Is massively collaborative mathematical physics possible?

In the spirit of the Polymath Project, I’d like to present some thoughts on a mathematical physics problem that could be interesting.  The problem is simple enough: compute the eigenvalues of the Hamiltonian for a perturbed quantum harmonic oscillator.  But before we get started, I’ll just state that we should try to follow Timothy Gowers’s 12 ground rules at the end of his Polymath kick-off post.  Also, I’ll credit Clark Alexander for walking me through this entire method, and encouraging this post.

The Unperturbed Quantum Harmonic Oscillator — A Survey of Notation

So, we have the quantum harmonic oscillator, easily represented and solved by using raising and lowering operators: where the notation and is employed, such that .

The spectrum of solutions to the Schrödinger Equation are familiar — the first three are shown in the following plot: The Quartic Perturbation to the Quantum Harmonic Oscillator — Rewriting the Hamiltonian

With the above notation, we can readily incorporate a perturbation of into our Hamiltonian:  can be normal ordered to the following result:

(1) Intuitively, we know that the result of this perturbation will be a slight change in the resulting eigenstates making up the solution.  A plot of the resulting change for an arbitrarily chosen lambda is shown below: Computing the Eigenvalues to First Order in in Easy Steps

Step 1: Identify All Elements of the Lie Algebra

Elements of the Lie Algebra at first order ( where ) are determined by performing commutations with and , as identified below.  At first order, terms of order are ignored, so only one commutation is required.

The first commutator:

(2) At this point, we can identify all the terms of the Lie Algebra to first order.

Nonperturbative Terms ( where )

(3) First Order Terms ( where )

(4) In this representation, we see the following:

(5) This representation is complete for our purposes because it satisfies two conditions:

1. can be completely represented by terms in the algebra.
2. No two terms can be commuted to create a third non-trivial term not shown in the group.  (Remember, ).

Step 2: Construct a General Lie Group Element

In principle, the Lie group element could be constructed from all terms in the Lie algebra, like so: But, by nature of the Hammard lemma, we can choose to exclude all terms that commute with .  So we construct as follows: This gives us 6 constants we tune in order make this Lie group element a transformation of basis between perturbed and unperturbed eigenstates.

Step 3: Use the Hammard Lemma to Compute our Lie Group Element

It is our goal to choose a such that the following is true:

(6) where  where .

To first order in this simplifies to:

(7) Performing the commutator of Equation 7 and normal ordering, we get the following:

(8) Step : Tune so is a Number Operator

From Equations 5 and 7,

(9) Now, using our knowledge that must commute with , we know that cannot have terms involving or .  Thus, the alphas must be tuned such that:

(10) Which leaves:

(11) Huzzah!  We’ve done it.  This is the perturbation to the eigenvalue introduced by the quartic term.

1. The solution came down to performing a few commutators (Step 1), expanding the Hammard Lemma (Step 3), and solving a simple system of equations, (Step ).
2. It stays just that simple for all orders in .
3. It stays just that simple for all perturbations of the form Perhaps there’s more here.  Perhaps we could find a general form for the perturbation to order (in ) to the harmonic oscillator?  Or more?