Quantum Mechanics: Finding Wavefunction

[CuriosLearner]

Homework Statement

The wave function Ψ of a quantum mechanical system described by a Hamiltonian H ̂ can be written as a linear combination of linear combination of Φ1 and Φ2 which are eigenfunctions of H ̂ with eigenvalues E1 and E2 respectively. At t=0, the system is prepared in the state Ψ0=4/5 Φ1+3/5 Φ2 and then allowed to evolve with time. The wave function at time T=1/2 {h/(E1-E2)} will be (accurate within a phase)
a) 4/5 Φ1 + 3/5 Φ2
b) Φ1
c) 4/5 Φ1 - 3/5 Φ2
d) Φ2
e) 3/5 Φ1 + 4/5 Φ2
f) 3/5 Φ1 - 4/5 Φ2

Homework Equations

Ψ(x,t) = \sumCn Φn(x) exp(-i En 2π t/h)

The Attempt at a Solution

I tried evaluate the Ψ for given value of time but couldn't get any relationship between E1 and E2 to simplify it further. I don't know if this approach is right.

 

[vela]

Show your work.

 

[CuriosLearner]

Ψ(x,T) = 4/5 Φ1 exp{-i π E1/(E1-E2)} + 3/5 Φ2 exp{-i π E2/(E1-E2)}
= 4/5 Φ1 exp{-i π (1+ E2/(E1-E2))} + 3/5 Φ2 exp{-i π E2/(E1-E2)}
= exp{-i π E2/(E1-E2)}[4/5 Φ1 exp{-i π} + 3/5 Φ2]
= exp{-i π E2/(E1-E2)}[-4/5 Φ1 + 3/5 Φ2]
I am stuck here.

 

[gabbagabbahey]

Ψ(x,T) = 4/5 Φ1 exp{-i π E1/(E1-E2)} + 3/5 Φ2 exp{-i π E2/(E1-E2)}
= 4/5 Φ1 exp{-i π (1+ E2/(E1-E2))} + 3/5 Φ2 exp{-i π E2/(E1-E2)}
= exp{-i π E2/(E1-E2)}[4/5 Φ1 exp{-i π} + 3/5 Φ2]
= exp{-i π E2/(E1-E2)}[-4/5 Φ1 + 3/5 Φ2]
I am stuck here.

exp{-i π E2/(E1-E2)} is a constant phase factor, so "accurate within a phase" your wavefunction is just the stuff in square brackets.

 

[CuriosLearner]

exp{-i π E2/(E1-E2)} is a constant phase factor, so "accurate within a phase" your wavefunction is just the stuff in square brackets.

Does this really solve the problem? I mean we can add a π further to the overall phase (can we?) and it will be the option C that is listed there. Is this correct?
Also I was wondering if E1<E2 always. Because in that case the time T mentioned would be negative. What would it imply?

 

[gabbagabbahey]

Does this really solve the problem? I mean we can add a π further to the overall phase (can we?) and it will be the option C that is listed there. Is this correct?

I'm not sure what you mean here by "add a π further to the overall phase". Do you understand what a phase factor is in this context? Do you understand why we can safely ignore a constant phase factor?

Also I was wondering if E1<E2 always. Because in that case the time T mentioned would be negative. What would it imply?

Negative time is nothing special, you only measure differences in time. That said, it is probably safe to assume that E1<E2 for this problem. You should also realize that both E1 &E2 are real-valued constants (why?) and thus exp{-i π E2/(E1-E2)} is just some complex-valued constant.

 

[CuriosLearner]

I'm not sure what you mean here by "add a π further to the overall phase". Do you understand what a phase factor is in this context? Do you understand why we can safely ignore a constant phase factor?

I think that is because any constant phase factor gets canceled out when you write the Schrödinger's equation. So ψ is only accurate within a constant complex phase. I actually meant 'iπ' to be added to that constant phase so as to make it equal to option C there. Also physical significance of the phase should be same as that of phase in any wave equation.

Negative time is nothing special, you only measure differences in time. That said, it is probably safe to assume that E1<E2 for this problem. You should also realize that both E1 &E2 are real-valued constants (why?) and thus exp{-i π E2/(E1-E2)} is just some complex-valued constant.

Yes E1 and E2 are real because they are eigenvalues Hamiltonian which is Hermitian. But I would like to know when E1>E2 particularly. Also regarding negative time, doesn't it mean that we are finding the wavefunction before t=0? i.e. before the system was prepared in such state?
Kindly excuse my lack of knowledge and understanding. I am only a beginner in quantum mechanics. Thanks a lot for your help.