Quantum Theory: Minimizing Energy Subject to Normalization Contraint

[WisheDeom]

Homework Statement

In a previous problem, I derived that for a given wavefunction $\Psi (x)$ in a potential, the energy of the state could be calculated as a functional of the wavefunction. I now need to minimize the energy, subject to the usual wavefunction normalization constraint, and show that the minimized wavefunction satisfies the energy eigenvalue equation.

Homework Equations

Energy functional:
$$E[\Psi(x)] = \int dx \left(\frac{\hbar^2}{2m} \left|\frac{\partial \Psi}{\partial x} (x)\right|^2 + V(x) |\Psi(x)|^2 \right)$$

Normalization constraint:
$$\int dx |\Psi(x)|^2 = 1$$

Energy eigenvalue equation:
$$\langle x | H | \Psi \rangle = E \langle x | \Psi \rangle$$
$$-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2}(x) + V(x) \Psi(x) = E \Psi(x)$$

The Attempt at a Solution

I know to minimize a functional like above I need the Euler-Lagrange equations, but I'm completely stumped as to how to include a functional constraint.

 

[_coyote_]

I know I need to use a Lagrange multiplier, but I'm not sure how to incorporate it into the Euler-Lagrange equations. Help with this would be greatly appreciated!