A string nailed at two points

[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?

 

[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...)

 

[denian]

I am always excited to see someone exploring new and interesting problems in their field. In this case, your problem of a string nailed at two points and adding delta-like potentials is a fascinating one. I can offer some thoughts and suggestions on how to approach this problem.

1) To define normal modes in this system, we can use the standard method of separation of variables. We can write the displacement of the string as a sum of two independent modes, one for each potential. These modes can then be quantized separately, using the standard approach of phonons in 1D systems. This would give us a set of independent harmonic oscillators, each with their own frequency. However, as you have pointed out, this may not be the most efficient way to approach the problem.

Another approach could be to use the Lagrangian for the system and solve the equations of motion to find the normal modes. This would involve finding the eigenvalues and eigenvectors of the Lagrangian matrix, which would then give us the normal mode frequencies. This method may be more straightforward and efficient in this case.

2) To compute the ground state/free energy as a function of x0, we can use the standard methods of quantum field theory. We can write down the Hamiltonian for the system and use perturbation theory to calculate the ground state energy. We can also use the path integral approach, where we integrate over all possible configurations of the field to obtain the partition function and then use it to calculate the free energy.

As for the crossover from Casimir to van der Waals force, this is an interesting application of your problem. To demonstrate this, we can calculate the force between the two potentials by taking the derivative of the free energy with respect to x0. We would expect to see a transition from attractive (Casimir) to repulsive (van der Waals) forces as x0 increases.

Overall, your toy field theory problem is a fascinating one and I am sure there will be many different approaches to solving it. I hope my suggestions are helpful and I look forward to seeing the results of your exploration.