I am working through some introductory quantum mechanics materials, and I am stuck on the following problem...
Suppose we are working on a single qubit and we are given the Hamiltonian to be $$H=\begin{pmatrix}\alpha & \beta^* \\\ \beta & \alpha\end{pmatrix}$$ where $\alpha \in \mathbb{R}, \beta \in \mathbb{C}$ and both $\alpha, |\beta| \neq 0$.
We are also allowed to assume that $|ψ(t = 0)⟩ = |0⟩$.
From this, I need to compute the expectation value of the Pauli-$x$ operator, i.e. $$⟨ψ(t)|σ_x|ψ(t)⟩$$ and determine whether it can be independent of the value of $t$.
I am pretty stuck with this. I know that the time evolution is determined through a unitary operator, so I have $$|ψ(t)⟩ = U(t, 0)|ψ(0)⟩ = U(t,0)|0⟩.$$ However, I do not understand how I am to find $U$ and actually perform any computations?
I then tried to find the eigenvalues of the Hamiltonian, $E_{\pm} = \alpha \pm 2\sqrt{|\beta|^2}$. This then let me diagonalise the Hamiltonian as $H|E_{\pm}⟩ = E_{\pm}|E_{\pm}⟩$, thus giving a second basis of quantum states. But again, I do not understand how this would tie in to performing the computation...
Any help on this would be greatly appreciated.
For a time-independent Hamiltonian $H$, it is a direct consequence of Schrödinger's equation that the time evolution operator is given by the unitary operator $U_{t} := \text{exp} (-i H t)$. Once an expression for this operator is found, the expectation value of an observable $S$ given an initial state $\psi_0$ is \begin{align} < \psi_t | S | \psi_t > \; = \; < \psi_0 | U_{t}^* S U_t | \psi_0 >. \end{align}