# Tunneling through a PE barrier

I'm still in Intro Physics, and I'm a little perplexed by the concept of tunneling. An alpha particle escapes the nucleus by overcoming a potential energy barrier created by Coulombic repulsion with the protons of the nucleus, but this is a spontaneous process and does not require E input, according to my textbook. Energy conservation is violated to allow the alpha particle to move through this E barrier for a brief time interval because unk(E)unk(t) >= hbar. What's going on here... I mean, is this really what happens... How can energy conservation be violated? I thought that despite all the quantum phenomena violating classical Newtonian physics, that energy conservation still is supposed to remain constant?

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Conservation of energy is not violated. What you think is kinetic energy, namely the classical kinetic energy of a point particle as it travels through the barrier, seems to be negative, i.e., $$K = E-U(x)$$. The total energy, namely $$E$$, remains constant, and so conserved.

So now you just need to come to grips with the fact that the kinetic energy is negative. In quantum mechanics, the kinetic energy is an operator acting on a wavefunction, and in 1D is given by $-\hbar^2 \frac{d}{dx}$. If the kinetic energy is negative, it means that the wavefunction is a growing or (usually) decaying, instead of oscillating. Think of this operator acting on $$\psi(x) = e^{-\lambda x}$$, versus $$\psi(x) = e^{i k x}$$.

dont understand what is Ibrits talking about... too complex -.-"
What do you mean by kinetic energy is an operator acting on a wavefunction" and "wavefunction is a growing or (usually) decaying, instead of oscillating"?

malawi_glenn