# Particle in a box with the finite depth

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1. Jul 23, 2015

### fricke

For particle in a box with the finite depth, is it traveling wave? or standing wave?

I am confused with its ability to pass through the potential walls that is classically forbidden area which makes me think it is traveling wave. But for particle in a box with infinite potential, I understand that it is standing wave since the presence of infinite potential walls makes a restriction towards the wave function.

So, I kind of have no idea if it is traveling wave or standing wave for particle in a box with the finite depth. Help me please, thank you.

2. Jul 23, 2015

### ShayanJ

At first lets see what is a standing wave. Maybe calling such a thing a wave is misleading, because a wave is, by definition, accompanied by propagation of energy but a standing wave doesn't propagate any energy. The equation of a standing wave is of the form $\psi(x,t)=\chi(t) \phi(x)$. The point in such a definition is that the spatial parts gives an amplitude for the oscillation at a particular point and the temporal part is responsible for that oscillation. So in a standing wave, you only have an infinite number of oscillators lined up that have nothing to do with each other.
Now by the criterion $\psi(x,t)=\chi(t) \phi(x)$, any energy eigenstate of a system with a time-independent potential, is a standing wave because the time dependence of the wave-function is always given by multiplying the spatial part by a $e^{-i\frac E \hbar t}$, so the wave-function of the energy eigenstate is always of the form $\psi(x,t)= e^{-i\frac E \hbar t} \phi(x)$.
But if you consider a state that is the superposition of several energy eigenstates, then you may have a travelling wave.
The point here is that when your problem is indicating that the world is divided into several regions each with a different potential, then you should solve the Schrodinger equation in each region separately and so the above considerations are different for each region.
Another point is that the penetration of the wave-function in the classically forbidden region is done via a exponentially decaying function which is not a wave. But even if the potential was something else that implied that the penetration was done via a wave, then we could have a standing wave in one region that connects to a travelling wave in another region. It would be no problem if you have the right interpretation in mind.

3. Jul 23, 2015