1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Quantum Theory: Minimizing Energy Subject to Normalization Contraint
Loading...