PHY456H1F: Quantum Mechanics II. Lecture 4 (Taught by Prof J.E. Sipe). Time independent perturbation theory (continued)
Posted by peeterjoot on September 23, 2011
Peeter’s lecture notes from class. May not be entirely coherent.
Time independent perturbation.
To recap, we were covering the time independent perturbation methods from section 16.1 of the text . We start with a known Hamiltonian , and alter it with the addition of a “small” perturbation
For the original operator, we assume that a complete set of eigenvectors and eigenkets is known
We seek the perturbed eigensolution
and assumed a perturbative series representation for the energy eigenvalues in the new system
Given an assumed representation for the new eigenkets in terms of the known basis
and a pertubative series representation for the probability coefficients
We rescaled our kets
The normalization of the rescaled kets is then
One can then construct a renormalized ket if desired
That’s as far as we got last time. We continue by renaming terms in 2.10
Now we act on this with the Hamiltonian
Expanding this, we have
We want to write this as
So we form
and so forth.
Zeroth order in
Since , this first condition on is not much more than a statement that .
First order in
How about ? For this to be zero we require that both of the following are simultaneously zero
This first condition is
From the second condition we have
Utilizing the Hermitian nature of we can act backwards on
We note that . We can also expand the , which is
I found that reducing this sum wasn’t obvious until some actual integers were plugged in. Suppose that , and , then this is
More generally that is
Utilizing this gives us
And summarizing what we learn from our conditions we have
Second order in
Doing the same thing for we form (or assume)
We need to know what the is, and find that it is zero
Again, suppose that . Our sum ranges over all , so all the brakets are zero. Utilizing that we have
From 2.34 we have
We can now summarize by forming the first order terms of the perturbed energy and the corresponding kets
We can continue calculating, but are hopeful that we can stop the calculation without doing more work, even if . If one supposes that the
term is “small”, then we can hope that truncating the sum will be reasonable for . This would be the case if
however, to put some mathematical rigor into making a statement of such smallness takes a lot of work. We are referred to . Incidentally, these are loosely referred to as the first and second testaments, because of the author’s name, and the fact that they came as two volumes historically.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.
 A. Messiah, G.M. Temmer, and J. Potter. Quantum mechanics: two volumes bound as one. Dover Publications New York, 1999.