Consider two identical masses, m, connected to opposite walls with identical springs with spring constants, k₀. The two masses are connected with a third spring with a spring constant, k₁.

In order to calculate the Lagrangian, we need to first calculate the kinetic and potential energies:

Lagrange's equations in generalized coordinates are:

This produces two coupled differential equations for x₁ and x₂, which could have been derived directly from F = ma:

Let us define the characteristic frequencies for the springs:

The differential equations thus become:

The normal modes each have a single harmonic time dependence for all the coordinates used to describe it and thus the x coordinates can be expressed as such (this amounts to the separation of position and time variables):

and thus the differential equations can be rearranged into a matrix eigenvalue equation (after canceling the exponential time dependence out) :

This is a degenerate coupled two level eigensystem problem. It is interesting to note that the system can be interpreted in two ways.

One is to consider the coupling spring as both coupling the unperturbed systems as well as adding an offset to them. That is, the matrix operator above consists of a diagonal unperturbed operator plus a perturbing operator that has both diagonal and off-diagonal terms:

This view has the problem of making it difficult to identify the effect of the coupling on the unperturbed system because of the diagonal term offsets.

One the other hand, it may be easier to reconsider what really is the original, unperturbed system. Rather than consider the coupling spring to be an addition, consider a system where each mass initially has two restoring springs, one of spring constant k₀ and the other with spring constant k₁. This is equivalent to putting one finger down and immobilizing one of the masses. Thus we have two identical systems where we alternately immobilize the each mass. The unperturbed system thus has each mass with a net restoring spring of size k₀+ k₁. The perturbing operator thus has only off-diagonal terms:

The analysis of this system is much cleaner and the splitting due to the coupling will be readily evident. See the web page on solving such systems (Two Level Eigensystems). The eigenvalues and eigenvectors are given by:

Including the time dependence, one gets:

Note that both the positive and negative frequencies are solutions (technically, degenerate). This means that there are four unknown amplitude coefficients that need to be established by the boundary conditions. The four equations can be satisfied by specifying the initial values of the two coordinates as well as their initial time derivatives (velocities). This is normal for a second order differential equation.

The symmetric state can be interpreted as both masses moving exactly in phase to the left and right. The coupling spring does not get compressed or stretched at all in this case. Hence the situation is as if each mass were affected only by one spring of spring constant, k₀.

The anti-symmetric state can be interpreted as each mass moving exactly 180 out of phase (hence the minus sign in the wavevector). Thus the coupling spring appears to be compressed twice as much as the movement in any given coordinate. Thus it contributes as if it had a spring constant that was twice as large. We can also see the advantage of the second viewpoint above on the unperturbed system. The new eigenvalues are a symmetric plus/minus split around ₀²+₁², not ₀².

How would the coupled system behave if it was originally prepared in one of the unperturbed eigenstates, which are not eigenstates of the coupled system?

The composition of a time evolving state in terms of the new eigenstates can be expressed via its overlap (Completeness Theorem with time dependence). Half the initial conditions can be satisfied by setting the expansion coefficients at t=0 (the initial velocity still needs to be satisfied):

If we want the initial velocity to be zero, then the exponentials reduce to the cosine solutions only:

To see the time evolution of a particular coordinate, we need to look at the projection along its direction at time, t:

For the other coordinate, the process is the same:

We can clearly see that the solutions consist of modulated cosine and sine waves. The fundamental frequency is approximately equal to the unperturbed frequency for small coupling. The modulation envelope frequency is equal to the coupling frequency.

While energy appears to be flowing back and forth between the |x₁> and |x₂> states, it is really just a beating between the different frequencies of the two stationary eigenstates.

Note that this system is conceptually similar to two atomic levels coupled by a harmonic photon field. This will cause an electron in the lower energy state to be absorbed and jump to the higher energy state. After a while though, the photon field will cause a stimulated emission and the electron will drop to the lower state.

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