Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical ThermodynamicsLecture 20
Perturbation Theory Expresses the solution to one problem in terms of another problem solved previously
-The idea behind perturbation theory is the following. Suppose that we are unable to solve the Schrödinger equation
(1)
-For somesystem ofinterest but that we do know how to solve itfor another system that is insome sense similar. We can write the Hamiltonian operator in equation 1 in the form
(2)
where
(3)
is the Schrödinger equation we can solve exactly. We call the first term in equation 2 the unperturbed Hamiltonianoperator and the additional term the perturbation.
-To apply perturbation theory to solution of equation 1 with given by equation 2, we write and E in the form
(4)
and
(5)
where and E(0) are given by the solution to unperturbed problem equation (3) and are successive correction to and E(1), E(2),…. are successive corrections to E(0). A baic assumption is that theses successive corrections become increasingly less significant. Although we will not do so here, we can derive explicit expressions for these corrections. The only one we will use is the expression for E(1) , which is
(6)
we say that E(1) is the first –order correction to E(0) , and we write
(7)
Equation 7 represents the energy through first order perturbation theory. If we were to evaluate (which we will not), then
(8)
Would represent through first order. Similarly,if we were to evaluate (which we will not), then
Would represent E through secondorder perturbation theory. In this book, we evaluate E to first order only, using equation 6.
Perturbation theory to calculate the energy of a particle in box
-We will use firstorder perturbation theory tocalculate the energy of a particle in box from x=0 to x=a with a slanted bottom, such that
In this case, the unperturbed problem is a particle in a box and so
Where V0 is a constant. The wave functions and theenergies for a particle in a box are
and
According to equation 6 , the first order correction of E(0) due to perturbation is given by
This integral occurs previously and is equal to a2/4. Therefore, we find that
For all values of n. the energy levels are given through first order by
Where the term O (V02) emphasizesthat terms of order V02 andhigher have been dropped. Thus, we see in this case that each of the unperturbed energy levels s shifted by V0/2.
Helium atom
-We can apply perturbation theory to the helium atom whose Hamiltonian operator is given by equation
For simplicity we will consider only the ground –state energy. If we consider the interelectronic repulsion term , to be the perturbation , then the unperturbed wave fnctions and energies are the hydrogenlike quantities given by
(9)
and
with Z= 2. Using Equation (6), we have
(10)
where
The final result is that
(11)
(12)
-Letting Z= 2 gives -2.750 compared with our simple variational result (-2.8477) and the experimental results of -2.9033.
-So we see that first order perturbation gives a result is about 5% in error. It turns out that second order perturbation theory gives -2.910
-Thus, we see that both the variational method and the perturbation theory are able to achieve very good results.
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