What Is the Tunnel Effect in Time-Independent Quantum Mechanics?

[LagrangeEuler]

In case of tunnel effect in quantum mechanics we often consider time independent Schroedinger equation with potential $0$, when $x<0$ then some $V_0$ when $0\leq x\leq a$ and $0$ when $x>a$ so potential barrier problem. And energy of particle that we send to barrier is $E<V_0$. In that case energy of the particles that past barrier will be the same as energy of particles before barrier. Why is that case? How we could even talk about tunneling in case of time independent problem?

 

[Jilang]

Conservation of energy. Why do you think it would be different?

 

[LagrangeEuler]

Yes I know that is conservation of energy. But why there do not exists some dissipation of energy when particles interact with potential $V_0$. When we have time dependent solution we have also uncertainty
relation $\Delta E \Delta t \approx\frac{\hbar}{2}$.

 

[Jilang]

The evolution of the wavefunction would not be reversible (or very highly unlikely to be reversible) if there were dissipation.