- #1
RJLiberator
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Homework Statement
Let |v(t)> ∈ℂ^2 by the time-evolving state of a qubit.
[tex]If |v(0)> =\begin{pmatrix}
0 \\
1
\end{pmatrix}[/tex]
, and the Hamiltonian of the system is [tex]H =
\begin{pmatrix}
0 & -iλ \\
iλ & 0
\end{pmatrix} (where λ∈ℝ)[/tex]
what is |v(t)>?
Homework Equations
Time dependent schrodinger Equation:
iħ*d/dt(|v(t)>=H*|v(t)>
iħ*d/dt(α_j (t)) = α_j(t)*λ_j
The Attempt at a Solution
We just learned this material at the end of last lecture and I need to apply it to a couple final homework problems.
Overall, this seems like a straightforward computation but I'm severely struggling to decipher what I need to do with the giving material.
I first calculated the eigenvalues of the Hamiltonian as the teacher stated. This came out to be +/- λ.
Next, the teacher suggested expanding |v(t)> to something, I'm not really sure what he means by this.
What do the alphas represent? I have no idea... All I have is:
α_j (t) = α_j(0) e^(-iλ_i*t/ħ)
Can some please help me decipher this / lead me to start?
Thank you.