exp(i (p • r - E t) / ħ),
where the energy E ≥ 0.
--> v a <---
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__/ \__ |
wave potential
packet wall
The wave packet collides with a potential wall which moves at an accelerating velocity.
Is there any reason why the decomposition of the reflected wave would not contain plane waves where E < 0? Such plane waves are called negative frequencies.
In our blog we have discussed this setup many times. The reflected wave packet probably is a "chirp", in which the main frequency of the wave changes with time. A chirp should contain negative frequencies. Hawking radiation is derived from the Fourier decomposition of a chirp.
Let us set V = 0 outside the potential wall. Let
ψ(x, t) = exp(i (p x - E t) / ħ).
We get
E ψ(x, t) = p² / (2 m) ψ(x, t).
The energy E cannot be negative, unless we allow p to be imaginary. But if p is imaginary, then ψ grows exponentially as x →-∞. That does not seem sensible.
We conclude that the Schrödinger equation does not allow a chirp to exist. This would imply that the equation does not have a solution for an accelerating potential wall?
What if we let the wave accelerate in a potential slope?
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\ V potential
\
\
\
Instead of accelerating the potential wall, we could let a potential slope accelerate the incoming wave packet.
This is a very basic configuration in which we would expect the Schrödinger equation to have a solution. The energy E of an incoming component wave is positive and fixed.
In what way is this different from the accelerating wall?
The total energy of the particle,
E = p² / (2 m) + V
is constant. There is no chirp.
What does gravity say about this? If we let the particle fall freely, we may be able to solve its wave function in the comoving frame. But how do we map it for a static observer who is accelerating in the gravity field?
The antiparticle
Paul Dirac realized in 1928 that in the Dirac equation, we have to allow negative energies, in order to be able to Fourier decompose an arbitrary wave packet. He interpreted the negative energy particle as the positron.
Our reasoning above suggests that the same problem already comes up with the Schrödinger equation. We should allow a particle to possess a negative energy – though then the particle does not satisfy the Schrödinger equation.
The Dirac equation has problems handling a large potential step. It is known as the Klein paradox. Could a modified Schrödinger equation handle it correctly?
Conclusions
We will not prove it here, but it is likely that the Schrödinger equation does not have solutions if there is an accelerating potential (wall). A solution should contain "negative frequencies", which are banned by the Schrödinger equation.
Since the Schrödinger equation is just a nonrelativistic approximation, it is not shocking that there are no solutions. A much more interesting thing is what happens when the Dirac equation contains an accelerating potential wall. Are electrons converted into positrons? We have written about that mystery in our blog.
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