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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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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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    $\begingroup$ I can't comment, yet. I'd just like to add to KDN's answer that this is called Quantum Fluctuation. $\endgroup$ – seb Feb 17 '13 at 19:11
  • $\begingroup$ 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? $\endgroup$ – Helios Feb 17 '13 at 19:53
  • $\begingroup$ related: physics.stackexchange.com/questions/53802/… $\endgroup$ – joshphysics Feb 17 '13 at 20:15
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    $\begingroup$ 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. $\endgroup$ – KDN Feb 18 '13 at 1:19
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Think about single eigenstate, energy just how the state change with time goes (well by factor of $i\hbar$) but it never say negative energy is not allowed . In fact negative energy mean growing and decay for single energy eigenstate and positive energy mean wave oscillation. Inside of the wall, it will grow or decay because energy is negative.

Energy always describe how things evolve (no matter it's QM or classical) but QM say it is evolve of wave function and it could be negative just like it could be positive. There is no bias in QM. Then you just sum up all the eigenstates and get the final function.

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Here is what Erwin Schrödinger did:

    Imagine you put a cat in a box, and leave it in there for a day. Now imagine 
you are going to take the cat out, but before you open the box, you don't know if 
it is dead or alive. So in that moment, the cat exists in two states at once, both
dead *and* alive. Therefore the cat is now in two states, in two places, and in two
forms of reality. This is known as quantum tunneling, as reality diverges into two 
types: one where the cat is dead and one where the cat is alive. This can apply to
particles and quantum fields, and allow for a gap to open in spacetime.

Does this make sense now?

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