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I understand that quantum tunneling is a pure example of the uncertainty principle but clearly transistor had to be powered to work properly, anyhow I like to know if it is true that particle must borrow energy to quantum tunnel then quickly pay it back? I thought this is supposed to be probabilistic phenomenon so how does energy come into play?

user6760
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1 Answers1

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Quantum tunneling arises because of probabilities, and what you are suggesting is that the particle must borrow energy in order to tunnel. This is incorrect. Let us consider a one dimensional case where we have a particle of energy $E$ at a position $x$ and a potential barrier

$$V(x)=\begin{cases}V_0, \quad x=[0, 1]\\ 0,\quad \text{otherwise}\end{cases}$$

where $E<V_0$ and $x<0$ at a time $t=t_0$. For $t>t_0$, there is some probability that the particle will be at a position $x>0$. You are suggesting that the particle must borrow energy if $x>0$: you believe that it must be that $E\geq V_0$ if $x>0$. If this was the case, then the particle could exist within the barrier for an infinite amount of time. What we actually find in quantum mechanics is that the particle has a probability of existing within $[0,1]$ that is proportional to a decaying exponential (or something of the sort), so that the particle is not certainly going to pass through the barrier. So energy is not borrowed, the energy is always $E$, it just happens to be the case that there is still some probability that the particle exists within the potential barrier, or that it passes right through it.

Kraigolas
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