What is $\frac{d}{d\psi}\langle\psi| \hat{O} | \psi\rangle$?

Question

I would like to know what is the derivative of an expectation value with respect to the molecular state

$$\frac{d}{d\psi}\langle\psi| \hat{\mathbf{O}} | \psi\rangle$$

Note that here $|\psi\rangle$ is a complex column vector of length $S$ where each of its components depend on the space $q$ and $\hat{\mathbf{O}}$ is a complex $S \times S$ matrix which also depends on the space.

I other words I want to know what is $$\frac{d}{d\mathbf{\Psi}(q)}\int dq \mathbf{\Psi}^{\dagger}(q)\hat{\mathbf{O}}(q) \mathbf{\Psi}(q)$$

Is the answer

$$\frac{d}{d\mathbf{\Psi}(q)}\int dq \mathbf{\Psi}^{\dagger}(q)\hat{\mathbf{O}}(q) \mathbf{\Psi}(q) = \int dq \mathbf{\Psi}^{\dagger}(q)\hat{\mathbf{O}}(q) $$

correct ?

Answer 1

What you have is called a functional derivative. The notation is usually somewhat different. It is defined by the relationship $$ \frac{\delta \psi(q)}{\delta\psi(q')} = \delta(q-q') . $$

In your example, we have $$ \langle\psi|\hat{O}|\psi\rangle = \int \psi^*(q)O(q,q')\psi(q') dqdq' . $$ Note that I have changed the way the $q$'s appear to make the operator a more general kernel function. Now we can apply the functional derivative: $$ \frac{\delta}{\delta\psi(q_0)} \langle\psi|\hat{O}|\psi\rangle = \int \psi^*(q)O(q,q')\delta(q'-q_0) dqdq' = \int \psi^*(q)O(q,q_0) dq . $$ Here we have a different variable $q_0$ for the function with respect to which the functional derivative is evaluated.

Hope this answers your question.