I have no idea how to even start this.

Problem 4 We have mentioned "stationary states," which are states where the probability distribution does not change with time. We can technically define such a wave function by making it the product of a pure phase term e'() and a function that depends only on space do(); then (y(x, t)? = (wo(r)| which does not change in time. For technical reasons o(t) has to be a lincar function of t: w(x, t) = exp( -ibt )wo(r) where b is a constant. By plugging this representation of @(x, () into the time-dependent Schrodinger equation (i.e., what we have been discussing in class), show that vo(r) must satisfy h2 02 hbyo() = - 2ma7240 - V(c)to(x) This is the time-independent Schrodinger equation for wo (notice there is no time derivative any more, the equation only involves space). The value of b needs to be found, since not all values will, in general, allow us to obtain a vo that satisfies the relation and which also satisfies _w/ dx = 1. For a vo that satisfies this differential equation, the value of hb turns out to be the total energy of the state represented by the wave function. Find- ing functions that satisfy the time-independent Schrodinger equation for various potential energies (i.e., stationary states) is one of the main things one does in Quantum 1