I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

$$ E = \int d^3r \left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right] $$ Then we are told to make the transformation

$$\Psi \to \Psi + \delta\Psi $$

Noting the perturbation is conventionally normalised

$$\int d^3r (\Psi + \delta\Psi)^2 = 1$$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$$\int d^3r \left[ \frac{\hbar}{2m} \nabla(\Psi+\delta\Psi)\cdot\nabla(\Psi+\delta\Psi) + V(\vec r)(\Psi+\delta\Psi)^2\right]$$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

$$ E = \int d^3r \left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right]. $$ Then we are told to make the transformation

$$\Psi \to \Psi + \delta\Psi . $$

Noting the perturbation is conventionally normalised

$$\int d^3r (\Psi + \delta\Psi)^2 = 1.$$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$$\int d^3r \left[ \frac{\hbar}{2m} \nabla(\Psi+\delta\Psi)\cdot\nabla(\Psi+\delta\Psi) + V(\vec r)(\Psi+\delta\Psi)^2\right].$$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

$$ E = \int d^3r \left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right] $$ Then we are told to make the transformation

$$\Psi \to \Psi + \delta\Psi $$

Noting the perturbation is conventionally normalised

$$\int d^3r (\Psi + \delta\Psi)^2 = 1$$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$$\int d^3r \ (1+\delta)^2\left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right]$$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

$$ E = \int d^3r \left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right] $$ Then we are told to make the transformation

$$\Psi \to \Psi + \delta\Psi $$

Noting the perturbation is conventionally normalised

$$\int d^3r (\Psi + \delta\Psi)^2 = 1$$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$$\int d^3r \left[ \frac{\hbar}{2m} \nabla(\Psi+\delta\Psi)\cdot\nabla(\Psi+\delta\Psi) + V(\vec r)(\Psi+\delta\Psi)^2\right]$$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

E = $\int d^3r [ \frac{\hbar}{2m} \nabla\Psi.\nabla\Psi + V(\vec r)\Psi^2]$

Then we are told to make the transformation

$\Psi \to \Psi + \delta\Psi $

Noting the perturbation is conventionally normalised

$\int d^3r (\Psi + \delta\Psi)^2 = 1$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$\int d^3r [(1+\delta)^2][ \frac{\hbar}{2m} \nabla\Psi.\nabla\Psi + V(\vec r)\Psi^2]$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector identities):

$$ E = \int d^3r \left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right] $$ Then we are told to make the transformation

$$\Psi \to \Psi + \delta\Psi $$

Noting the perturbation is conventionally normalised

$$\int d^3r (\Psi + \delta\Psi)^2 = 1$$

The question asks to show the change in the energy functional vanishes to first order in $\delta\Psi$. I'm assuming that means show the $\delta$ terms of order 1 vanish but I have no idea how to go about that. If I straight up plug in the transformation into the energy functional I think we get the below expression on RHS:

$$\int d^3r \ (1+\delta)^2\left[ \frac{\hbar}{2m} \nabla\Psi\cdot\nabla\Psi + V(\vec r)\Psi^2\right]$$

Any tips from here? We haven't covered perturbation theory or calculus of variations.. Do I need to look at the small change in the LHS as well ($E + \delta E$) and express that as a power series? I'm still not sure how that would help.