For an infinite square well that starts at x = 0 and has a width of L, we know that the spatial part of the wave function for energy level n can be described as ψn(x) = √(2/L) * sin((nπx)/L), where kn = nπ/L.
The temporal part of the wave function should have the same form as in the previous question, but now we know that if two different wave functions ψa(x,t) and ψb(x,t) are both valid solutions to the Schrodinger equation for our infinite square well, then a superposition of these two wave functions must also be a solution ψ(x,t) = Aψa(x,t) + Bψb(x,t), where a and b are two different positive integers, and A and B are normalization constants. Here, we will assume A = B = 1/√2, which will satisfy the normalization condition.
Write the expression for the ratio of the probability density function at t = 2A to t = 0, where P(x,t0), answer in terms of sin((nπx)/L), sin((mπx)/L) and their squares.
Note that if our wave function was only comprised of ψa(x,t) or ψb(x,t), then we would find the probability density function does not change in time, and the ratio above would equal one, regardless of the two times compared. If your answer to this problem is not equal to one, it tells us that the probability density function is changing in time for the case where our solution is a superposition of two different wave functions.
