Free Particle Classical Action and Transition Amplitude

Table of Contents:

This heading examines some specific systems of particular interest in our investigation. These include the free particle and the oscillator in addition to others that will provide an avenue for the evaluation of the path integrals. For brevity, we consider the case of particles moving in one dimension labelled by the position coordinate q.

Free Particle Classical Action

We begin first with the free particle where the particle is associated with the Lagrange function:

Here, the mass of the particle is m . Assuming only those paths quite near to the classical path q(t) are important then for the new path we select a fluctuating pathy(f) about the classical one (Figure 5.2):

Here, q(t) is the classical path that connects the space-time points (q„,t„) and

It is important to note that for

So, follows the Dirichlet boundary conditions:

From the equation of motion for the free particle considering 5.59 then

and

This shows that the acceleration is zero,

FIGURE 5.2 The thick line represents the classical path q and the thinner lines associated with quantum fluctuations y.

So, from here and

So, the velocity of the classical path is constant:

We can then find the action integral S[q(t) via an action integral over the path y(t) relative to the classical path q[t). One obtains

From here, the first integrand is the contribution of the classical path to the free particle action or

The second integrand gives

The third integrand is the contribution of the fluctuation to the action:

So, we observe the contribution of only a quadratic fluctuation to the action.

Free Particle Transition Amplitude

We show, as already noted by Dirac [7], that the quantum mechanical propagator and the classical properties of a free particle are closely related. So, we now evaluate the path integral for the transition amplitude of a free particle following the algorithm introduced above considering the action has contributions from the classical path q as well as the fluctuation y(t) to the classical path:

Here, there is no linear term in y{t) and no higher than quadratic terms because the free particle action itself is quadratic. So, in the path integral, instead of integrating over all paths q(t) with boundary conditions 5.62 we now integrate over all paths y{t) with the boundary conditions 5.63. So, the path integral expression for the transition amplitude for the free particle can be evaluated via 5.58:

We observe from 5.75 that the action is a sum of two terms and so the amplitude factorizes into the product of a classical amplitude and a fluctuation factor. We rewrite the Gaussian integral around the classical path (reduced propagator or fluctuation factor) for the free particle as

Due to the vanishing of y(t) at the end points, this integral is independent of q_a and q_b but only on the initial and final times t_a and t_b with the time translational invariance reducing this dependence further to the time difference T=t_b-t_a.

The path integral in 5.77 is that of an oscillator with frequency a>= 0. So, we can therefore more explicitly expand y(t) in normalized eigenmodesy„(f) of some differential operator for the free particle:

with, respectively, the conditions of orthonormality and those at end points of the path:

where, the first equality shows that the expansion functions are orthogonal while the second gives the Dirichlet boundary conditions (conditions at end points of the paths).

From the conditions 5.79, asy,,(0 vanishes at the initial time t = t_a, the expansion can be constrained to the sine functions while the vanishing at the final time t = t_b is imposed by a constraint of the frequencies a>„ with discrete values:

So, the expansion ofy(f) in terms of eigenmodes is synonymous to diagonalizing the operator and path integral over ally(f) equal integral over the real Fourier coefficients a„:

We evaluate exponential function with the fluctuation y(t) in 5.77:

So, from the first condition in 5.79 we have

As,

then

and

So,

The first term is nothing but the classical action and then the path integral reduces to an infinite collection of Fresnel integrals:

So,

This may be simply represented by

The normalization constant / can depend only on the time interval T determined from the requirement that

and so,

Hence, the reduced propagator:

and multiplying the reduced propagator (fluctuation factor) with the classical amplitude, then the transition amplitude for the free particle:

From here we observe that the quantum mechanical propagator and the classical properties of a free particle are closely related [7]. So, S_d[q_b, t_b; q„, fj , is the classical action along the classical path, with the condition that q(0) = q_a and q{T) = q_b. The absolute square of К :

This gives the probability distribution per unit length that the particle will be found around position q_b, given that it was known to be at q„ at the start of the time interval T. This is simply a diffusive motion or spreading of a localized wave packet that began, as a Dirac delta-function centered at q_a which corresponds in momentum space to an exactly flat distribution of momenta. We note that the wave packet width (in probability) increases linearly with the time and one might suspect that it is due to a peculiarity of the free particle. However, this result in 5.94 may indeed be employed to construct a representation of the propagator where the classical action appears in the exponent. This is because, the propagation in a general potential (in the absence of delta-function contributions) may be decomposed into a series of short-time propagations of a free particle. The type of action S_d[0, T], appears only within the quasi-classical approximation or for potentials where the given approximation turns out to be exact.

From Path Integrals to Quantum Mechanics

From equation 4.99 chapter 4, we can find the link between the coordinates and the wave parameters. The type of the coordinates q_ifl and q_: contributing most to the transition amplitude are those satisfying the relation in 5.70 and tailor the phase of the transition amplitude to have an extremal (or stationary) value where sum over neighbouring trajectories tend to interfere constructively as their phases are nearly equal. From here, considering