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Quantum Theory: Minimizing Energy Subject to Normalization Contraint

  1. Oct 29, 2012 #1
    1. The problem statement, all variables and given/known data
    In a previous problem, I derived that for a given wavefunction [itex]\Psi (x)[/itex] 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.

    2. Relevant equations
    Energy functional:
    [tex]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)[/tex]

    Normalization constraint:
    [tex]\int dx |\Psi(x)|^2 = 1[/tex]

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

    3. 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.
    Last edited: Oct 29, 2012
  2. jcsd
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