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Linear Combination of 2 particular solutions of the TDSE

  1. Aug 23, 2013 #1
    1. The problem statement, all variables and given/known data

    1. A particular solution [itex]\Psi[/itex]n(x,t) of the TDSE, -iℏ∂[itex]\Psi[/itex]/∂t=[itex]\hat{H}[/itex][itex]\Psi[/itex], can be constructed by taking [itex]\Psi[/itex]n(x,t)=ψn(x)exp(-iEnt/ℏ) where the ψn are solutions of the TISE.

    a) Show that a linear combination of 2 such solutions c1[itex]\Psi[/itex]1(x,t) + c2[itex]\Psi[/itex]2(x,t) is also a solution of the TDSE but not a solution of the TISE.

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

    c) What's the expectation value of the energy [itex]\langle[/itex]E[itex]\rangle[/itex] at an arbitrary later time t for the case c1=c-1.

    d) Consider now the 2 (also orthonormal) quantum states: v±(t)=([itex]\phi[/itex]1(t)±[itex]\phi[/itex]-1(t))/√2. Find [itex]\phi[/itex]1(t) and [itex]\phi[/itex]-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+

    2. Relevant equations



    3. The attempt at a solution

    a) Not sure if this is right

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

    [itex]|[/itex][itex]\Psi[/itex](x,t)[itex]|[/itex]2= [itex]|[/itex]c1[itex]|[/itex]2[itex]|[/itex]ψ1[itex]|[/itex]2+[itex]|[/itex]c2[itex]|[/itex]2[itex]|[/itex]ψ2[itex]|[/itex]2+c1*c2ψ12exp(-i(E2-E1)t/ℏ)+c1c21ψ2*exp(-i(E1-E2)t/ℏ)

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

    b) Not sure how to finish the question

    [itex]\Psi[/itex](x,t)=c1[itex]\phi[/itex]1(x)exp(-iE1t/ℏ)+c-1[itex]\phi[/itex]-1(x)exp(-iE-1t/ℏ)

    =c1[itex]\phi[/itex]1(x)exp(-iωt)+c-1[itex]\phi[/itex]-1(x)exp(iωt)
     
  2. jcsd
  3. Aug 27, 2013 #2
    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)?
     
  4. Aug 28, 2013 #3
    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
     
  5. Aug 28, 2013 #4
    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.
     
  6. Sep 1, 2013 #5
    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.
     
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