I am trying to perform a perturbation for a system but I get really confused when trying to calculate an expectation for a column vector wave function. Hamiltonian is a 2×2 diagonal matrix and I am trying to perform a perturbation.
Perturbation matrix: \begin{equation} H = \begin{bmatrix} 0 & \lambda\\ \lambda & 0 \end{bmatrix}\qquad 2 \times 2 \;\text{matrix} \end{equation}
This is the perturbation of the Hamiltonian. By diagonal system Hamiltonian, I come up with wave functions like $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ column vector wave functions (2×1).
What I am trying to perform is this: \begin{equation} \langle\Psi_0| H | \Psi_1\rangle \end{equation}
Now, isn't there a dimensional problem? $\Psi_0$ and $\Psi_1$ are 2×1 matrices and $H$ is 2×2 matrix. So, left multiplication will not work.
In fact, this is a two-state system with defined Hamiltonians for Interaction and system energies. The interacted system is a CLASSICAL oscillator. I am trying to solve the spontaneous emission possibility.