8.1: Variational Methods
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So far, we have concentrated on problems that were analytically solvable, such as the simple harmonic oscillator, the hydrogen atom, and square well type potentials. In fact, we shall soon be confronted with situations where an exact analytic solution is unknown: more general potentials, or atoms with more than one electron. To make progress in these cases, we need approximation methods. The best known method is perturbation theory, which has proved highly successful over a wide range of problems (but by no means all). We shall soon be discussing perturbation methods at length. First, though, we shall review two other approximation methods: in this lecture, the variational method, then in the next lecture the semiclassical WKB method. The variational method works best for the ground state, and in some circumstances (to be described below) for some other low lying states; the WKB method is good for higher states.
Variational Method for Finding the Ground State Energy
The idea is to guess the ground state wave function, but the guess must have an adjustable parameter, which can then be varied (hence the name) to minimize the expectation value of the energy, and thereby find the best approximation to the true ground state wave function. This crude sounding approach can in fact give a surprisingly good approximation to the ground state energy (but usually not so good for the wave function, as will become clear).
We’ll begin with a single particle in a potential,
To gain some insight into what we’re doing, suppose the Hamiltonian
Since the Hamiltonian is Hermitian, these states span the space of possible wave functions, including our variational family, so:
From this,
for any
We can see immediately that this will probably be better for finding for the ground state energy than for mapping the ground state wave function: suppose the optimum state in our family is actually
To get some idea of how well this works, Messiah applies the method to the ground state of the hydrogen atom. We know it’s going to be spherically symmetric, so it amounts to a one-dimensional problem: just the radial wave function. Using standard notation,
and for a trial wave function u
(we’re going to take u real).
Messiah tries three families:
and finds
For the three families, then the energy of the best state is off by 0, 25%, 21% respectively.
The wave function error is defined as how far the square of the overlap with the true ground state wave function falls short of unity. For the three families,
Variational Method for Higher States
In some cases, the approach can be used easily for higher states: specifically, in problems having some symmetry. For example, if the one dimensional attractive potential is symmetric about the origin, and has more than one bound state, the ground state will be even, the first excited state odd. Therefore, we can estimate the energy of the first excited state by minimizing a family of odd functions, such as
Ground State Energy of the Helium Atom by the Variational Method
We know the ground state energy of the hydrogen atom is -1 Ryd, or -13.6 ev. The He+ ion has
To get a better value for the ground state energy still using tractable wave functions, we change the wave functions from the ionic wave function
To find the potential energy from the nuclear-electron interactions, we of course use the actual nuclear charge
This could have been figured out from the formula for the one-electron ion, where the potential energy for the one electron is
The kinetic energy is even easier: it depends entirely on the shape of the wave function, not on the actual nuclear charge, so for our trial wave function it has to be
The tricky part is the P.E. for the electron-electron interaction. This is positive.
Each electron has a wave function
Denoting charge probability density by
Collecting terms, the total energy (for
and this is minimized by taking