The three lowest energy states of an infinitely deep square well (of width L, between x=0 and x=L) are:
Ψ1(x,t) = N sin(πx/L) e-iω1t
Ψ2(x,t) = N sin(2πx/L) e-iω2t
Ψ3(x,t) = N sin(3πx/L) e-iω3t
- N = sqrt(2/L) is the normalization, to make the total probability = 1.
- Each wave function oscillates with a different frequency, ωi = 2πfi = Ei/ħ. This relation between energy and frequency is the same as for photons.
Suppose a particle in the well is described by the wave function Ψ2. If you measure its energy, what result will you obtain, as a multiple of the ground-state energy E1?
E2 is a factor of (2)2 or 4 larger than the ground-state energy E1
Suppose the particle's wave function isΨ = 0.616Ψ1 + 0.7Ψ2 + 0.361Ψ3. If you measure the energy of the particle, what is the probability that you will obtain these results:
P(x) = |ψ(x,t)|2
The Attempt at a Solution
I tried computing |ψ(x,t)|2 but the expression is nasty and I don't see how I can shrink this expression down. I honestly have no clue how to interpret most of these symbols or even how to start the problem itself.