How is quantum tunneling possible?

Think about single eigenstate, energy just how the state change with time goes (well by factor of $ihbar$) 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.

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?

Apparently, extra energy can be extracted out of the universe for a very short period of time, corresponding to the uncertainty relation given by $$Delta Etimes Delta tgefrac{h}{4pi}$$ This is what quantum electrodynamics (the quantum theory explaining electrostatic force between charges) is based on. Consider this A electron A is present in space, another electron B is placed in the first electrons electric field. Now this electron A would release a photon(apparently borrowing the energy for it from the universe), the other electron B would absorb it, and send another photon back to electron A and hence experience the electron A's presence there (force) through the photons(the first electron would also experience electron B's presence similarly). This energy borrowed by electron A for its photon release must be returned within an interval $Delta t$,(by the returning photon it absorbs from electron B). This is how electrons sense each other's presence. These photons are called virtual ones because we can't detect them. This might not be literally physically true, but the theory does involve violating the energy conservation law through the Heisenberg uncertainty principle.

The argument in the question is as follows:

• There is a finite probability of finding the particle inside of the barrier
• If the particle is inside the barrier its energy is negative
• States with negative (kinetic) energy are impossible

The seeming paradox here is due to mixing classical and quantum concepts. Let us first note that there is no state inside the barrier. The state of a particle in this case a scattered wave, extending to infinity. The probability of measurement finding a particle inside the barrier is indeed non-zero. On the other hand in classical physics the state is identified with a specific location and momentum, i.e. the measured position of the particle is its state. Once we localized a quantum particle at a specific location, it is no more in the scattering state, and has undetermined momentum and energy.

Secondly, let us consider what the conservation of energy means in quantum mechanics.

• We could view it as a fact that a particle in an energy eigenstate remains forever in this eigenstate. But this is not relevant here, since we perform a measurement, collapsing the wave function to a position eigenstate.
• It can be viewed from the point of view of the uncertainty relation, as discussed in the answer by KDN.
• Finally, it can be viewed from the point of view of the correspondence principle and the Ehrenfest theorem, where it will be conservation of the energy average in the quantum mechanical sense: $$langlefrac{p^2}{2m}rangle + langle U(x)rangle neq frac{langle prangle^2}{2m} + U(langle xrangle),$$ where on the left we have the average energy in QM sense, whereas on the right is its classical estimate implied by the argument in the question. These are not the same.

To summarize: quantum mechanics appear paradoxial only as long as one relies on the intuition, grounded in classical phsyics.

In quantum mechanics you have to be careful what you mean by 'the energy'. In QM the energy is calculated using the Hamiltonian operator $hat H=-frac{hbar^2}{2m}nabla^2+U(x)$. Naively you would expect you can just apply it to wavefunctions $hat Hpsi(x)$ and wherever this gives you a larger number the energy is high. But the only way to calculate the energy that makes sense is to calculate the expectation value of the Hamiltonian $langle hat Hrangle=int text{d}x psi^*(x)hat Hpsi(x)$. For this expectation value you can't just look at the energy at a point you have to consider the entire space.

What you are trying to calculate is the energy at a specific point $x$. This only makes sense when you are talking about energy eigenstates. Or states $psi(x)$ that satisfy $$-frac{hbar^2}{2m}nabla^2psi(x)+U(x)psi(x)=Epsi(x)$$ This means you can talk about 'the energy' without confusion: the energy is given by $E$ and likewise $langlehat Hrangle$=E. If you take a look at this last equation you see that there are two regions: $E>U(x)$, the classically allowed region, and $E<U(x)$, the classicaly forbidden region. If you work it out you'll find that in the allowed region the wavefunction curves towards the zero axis like a sine-wave and in the forbidden region the wavefunction curves away from the zero axis like an exponential function.

These two regions just tell you in which way the wavefunction curves. They both have the same energy since they are part of the same energy eigenstate. The best you can do is say that the kinetic energy $hat Tpsi(x)=-frac{hbar^2}{2m}nabla^2psi(x)$ is negative in the forbidden region but again this is not really a useful quantity because ultimately you are interested in expectation values $langlehat Trangle$