Schrodinger Equation from Ritz Variational Method

In summary, The derivatives in equations 11.35a and 11.35b are functional derivatives. The integral expressions can be simplified because the functional derivatives do not vary ##\psi## or ##H##.
  • #1
Samama Fahim
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ritz variationalk principle.JPG

(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?
 
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  • #2
1) Those are functional derivatives, not regular derivatives.

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