will prove that energy is not conserved.
From the uniform part of the string there is no reflection, because there is a symmetry by translation in the space coordinate x. There is no reason why a reflected wave of a specific phase φ should originate from a uniform section of the string.
Diagram 1.
Ψ ---> Ψ_A' --->
--------------------=====================
<--- Ψ_1'Diagram 2.
Ψ_A' ---> Ψ_B' --->
======================XXXXXXXXX
<--- Ψ_2'
Diagram 3.
Ψ_A'' --->
-------------------=======================
<--- Ψ_1'' <--- Ψ_2'
Diagram 4.
Ψ_A'' ---> Ψ_B'' --->
=====================XXXXXXXXXX
<--- Ψ_2''
We can continue drawing such diagrams where the upper one tells what wave we feed to the lower one.
Let us sum all the solutions in diagrams 1, 3, 5, 7, ..., and respectively in 2, 4, 6, ....
Odd numbers:
Ψ ---> Ψ_A --->
-------------==================
<--- Ψ_1 <---- Ψ_2
Even numbers:
Ψ_A ---> Ψ_B --->
====================XXXXXXX
<--- Ψ_2
Then we can use the locality to glue together the two diagrams and we get the first diagram of this section.
For energy streams, we get from the Even diagram
E_B = E_A - E_2.
We assumed that there is a significant destructive interference in E_1' and E_1''. We can assume that E_1 is much smaller than E_2', and therefore smaller than E_2:
0 > E_1 - E_2.
Diagram 1:
E = E_A' + E_1'.
E_A'' and so on are very small compared to E_1'. Therefore,
E > E_A.
The outgoing energy flux is
E_B + E_1 = E_A + E_1 - E_2
which is < E. QED.
But energy is conserved in classical mechanics. Which of our assumptions was wrong? Break of locality would be spooky. We conclude that the solutions are not closed under sums. The wave Ψ_2 at A is probably affected by the incoming wave in such a fashion that Ψ_2 reflects completely back.
If a string is uniform, we believe that solutions are closed under sums, but if the weight changes at A, that probably is not true.
Path integral approach?
If solutions of quantum mechanical equations are closed under sums and equations are local, how do we salvage conservation of energy?
What if we think of the quantum mechanical experiment as a path integral? The particle may have different paths. The reflected paths end up with a probability amplitude zero. What rule then inflates the amplitude of transmitted paths? Maybe we need to add a rule of normalization:
Normalization rule 5. If the end result of a path integral would not conserve the probability of finding a particle, we have to normalize the end result by multiplying it by a suitable real number C.
In the many-worlds interpretation of quantum mechanics, some branches have a probability amplitude zero. If we claim that an observer can only exist in a non-zero branch, then the normalization results from this fact. An observer can never see a particle disappear.
The observer must choose one of the branches that have a non-zero probability amplitude. The Born rule states that the probability of each branch has a weight
|Ψ|^2,
where the complex number Ψ is the probability amplitude of the branch.
What about a converse rule? An observer cannot exist in a branch where a particle appeared from empty space. The time-reversed process would make the particle to disappear. The probability amplitude of the history of such an observer would be zero.
Destructive interference happens only in the head of the one scientist?
How does the wave function "know" that there is a destructive interference at A, so that it knows to send more particles forward at B?
Conjecture 5 of
http://meta-phys-thoughts.blogspot.com/2018/10/huygens-principle-smoothens.html
is relevant here. When we say that there is destructive interference at A, we kind of sum path probability amplitudes before any observation is made. Conjecture 5 states that such summing only makes sense in the head of (just one) observing scientist. In his head, the wave function may "collapse" without any spooky action at a distance.
The collapse happens according to the Born rule. The destructive interference inside his head sets an amplitude zero on any observation where he would see the particle number not conserved in the process. We may think that we also normalize the wave function before applying the Born rule, so that the sum of weights is one.
The concept of a "flux" of particles is wrong, because in it we are summing intermediate probability amplitudes without an observation.
The non-conservation of the "fluxes" in destructive interference is a symptom of using a wrong concept.
Wave equations generally assume that one can sum the probability amplitudes at a spacetime point. In which cases does that work and in which not? Our treatment of regularization in Feynman diagrams suggests that in some cases intermediate sums can be infinite.
Make quantum mechanics nonlinear: parallel universes interact?
If we use a path integral approach, is there some way to restore conservation of energy without normalization?
We may assume that just one particle at a time enters the reflection experiment. If reflected back from B, it should somehow know that it must reflect from A to conserve energy. It must somehow interact with other paths.
That is a new concept. The whole path integral approach with propagators assumes that we can calculate the probability amplitudes of paths individually.
In the double slit experiment we assume that different paths interfere - we have to sum their probability amplitudes at each point on the screen. If the paths also interact with each other, that means that the same photon can communicate with its own copies in different "parallel universes".
Since different particles are typically indistinguishable, it makes sense to allow the same particle in different universes to interact with itself. Banning such interaction would be hard, except in the case where we know that there is just one particle.
That is a new concept. The whole path integral approach with propagators assumes that we can calculate the probability amplitudes of paths individually.
In the double slit experiment we assume that different paths interfere - we have to sum their probability amplitudes at each point on the screen. If the paths also interact with each other, that means that the same photon can communicate with its own copies in different "parallel universes".
Since different particles are typically indistinguishable, it makes sense to allow the same particle in different universes to interact with itself. Banning such interaction would be hard, except in the case where we know that there is just one particle.
We do not observe people communicating with their copies in other universes (?). Maybe decoherence makes such communication impossible for large objects?
Claim 6. Different paths in a path integral must interact, if we want to model the destructive interference at a potential step with a path integral.
Normalization does not really work
Light is also a classical wave. Energy fluxes have to be conserved at each stage. It is not enough to normalize them at the final stage.
Suppose that we have an experiment where the reflected energy stream would be 20%, but destructive interference reduces it to 10%.
Energy conservation requires that 90 % is transmitted.
On the other hand, if we normalize 10% + 80% to 100%, then transmission is only 89%.
Quantum mechanical processes should have classical physics as the limit when a large number of particles is concerned.
This shows that the normalization approach is wrong.
Are there any practical experiments about the energy flux in antireflective coatings?
Nonlinear Schrödinger equation
Are there any practical experiments about the energy flux in antireflective coatings?
https://en.m.wikipedia.org/wiki/Nonlinear_Schrödinger_equation
There is ample literature on a nonlinear Schrödinger equation, but at first sight, destructive interference in reflection is not mentioned on the Internet.
A comment on the Physics Stack Exchange links an antireflective coating and destructive interference to retarded and advanced solutions of Maxwell's equations. But if Maxwell's equations are linear, we already proved that they cannot explain the phenomenon.
There is ample literature on a nonlinear Schrödinger equation, but at first sight, destructive interference in reflection is not mentioned on the Internet.
A comment on the Physics Stack Exchange links an antireflective coating and destructive interference to retarded and advanced solutions of Maxwell's equations. But if Maxwell's equations are linear, we already proved that they cannot explain the phenomenon.
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