# A string nailed at two points

1. Sep 7, 2007

### Slaviks

I have been playing around with the following rather elementary 1d field theory problem and got stuck. May you have some good ideas on it.

Let us consider an ideal 1-D elastic spring with just one polarization direction (say, transverse displacement in y-direction while the unperturbed string is along x).
The string is of length L >> than anything else in the problem, boundary conditions are as you like. Apart from the boundary conditions, it is a free boson field in 1+1 dimension, arguably the simplest translationally invariant action in 1D.

Now let me add to my ideal spring two delta-like constant potentials separated by a distance x0. The corresponding term in the Lagrangian / Hamiltonian is

V_1 \phi(-x_0/2) +V_1 \phi(+x_0/2)

1) How to define the normal modes in this system to that these independent modes can be quantized along the line of any introductory theory of phonons?

My problem started with the realization that in classical mechanics, the eigenvalue problem which determines the normal mode frequencies of a system is different form the usual (for me) diagonalization of a Hermitian matrix (that is, of the Hamiltonian).
So I failed to find such a transformation of \phi(x) and the corresponding conjugate momenta that would give me a sum of independent linear harmonic oscillators already at the classical level. May be can suggest a better way to quantize this toy field theory.

2) My aim is to compute exactly the ground state / free energy as a function of x0,
and use it to demonstrate the crossover from Casimir to van der Waals force.
May be some of you have seen such an exercise before?

2. Sep 9, 2007

### Slaviks

This is not a homework

Just to make sure there is no misunderstandning: this a research problem, not a homework. (Although what was yesterday's research may turn to be today's homework, but that's a different issue...)