Matrix Representation of the Transition Amplitude of a Forced Harmonic Oscillator

We analyze a single harmonic oscillator linearly coupled to some external force y(t) described by the Lagrangian:

We assume for brevity that an oscillator with the force y(f) in state |Ф„(д, 0) is turned on from time moment t_a = 0 and later on at the state |Ф,„( 0) is turned off at time moment t_b = T. Tire transition amplitude G„,„ describes the oscillator initially in the state with |Ф„(д, f)) at time t_a and now found in the state with |Ф,„(<7, f)> at time t_b:

Inserting identity operators over position states then

Supposing |Ф,Д,)) and |Ф,„(^)) are Schrodinger state vectors at the indicated times, then from 7.36 we obtain

From 5.134, 5.143 and 5.166 then for easy computation, we find the expression of G„,„ by considering the situation m = n = 0, where G₀₀ is the amplitude for which an external perturbation does not change, the oscillator’s ground state:

Here,

and considering equations 5.120 and 5.121 we have: with

Computing the integral in 10.20 we find the amplitude to remain in the ground state,

Let us now calculate a general case. We make use of the generating function for the Hermite polynomials:

On this basis we can construct a generating function for the matrix elements G„,„. Let us introduce the generating function from definition:

Considering

therefore, it follows from 10.27 and 10.28 that

Further, we suppose that t_b =T,t_a = 0. From these expressions we have the following: where

Considering,

then

Considering the sum as a Newton binomial then

Since from definition

Then

Now that all matrix elements between the energy eigenstates are determined, we can say that these represent the complete solution of the problem of a forced harmonic oscillator. Depending on the particular driving function y(t), the amplitudes for any particular transitions, can, in principle, be found.

Charged Particle Interaction with Phonons

We apply the knowledge of the transition amplitude matrix to the case of an electron interacting with lattice vibrations (phonons) in a polar medium. The electron in its motion carries with itself, the self- created polarization and this constitutes a many-body problem. More details on a charge particle interacting with lattice vibrations in a polar medium will be seen later with the polaron theorywhere we will find the polaron ground state energy (variational upper-bound ground state energy approximation) applying the Feynman variational principle..

If the wave function at an initial time moment f, is 4^ (Я,(r₂,q₂) at an arbitrary time moment f, can be written as:

with the transition amplitude К being

Here, the action functional for the composite particle sj^r,r ,q,q'^ :

is considered linear in the electron-phonon-coupling-amplitude and quadratic in the lattice vibration coordinates where S_e [r ], the electronic action functional; S, [q’] and S_e^_L [r,q] are respectively, the lattice and electron-lattice action functionals. The transition amplitude considering 7.42 or 7.43 is rewritten

Here, r and q are, respectively, the electronic and the polarization field oscillator coordinates, Ч¹ (r,q) a complete orthonormal set of eigenfunctions of the full Hamiltonian of the composite particle:

and E„ represents the corresponding energy eigenvalues.

The ground-state energy, E₀, can be conveniently obtained from equation 10.45 by multiplying both sides of 10.45 by i//o" (<^2 )v^/o (fZi) and then integrate over the variables and q₂, giving

where,

and y/₀{q) is a function of q that is not orthogonal to the full ground state eigenfunction У₀(г,^).

From 10.17 to 10.21 and 10.47 then the amplitude for which an external perturbation does not change, the oscillator’s ground state, is:

Evaluating equation 10.47, by considering the analytic continuation t = - ifir (where г is a real number):

then

and letting

so, from 10.47 we have

So, the exact expression for the composite’s particle ground-state energy in terms of the transition amplitude:

Here,

and the initial bonafide many-body problem via the Feynman approach is reduced to a one-body problem (because the phonon coordinates are adequately eliminated) and described by the action functional:

with the first term being the kinetic energy of the electron, Ф_шГ?>г'1, considering 10.26 describes the full functional of the electron-phonon interaction (influence phase):

This functional is responsible for the phonon-mediated attractive interaction of the electron at time r with itself at an earlier time r' and also depend on the quantity |r - r'| indicative of a retarded function depending on the past, thereby signifying interaction with the past where the perturbative motion of the electron takes “time” to propagate in the crystal lattice. This suggests the self-interaction to be stronger in the near past than it is in the distant past.

The ground state energy in 10.54 could not be possibly evaluated with ease via path integral as it is non-Gaussian. The Feynman solution to the composite particle problem provides a technical mathematical tool on how to calculate this non-Gaussian path integral. This tool allows the building of a class of exactly solvable models corresponding to quadratic functionals on whose bases is built a variational technique that gives a good upper bound to the composite particle’s ground state energy. For this, Feynman introduced the notion of a model system with a fictitious particle that we examine together with the coupling amplitude y(t) in chapter 18 on the polaron theory.