Undergrad Schrodinger Equation from Ritz Variational Method

[Samama Fahim]

(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method)

1) Are $\frac{\delta}{\delta \psi^{*}}$ derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get $\int d^3x (\hat{H}\psi)$ in equation 11.35a and $\int d^3x \psi$ in 11.35b, but why would $\int d^3x (\hat{H}\psi)$ be equal to just $\hat{H}\psi$ and $\int d^3x \psi$ to just $\psi$?

2) Moreover, we replace $\delta \bra{\psi}\hat{H}\ket{\psi}$ in 11.34 with $\hat{H}\psi$ and $\delta \bra{\psi}\ket{\psi}$ with $\psi$ resulting in the equation at the bottom. Why is that? Is it that $\delta (\psi^{*}\hat{H}\psi) = \hat{H}\psi \delta \psi^{*}$ because we are looking at variation in $\psi^{*}$ and so we can take $\hat{H} \psi$ out?

 

[Orodruin]

1) Those are functional derivatives, not regular derivatives.

2) The functional derivative wrt $\psi^*$ does not vary $\psi$ or $H$.