Particle in a box Schrodinger equation

[gkiverm]

I'm going though the particle in a box lesson in my physics textbook right now. I understand all the math, but don't understand a lot of the physics behind it. Also this is an intro physics course, we're only covering the basics of quantum mechanics and not going into too much detail.

How come we use the time independent Schrödinger equation for the particle in a box problem? In general, do you always use the time independent Schrödinger equation for stationary state (energy is known with no uncertainty)?

Why is the potential energy of the particle inside the box always equal to zero and infinite outside the box?

Thanks

 

[WannabeNewton]

The problem involves a time-independent potential which is why we resort to the time-independent Schrödinger equation. When you have a time-independent potential you can use separation of variables to write the coordinate representation of the state vector (i.e. the position wave-function) as $\Psi(\mathbf{x},t) = \psi(\mathbf{x})\varphi(t)$. Plugging this into the time-dependent Schrödinger equation it is easy to show that $\varphi = e^{-iEt/\hbar}$ and $\frac{-\hbar^2}{2m}\nabla^2 \psi + V\psi = E\psi$ where $E$ is the separation constant; the latter of these is the time-independent Schrödinger equation.

The solutions $\psi$ are eigenstates of the Hamiltonian operator that you see on the left hand side of the equation of motion for $\psi$; we identify $E$ as the total energy since this is the observable associated with the Hamiltonian operator.

For the particle in a box problem we have $V = 0$ inside the box and $V = \infty$ at the walls of the box because that's how the system is setup. We want the particle to be confined to a finite volume in space but be free within this finite volume so we have zero potential inside (hence it is free inside) while placing an infinite potential barrier in the form of a box to keep the particle confined inside.