Linear Combination of 2 particular solutions of the TDSE

[A9876]

Homework Statement

1. A particular solution \Psi_n(x,t) of the TDSE, -iℏ∂\Psi/∂t=\hat{H}\Psi, can be constructed by taking \Psi_n(x,t)=ψ_n(x)exp(-iE_nt/ℏ) where the ψ_n are solutions of the TISE.

a) Show that a linear combination of 2 such solutions c₁\Psi₁(x,t) + c₂\Psi₂(x,t) is also a solution of the TDSE but not a solution of the TISE.

b) Consider a Hamiltonian \hat{H} with eigenvectors \phi₁ and \phi_-1 and corresponding energy eigenvalues E₁=ℏω and E_-1=-ℏω. If at time t=0, the state of the system is \Psi(t=0)=c₁\phi₁(t=0)+c_-1\phi_-1(t=0), give the state of the system at an arbitrary later time t. This system demonstrates recurrences, where \Psi(t)=\Psi(0). At what time will this occur?

c) What's the expectation value of the energy \langleE\rangle at an arbitrary later time t for the case c₁=c_-1.

d) Consider now the 2 (also orthonormal) quantum states: v_±(t)=(\phi₁(t)±\phi_-1(t))/√2. Find \phi₁(t) and \phi_-1(t) in terms of the v_±(t)

e) If at time t=0, the state of the system is found to be ψ(0)=v_-, find the probability as a function of time, that ψ(t) will be found in the state v₊

Homework Equations

The Attempt at a Solution

a) Not sure if this is right

\Psi(x,t)=c₁ψ₁(x)exp(-iE₁t/ℏ)+c₂ψ₂(x)exp(-iE₂t/ℏ)

|\Psi(x,t)|²= |c₁|²|ψ₁|²+|c₂|²|ψ₂|²+c₁*c₂ψ₁*ψ₂exp(-i(E₂-E₁)t/ℏ)+c₁c₂*ψ₁ψ₂*exp(-i(E₁-E₂)t/ℏ)

Not a stationary state since |\Psi(x,t)|²≠ |\Psi(x,t=0)|² ∴ solution of TDSE but not of TISE

b) Not sure how to finish the question

\Psi(x,t)=c₁\phi₁(x)exp(-iE₁t/ℏ)+c_-1\phi_-1(x)exp(-iE_-1t/ℏ)

=c₁\phi₁(x)exp(-iωt)+c_-1\phi_-1(x)exp(iωt)

 

[voko]

For a), you merely need to substitute the linear combination into the equations, and show that one is satisfied, and the other is not.

For b), what is the relation between the solutions of the time-dependent and time-independent equations? Can you use the result of a)?

 

[A9876]

For a), you merely need to substitute the linear combination into the equations, and show that one is satisfied, and the other is not.

For b), what is the relation between the solutions of the time-dependent and time-independent equations? Can you use the result of a)?

For a) what equations do I substitute the linear combination into? I tried to sub it into the TDSE but it didn't seem to work :s

 

[voko]

For a) you have to sub the combo into both equations, and TDSE must be satisfied identically, while TISE must not. Unless you show exactly what you do, it is hard to say what goes wrong.

 

[mathskier]

For (b), you just need to find when the complex exponentials have argument equal to a multiple of 2*pi, and then they will equal 1. It's very simple from where you got.