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:

- can be completely represented by terms in the algebra.
- 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.

### What’s So Special About This Approach to Perturbation Theory?

While we were methodically plodding through, you might not have noticed, but there’s a few special aspects to this approach:

- 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 ).
- It stays just that simple for all orders in .
- 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?

Feel free to comment with questions and thoughts. Where would you go from here?

There are already many reductions that can be made to simplify using commutators. Perhaps we can use d/dx and x as our other coordinate system. There is probably someway to move (large) commutators back and forth between fourier space(s).

So what we need is the following… If we want $Lambda_n$ to commute we need to find some operator $U$ so that

$U^* H_0 U = H_n + U^* f(N) U $

In this case f(N) will be a function of a number operator and we’ll have eigenvectors given by $U^* |j>$ where $N|j> = j |j>$ so that $|j>$ are the eigenvectors of the unperturbed oscillator.

In this case by the functional calculus (at least formally) $f(N)|j> = f(j) |j>$.

So we should be able to read off perturbations directly.

Now H_0 = N + 1/2 which means it is already a function of N.

So we can move everything as a function of N to one side.

U^* (N+1/2 + f(N)) U = H_n.

So we need to figure out a few things…

1) How U^* N U works in general:

2) How to “build” an appropriate U

So our reduction now becomes:

g(N) = f(N) + N

then

U^* g(N) U = N + lambda x^n

Calling all the topologists out there:

Is there a way to setup a spectral sequence structure on the perturbed eigenvalues?

Consider that we need to compute these values…

1) lambda^k (in the power series)

2) x^n (the perturbation in the potential)

3) the m^th energy level for the x^n potential.

Therefore we’re looking for the terms E^k_{n,m}.

What’s known so far are

i) E^0_{n,m} all n,m

ii) E^1_{n,m} all n,m (solved by A.)

iii) E^k_{0,m} all k,m

iv) E^k_{1,m} all k,m

v) E^2_{4,m} all m (solved by A.)

What’s also known is that

E^k_{n,m} implies E^{k+1}_{n,0}

we get the next ground state if we know the entire previous solution.

Can we set up some sort of differential structure here?

Perhaps also we won’t have a d^2=0 but perhaps d^j = 0 for some other j…

Maybe

d^j : E^k_{n,m} rightarrow E^{k+1}_{n,m-j} or something like this…

If we move the U operators over we have

In this case we may be able to convert from being in terms of and

to being in terms of and . Luckily (let’s call for the moment)

We can mover the piece furthest to the right into

But

and if then

which makes our commutators a little easier.

In the same way that we can view and as differential operators in “standard coordinates” we can realize and as “differential operators” of and respectively.

means and .

Now that I can write these things a little more nicely:

Let me write these three term energy values as

where corresponds to the correction, corresponds to the power of the overall perturbation, and corresponds to the energy level of the eigenstate.

So here we have:

when and otherwise.

when and when otherwise.

if is odd.

when .

Furthermore, we know

.

The question becomes whether we can define some function (or sequence of functions) which may function as “differentials” so that

.

Here’s a guess at the energy levels for .

In this case will be our ground state correction for .

One thing we can see easily is that .

In fact we know from Bender and Wu (ca.1963) .

For the ground state of the the quartic oscillator, it appears we can make some rapid progress…

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this is in good standing for resummability if the grow larger than this.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that might diverge incredibly fast.

For the ground state of the the quartic oscillator, it appears we can make some rapid progress…

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this is in good standing for resummability if the grow larger than this.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that might diverge incredibly fast.

For the ground state of the the quartic oscillator, it appears we can make some rapid progress…

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this will be difficult to resume if the diverge at all.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that appear to diverge but at some algebraic rate, perhaps .

Not sure…

Sorry for the three posts, I obviously made a few typing mistakes the first two times.

If we simply set up the differential equation as a power series we arrive at the recurrence relation

Sadly this is a 6th order equation rather than a second order equation.

So it seems we need to know through …