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How is quantum tunneling possible?

According to quantum mechanics, each particle is represented by a probability density function. This function must be continuous, and therefore when we look at a particle near a potential barrier, we deduce that there is a finite probability for finding the particle inside the barrier (and as a result, beyond the barrier). If the particle can be found inside the barrier, his energy will be negative. This state sounds impossible. Where does the extra energy come from?

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Related: physics.stackexchange.com/q/34091/2451 –  Qmechanic Sep 12 at 17:45

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The instantaneously computed value for the kinetic energy would appear to be negative, as you propose. However, the reason for quantum tunneling has to do with Heisenberg uncertainty relations. You may find a particle in a classically forbidden region, but you are much less likely to find it here. The length of time of any departure from this energy conservation condition is limited by the energy departure, so that \begin{equation} \Delta E \Delta t \ge \hbar/2. \end{equation} The larger the apparent energy violation, the more fleeting the event. The uncertainty principle essentially allows the system to momentarily have enough energy for the particle to be in the forbidden region, with the proviso that it may not do so for very long. (This weirdness is also responsible for the Casimir effect.)

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I can't comment, yet. I'd just like to add to KDN's answer that this is called Quantum Fluctuation. –  lomppi Feb 17 '13 at 19:11
    
Thanks! this uncertainty relation is quite unique, because there is no time operator in quantum mechanics. I googled it and found this proof. it seems that delta t means evolution time of some arbitrary operator rather than simple clock. I'm not really sure what is the meaning of this relation. can energy preservation principle be broken for a short time? –  Helios Feb 17 '13 at 19:53
    
related: physics.stackexchange.com/questions/53802/… –  joshphysics Feb 17 '13 at 20:15
    
It's not so much can energy conservation be broken, It's more that defining the energy of a state takes time. The larger interval you take to measure the energy of a system, the more certain you are about it's energy. If you measure over a very short time and happen to find the particle tunneling, you have only a rough idea of the energy of the system. Measuring the same system over a long time (so that $\Delta t$ is large) yields a much smaller uncertainty in the energy. However, measuring the particle over a long time, you will only rarely find it tunneling. –  KDN Feb 18 '13 at 1:19

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