1
$\begingroup$

I saw something very strange when I was studyng about the variational method. In the text, to minimize the functional $$E[\psi] = \frac{\langle \psi |\hat{H}|\psi \rangle}{\langle \psi |\psi \rangle},$$the author derive it with respect to the ket $|\psi \rangle$, and write $$\frac{\partial E}{\partial |\psi \rangle} = 0,$$ and the same for the bra. But does it make sense? I have never seen it before.

$\endgroup$
4
$\begingroup$

Yes, it does make sense. Remember that a ket is just a vector. A derivative with respect to a vector simply stands for a gradient. For example, in ordinary calculus notation, $$\frac{\partial f(\mathbf{r})}{\partial \mathbf{r}} \equiv \nabla_{\mathbf{r}} f(\mathbf{r}).$$ One easy way to carry out the derivative is to work in components. Let $|\psi \rangle = \sum_i \psi_i |i\rangle$. Then $$\frac{\partial E(|\psi\rangle)}{\partial |\psi \rangle} = \sum_i \frac{\partial E}{\partial \psi_i} |i \rangle.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.