- 9.1.1 Basic variational statement
- 9.1.2 Differential form of the statement
- 9.1.3 Using Lagrangian multipliers

9.1 The Variational Method

Solving the equations of quantum mechanics is typically difficult, so approximations must usually be made. One very effective tool for finding approximate solutions is the variational principle. This section gives some of the basic ideas, including ways to apply it best.

9.1.1 Basic variational statement

Finding the state of a physical system in quantum mechanics means
finding the wave function

However, the expectation value of energy is just a simple single
number for any given wave function. It is defined as

where

That means that if you would find

Of course, finding the expectation value of the energy for all
possible wave functions is still an impossible task. But you may be
able to guess a generic type of wave function that you would expect to
be able to approximate the ground state well, under suitable
conditions. Normally, suitable conditions

means that
the approximation will be good only if various parameters appearing in
the approximate wave function are well chosen.

That then leaves you with the much smaller task of finding good values
for this limited set of parameters. Here the key idea is:

The variational method as described above has already been used earlier in this book to find the ground state for the hydrogen molecular ion, chapter 4.6, and the one for the hydrogen molecule, chapter 5.2. It will also be used to find the ground state for the helium atom, {A.38.2}. The method works quite well even for the crude approximate wave functions used in those examples.

Of course, it is not obvious why getting the best energy would also
produce the best complete wave function. After all,
best

is a somewhat tricky term for a complex object
like a wave function. To take an example from another field, surely
you would not argue that the best sprinter in the world must
also be the best person in the world. But for approximate
ground state wave functions, if the approximate energy is close to the
exact energy, then the entire approximate wave function will
also be close to the exact one. (This does assume that the ground
state is unique; there must not be other wave functions with the same,
or almost the same, energy. See addendum {A.7} for
details.)

And there are other benefits to specifically getting the energy as accurate as possible. One problem is often to figure out whether a system is bound. For example, can you add another electron to a hydrogen atom and have that electron at least weakly bound? The answer may not be obvious. But if using your approximate solution, you manage to show that the approximate energy of the bound system is less than that of having the additional electron at infinity, then you have proved that the bound state exist. Despite the fact that your solution has errors. The reason is that, by definition, the ground state must have lower energy than your approximate wave function, so is even more tightly bound together than your approximate wave function says.

Another reason to specifically getting the energy as accurate as possible is that energy values are directly related to how fast systems evolve in time when not in the ground state, chapter 7.

For the above reasons, it is also great that the errors in energy turn out to be unexpectedly small in a variational procedure, when compared to the errors in the guessed wave function, {A.7}.

To get the second lowest energy state, you could search for the lowest energy among all wave functions orthogonal to the ground state. But since you would not know the exact ground state, you would need to use your approximate one instead. That would involve some error, and it is no longer sure that the true second-lowest energy level is no higher than what you compute, but anyway. The suprising accuracy in energy will still apply.

If you want to get truly accurate results in a variational method, in
general you will need to increase the number of parameters. The
molecular example solutions were based on the atom ground states, and
you could consider adding some excited states to the mix. In general,
a procedure using appropriate guessed functions is called a
Rayleigh-Ritz method. Alternatively, you could just chop space up
into little pieces, or elements,

and use a simple
polynomial within each piece. That is called a finite-element method.
In either case, you end up with a finite, but relatively large number
of unknowns; the parameters and/or coefficients of the
functions, or the coefficients of the polynomials.

9.1.2 Differential form of the statement

You might by now wonder about the wisdom of trying to find the minimum energy by searching through the countless possible combinations of a lot of parameters. Brute-force search worked fine for the hydrogen molecule examples since they really only depended nontrivially on the distance between the nuclei. But if you add some more parameters for better accuracy, you quickly get into trouble. Semi-analytical approaches like Hartree-Fock even leave whole functions unspecified. In that case, simply put, every single function value is an unknown parameter, and a function has infinitely many of them. You would be searching in an infinite-dimensional space, and could search forever. Maybe you could try some clever genetic algorithm.

Usually it is a much better idea to write some equations for the
minimum energy first. From calculus, you know that if you want to
find the minimum of a function, the sophisticated way to do it is to
note that the partial derivatives of the function must be zero at the
minimum. Less rigorously, but a lot more intuitive, at the minimum of
a function the changes in the function due to small changes in
the variables that it depends on must be zero. In the simplest
possible example of a function small

does
have an unambiguous meaning: it means that you must ignore everything
that is of square magnitude or more in terms of the
small

quantities.)

In physics terms, the fact that the expectation energy must be minimal
in the ground state means that you must have:

constrained minimization;you cannot make your changes completely arbitrary.

9.1.3 Using Lagrangian multipliers

As an example of how you can apply the variational formulation of the
previous subsection analytically, and how it can also describe
eigenstates of higher energy, this subsection will work out a very
basic example. The idea is to figure out what you get if you truly
zero the changes in the expectation value of energy

The differential statement is:

But

acceptableis not a mathematical concept. What does it mean? Well, if it is assumed that there are no boundary conditions, (like the harmonic oscillator, but unlike the particle in a pipe,) then acceptable just means that the wave function must remain normalized under the change. So the change in

But how do you crunch a statement like that down mathematically?
Well, there is a very important mathematical trick to simplify this.
Instead of rigorously trying to enforce that the changed wave function
is still normalized, just allow any change in wave function.
But add penalty points

to the change in expectation
energy if the change in wave function goes out of allowed bounds:

Here

You do not, however, have to explicitly tune the penalty factor yourself. All you need to know is that a proper one exists. In actual application, all you do in addition to ensuring that the penalized change in expectation energy is zero is ensure that at least the unchanged wave function is normalized. It is really a matter of counting equations versus unknowns. Compared to simply setting the change in expectation energy to zero with no constraints on the wave function, one additional unknown has been added, the penalty factor. And quite generally, if you add one more unknown to a system of equations, you need one more equation to still have a unique solution. As the one-more equation, use the normalization condition. With enough equations to solve, you will get the correct solution, which means that the implied value of the penalty factor should be OK too.

So what does this variational statement now produce? Writing out the
differences explicitly, you must have

Multiplying out, canceling equal terms and ignoring terms that are quadratically small in

That is not yet good enough to say something specific about. But
remember that you can exchange the sides of an inner product if you
add a complex conjugate, so

Also remember that you can allow any change

or using the fact that numbers come out of the left side of an inner product as complex conjugates

If you divide out a

You can now combine them into one inner product with

If this is to be zero for any change

So you see that you have recovered the Hamiltonian eigenvalue problem
from the requirement that the variation of the expectation energy is
zero. Unavoidably then,

Indeed, you may remember from calculus that the derivatives of a
function may be zero at more than one point. For example, a function
might also have a maximum, or local minima and maxima, or stationary
points where the function is neither a maximum nor a minimum, but the
derivatives are zero anyway. This sort of thing happens here too: the
ground state is the state of lowest possible energy, but there will be
other states for which