- 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:

So this procedure gives you the best possible approximation to the true energy, and energy is usually the key quantity in quantum mechanics. In addition you know for sure that the true energy must be lower than your approximation, which is also often very useful information.

The variational method as described above has already been used earlier in this book to find an approximate ground state for the hydrogen molecular ion, chapter 4.6, and for the hydrogen molecule, chapter 5.2. It will also be used to find an approximate 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.

To be sure, it is not at all obvious that getting the best energy will
also produce the best 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 in this case, your wave function will in fact be close to the exact wave function if you manage to get close enough to the exact energy. More precisely, assuming that the ground state is unique, the closer your energy gets to the exact energy, the closer your wave function gets to the exact wave function. One way of thinking about it is to note that your approximate wave function is always a combination of the desired exact ground state plus polluting amounts of higher energy states. By minimizing the energy, in some sense you minimize the amount of these polluting higher energy states. The mathematics of that idea is explored in more detail in addendum {A.7}.

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 is not obvious. But if using a suitable 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 the ground state 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 atomic 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 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 might search forever.

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 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. Mathematicians may not
like that, since the word small

has no rigorous
meaning. But unless you misuse your small quantities, you can always
convert your results using them to rigorous mathematics after the
fact.

In the simplest possible example of a function

In variational procedures, it is common to use

So in quantum mechanics, the fact that the expectation energy must be
minimal in the ground state can be written as:

constrained minimization;you cannot make your changes completely arbitrary.

9.1.3 Using Lagrangian multipliers

As an example of how the variational formulation of the previous
subsection can be applied 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
Lagrangian multiplier

method can deal with the
constraints.

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

Remarkably, you can throw away the second of each pair of inner
products in the expression above. To see why, remember that you can
allow any change

The two additional minus signs arise because an

You can now combine the remaining two terms 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