10 - Variational Principle

Physics Theory II PHYS2001
eugene.hickey@tudublin.ie

Technological University Dublin

Summary


Unlike particles in boxes, SHO’s and hydrogen-like ions, almost all problems in quantum mechanics are not exactly solvable. But there are a number of techniques to get sensible, if not exact, solutions for quantum systems. We’ll look at two of these. The first is the Variational Principle in which we guess, and then optimise, a wavefunction in such a way to give an upper bound for the energy levels.

Contents

  1. Variational Method
  2. Application to Particle-in-a-Box
  3. Application to Simple Harmonic Oscillator
  4. Equations
  5. References

More Complex Problems

  • Hydrogen atom can be solved exactly by time independent Schrödinger equation

  • Other problems aren’t so kind

    – Multi-electron atoms

    – Molecules

  • Need to cut some corners

    – Approximations

    – Astute guesses

  • We’ll begin with two foundation methods

    – Variation method

    – Perturbation theory

  • In practice, modifications / combinations of these two used

Variational Method

  • Get approximate energy of ground state without solving Schrödinger equation

  • Gives upper bound for ground state energy

  • Guess a trial wavefunction, \(\phi\)

    – \(\phi\) must be normalisable

    – \(\phi\) must satisfy relevant boundary conditions

    – Symmetry of problem leads us to \(\phi\)

    – Trying to make \(\phi\) as close to what we think the real wavefunction, \(\psi\), is as possible

  • Bound on ground state energy given by:

\(\int_{-\infty}^{+\infty} \phi^* H \phi \; dx^3 \ge \; E_0\)

  • (or \(\int_{-\infty}^{+\infty} \phi^* H \phi \; dx \ge \; E_0\) for 1-D problems)

  • H is the Hamiltonian, the energy operator for the system

  • Can be proved by expanding \(\phi\) in terms of the (unknown) true wavefunctions \(\psi\)

  • Equation above assumes \(\psi\) normalised

  • In practice, skill in using variation method is astute choice of trial wavefunction

    • often start with something like the SHO ground state: \(\psi(x) = C e^{-\alpha x^2}\)

  • If we happen to chance upon actual ground state wavefunction, \(\psi_0\), then we’ll get an energy = \(E_0\)

  • Best way to use variation method

    – construct trial wavefunction with parameters that can be adjusted

    – Minimise value of integral by adjusting parameters


  • Take particle in one dimensional box, problem we solved exactly before

    • Chose \(\phi = x(x-L)\) as the trial wavefunction

    – Satisfies boundary conditions

    – Not normalised yet

    – Remember hamiltonian:

inside the box
\(H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}\)
outside the box
\(H = 0\)

  • \(H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}\)

  • Remember \(\phi = x(x-L)\) is the trial wavefunction

  • integral gives us

  • \(\int_{-\infty}^{+\infty} \phi^* H \phi \; dx\), but really \(\int_{0}^{+L} \phi^* H \phi \; dx = \frac{\hbar^2 L^3}{6m}\)

  • to normalise we get \(\int_{0}^{+L} \phi^* \phi \; dx = \frac{L^5}{30}\)

  • putting together we get:

\(E_1 \le \frac{5 \hbar^2}{mL^2}\)

  • compare to \(E_1 = \frac{\pi^2 \hbar^2}{2mL^2}\)

  • agrees to better than 2% (\(\frac{\pi^2}{2} = 4.935\))

  • Previous example didn’t have adjustable parameters

    – In general not the case

    • to be honest, result was kind of lucky

  • Suppose we don’t know solution for SHO problem

  • Guess that it’s similar to 1-D box we did

  • \(\phi = \sqrt{ \frac{2}{L} } \sin \frac{\pi x}{L}\) for \(0 \le x \le L\)

  • need to make box from \(-L \le x \le +L\) by symmetry, and use cosine functions instead of sine

  • \(\psi(x)= \frac{1}{\sqrt{L}} \cos(\frac{\pi x}{2L})\)

  • get \(E_0 \le \frac{1}{L} \int_{-L}^L \cos \frac{\pi x}{2L}\left(\frac{- \hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} m \omega^2 x^2\right) \cos \frac{\pi x}{2L} dx\)

  • \(E_0 \le \frac{\hbar^2 \pi^2}{8 m L^2} + m \omega^2 L^2 (\frac{1}{6} - \frac{1}{\pi^2})\)

  • set \(\frac{dE_0}{dL} = 0\) to get minimum of this and find \(L = 1.302 \sqrt{\frac{\hbar}{m \omega}}\)

  • then \(E_0 \le 0.839 \hbar \omega\)

  • not a great choice of initial wavefunction

    • \(\psi(x)\) is too heavy when you get towards \(\pm L\)

  • but still an instructive result

  • To find higher energy states, above ground state

    – Use trial wavefunctions that are orthonormal to ground state

    • for example, for the SHO we could use \(\psi_1(x) = A x \cos(\frac{\pi x}{2 L})\)

    • \(\int_{-\infty}^{+\infty} x \cos^2(\frac{\pi x}{2 L}) dx = 0\) because \(x \cos (\frac{\pi x}{2 L})\) and \(\cos (\frac{\pi x}{2 L})\) are orthogonal

  • Note that while estimations of energy levels can be pretty good, harder to get good picture of wavefunctions from adjusted trial wavefunctions

  • Variational Method pretty useful for the quantum mechanics of \(He\) and of the \(H_2\) molecule

Equations

  • \(\int_{-\infty}^{+\infty} \phi^* H \phi \; dx^3 \ge \; E_0\)

References