Solving Schrodinger's Equation: Quantum Mechanics Assignment

[doublemint]

Hi,

I am working on my quantum mechanics assignment and I and trying to determine the state of a system at an arbitrary time using two different methods: solving the differential equation (Schrödinger Equation) and evolution operator.
I determined the final results using both methods, however, the solutions are different BUT when I sub in the initial conditions, I get the correct values...
Ive been staring at my work for hours, trying to find a mistake..
Ive attached my work, so if anyone can spot what i did wrong, much thanks!
DoubleMint

The question is in this http://qis.ucalgary.ca/quantech/443/2011/homework_three.pdf" . It is part h).

 

[sgd37]

would it not be more prudent to to calculate $$e^{\hbar \omega L_y} \left| v_3 \right\rangle = \sum_{i=1}^{3} e^{\hbar \omega L_y} \left| v'_i \right\rangle \left\langle v'_i \left\rigt| v_3 \right\rangle$$ where the primed vectors are eigenvectors of Ly. That way you don't have to deal with unnecessary work and matrices and to be honest I'm not even sure what you did there.

For the differential method again I'd work in the eigenbasis of Ly

 

[doublemint]

Hey sgd37,
I did my calculations using the eigenvectors and eigenvalues of Ly. Unless i solved for those incorrectly...

edit: as for the differential method, my professor taught us only the way I've done it. I am not sure how to use the eigenbasis of Ly.

 

[doublemint]

I just did the calculate using your summation notation that you posted and i get the same answer. So its possible that I did the differential method incorrectly...

 

[sgd37]

it isn't a question of wrong eigenvectors. Anyway using my method and your eigenvectors

$$e^{-i \omega L_y t} \left| v_3 \right\rangle = \frac {1}{\sqrt{2}} \left| v'_1 \right\rangle + e^{-i \omega \sqrt{2} t} \frac {1}{2} \left| v'_2 \right\rangle + e^{i \omega \sqrt{2} t} \frac {1}{2} \left| v'_3 \right\rangle = \begin{pmatrix} \frac {1}{\sqrt{2}}-cos(\omega \sqrt{2} t) \\ - \sqrt{2} sin(\omega \sqrt{2} t) \\ \frac {1}{\sqrt{2}}+cos(\omega \sqrt{2} t) \end{pmatrix}$$

where I've corrected the exponents from my previous post

 

[doublemint]

shouldnt the eigenvalues be squared since there are two eigenvectors?

 

[vela]

You made a couple of mistakes when solving the differential equations.

First, you need to get the initial conditions correct. When t=0, you have x=y=0 and z=1, so the Schrödinger equation

$$\begin{pmatrix}\dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix}= \omega\begin{pmatrix}-y \\ x-z \\ y \end{pmatrix}$$

tells you $\dot{x}(0)=\dot{z}(0)=0$ and $\dot{y}(0)=-\omega$.

Start by solving for y(t). You may find it more convenient to write y(t) in terms of sine and cosine instead of complex exponentials. The initial conditions for y(t) and its time derivative will allow you to solve for both arbitrary constants.

Then integrate to find x(t) and z(t). Don't forget the constant of integration. (You left it out in your earlier attempt to find z(t)). The initial conditions will allow you to determine those constants. You should get the solution you're looking for.

 

[doublemint]

I got it!
Thanks for the help Vela and sgd37!