## PHY456H1F: Quantum Mechanics II. Lecture 3 (Taught by Prof J.E. Sipe). Perturbation methods

Posted by peeterjoot on September 19, 2011

Peeter’s lecture notes from class. May not be entirely coherent.

# States and wave functions

Suppose we have the following non-degenerate energy eigenstates

and consider a state that is “very close” to .

We form projections onto “direction”. The difference from this projection will be written , as depicted in figure (\ref{fig:qmTwoL3fig1}). This illustration cannot not be interpreted literally, but illustrates the idea nicely.

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL3fig1}

\caption{Pictorial illustration of ket projections}

\end{figure}

For the amount along the projection onto we write

so that the total deviation from the original state is

The varied ket is then

where

In terms of these projections our kets magnitude is

Because this is

Similarly for the energy expectation we have

Or

This gives

where

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL3fig2}

\caption{Illustration of variation of energy with variation of Hamiltonian}

\end{figure}

“small errors” in don’t lead to large errors in

It is reasonably easy to get a good estimate and , although it is reasonably hard to get a good estimate of . This is for the same reason, because is not terribly sensitive.

# Excited states.

Suppose we wanted an estimate of . If we knew the ground state . For any trial form

We are taking out the projection of the ground state from an arbitrary trial function.

For a state written in terms of the basis states, allowing for an degeneracy

and

(note that there are some theorems that tell us that the ground state is generally non-degenerate).

Often don’t know the exact ground state, although we might have a guess .

for

but cannot prove that

Then

FIXME: missed something here.

Somewhat remarkably, this is often possible. We talked last time about the Hydrogen atom. In that case, you can guess that the excited state is in the orbital and and therefore orthogonal to the (?) orbital.

# Time independent perturbation theory.

See section 16.1 of the text [1].

We can sometimes use this sort of physical insight to help construct a good approximation. This is provided that we have some of this physical insight, or that it is good insight in the first place.

This is the no-think (turn the crank) approach.

Here we split our Hamiltonian into two parts

where is a Hamiltonian for which we know the energy eigenstates and the eigenkets. The is the “perturbation” that is supposed to be small “in some sense”.

Prof Sipe will provide some references later that provide a more specific meaning to this “smallness”. From some ad-hoc discussion in the class it sounds like one has to consider sequences of operators, and look at the convergence of those sequences (is this L2 measure theory?)

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL3fig3}

\caption{Example of small perturbation from known Hamiltonian}

\end{figure}

We’d like to consider a range of problems of the form

where

So that when we have

the problem that we already know, but for we have

the problem that we’d like to solve.

We are assuming that we know the eigenstates and eigenvalues for . Assuming no degeneracy

We seek

(this is the case).

Once (if) found, when we will have

This we know we can do because we are assumed to have a complete set of states.

with

where

There’s a subtlety here that will be treated differently from the text. We write

Take

where

We have:

FIXME: I missed something here.

Note that this is no longer normalized.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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