# How do I calculate the expectation value $\langle\Psi_0| H | \Psi_1\rangle$ for column vector wave functions?

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.

• transpose $\psi_0$? Jun 11, 2021 at 8:40
• Your $H$ seems to be anti-diagonal rather than diagonal. Is it correct? Jun 11, 2021 at 8:58

Let us represent the state vectors as $$\left| \psi_0 \right> = a\left| \phi_1 \right> + b\left| \phi_2 \right> \equiv \begin{pmatrix} a \\b \end{pmatrix}$$ and $$\left| \psi_1 \right> = c\left| \phi_1 \right> + d\left| \phi_2 \right> \equiv \begin{pmatrix} c \\ d \end{pmatrix}$$, where $$\left| \phi_1 \right>$$ and $$\left| \phi_2 \right>$$ are the basis states, the "bra"s would be the corresponding conjugate transpose, i.e. take the complex conjugate of all the elements and tranpose the matrix, $$\left< \psi_0 \right| \equiv \begin{pmatrix} a^* & b^* \end{pmatrix}$$. The desired expression can then be represented as
$$\left< \psi_0 \right| H \left| \psi_1 \right> = \begin{pmatrix} a^* & b^* \end{pmatrix} \begin{pmatrix} 0 & \lambda \\ \lambda & 0 \end{pmatrix} \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a^* & b^* \end{pmatrix} \begin{pmatrix} \lambda d \\ \lambda c \end{pmatrix} = \lambda \left( a^* d + b^* c\right)$$