Do All Lower Energy States Exist When n=3 in an Infinite Potential Well?

[victoria13]

ok so i have a question on infinite potential wells... if you have the energy state when n=3, does that mean that you have to have the energy state when n=2 and n=1?

Ok basically the Q we are given is:

A particle of mass m is trapped in an infinitely deep one-dimensional potential well between x = 0 and x = a, and at a time t = 0 is described by the wave function

psi(x,t=0) = sin(pix/a)cos(2pix/a)

(i) What possible values may be found for the energy of the particle?
(ii) What is the expectation (or average) value of the energy of the particle?
(iii) Give an expression showing the time dependence of the wave function, psi(x,t).
[Hint: Use sinAcosB = 1/2(sin(A + B) + sin(A-B))].

so when i use the hint and find the equation in terms of just sin i get

psi(x) = 0.5sin(3pix/a) + 0.5sin(-pix/a)

so does this mean that there is just the 3rd energy state, or the third and first, or all 3?

thanks

 

[vela]

The particle is in a superposition of the n=3 state and the n=1 state.

 

[victoria13]

so the energies are when n=3 and n=1?

 

[vela]

Yes, the measurement can only result in energies of states which you have some chance of finding the particle in.