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Evanescent waves are the mechanism beind both quantum tunneling and frustrated total internal reflection in @SteveB's answer. Evanescent waves and frustrated total internal reflection are not limited to light, but can occur in any phenomena governed by the wave equation, including sound and water waves.


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As for "a classical analog to quantum mechanical tunneling", theoretically one can jump over a classical barrier having lesser kinetic energy than the potential energy one's mass would have at the top of the barrier. In fact, in the course of a high jump, one can bend over the barrier in such a way that one's center of gravity will be outside of the body and ...


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First of all what you need to understand about quantum physics is that it's a theory of probability, not realist classical probability, but it is still a probability theory. The second thing you need to understand is that realism is wrong. Conservation of energy in quantum physics simply means that the Hamiltonian is not time dependent. That's it. From this ...


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Of course, one usual intepretation of quantum tunneling is that the particle will borrow some energy from the vaccum in order to pass an unsurmountable barrier otherwise and then restitute it asap after crossing the barrier. As many others have said, this is a valid interpretation. I am not sure it is necessary though. In fact, in quantum tunneling, what ...


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There are two ways to see the analogy between the "quantum diffusion" and classical diffusion. The first one, I think the easier one is comparing the Schrödinger equation with the diffusion equation: $$i \partial_t \psi = -\sum \partial_{xx} \psi$$ (forgetting all the $\hbar,m$ factors) When you transform $t \to -i \tau$ you get the usual diffusion equation ...


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Consider the path integral: $$\int Dx \exp\left(i\int\left(\frac{m\dot{x}^2}{2}-V\right)dt\right)$$ You can consider paths in "imaginary time" by performing a Wick rotation $t\to i\tau$ $$\int Dx \exp\left(-i^2\int\left(\frac{m}{2}\left[\frac{1}{i}\frac{dx}{d\tau}\right]^2-V\right)d\tau\right)$$ $$\int Dx ...


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Frustrated total internal reflection is an optical phenomenon. It's such a close analogue to quantum tunneling that I sometimes even explain it to people as "quantum tunneling for photons". But you can calculate everything about it using classical Maxwell's equations.


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You are right to be confused because these processes do not conserve energy when taken individually. If you are lucky a couple of particles may appear close to you and you can collect their energy for free. However, the inverse is also equally likely. Imagine you spend some energy and store it in a couple of particles, then there is a chance that these will ...


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Temporary violations of conservation of energy is allowed in quantum mechanics. A system can make a transition to a state that violates conservation energy by an amount $\Delta E$ as long as it stays in that state for a time shorter than $\Delta t$ where $\Delta E\Delta t \leq \hbar/2$. This is a form of Heisenberg's uncertainty principle.



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