Free Particle Transition Amplitude

We apply the generalized Feynman representation to the case of a free particle with the Lagrangian:

The transition amplitude for a free particle can be written via Feynman path integration with the help of relations 4.6,4.21 and 4.22 of chapter 4:

Here, the integration measure: and the action functional S(q_N, f, t₀):

The transition amplitude 6.23 considering 6.25 represents a set of Gaussian integrals. The integral of a Gaussian is a Gaussian. So, the integrations in 6.23 can be carried out on one variable after the other and when completed then the limit N -> oo is applied. We therefore start first when к = 0 and we perform integration over q, then computing the integral

yields

This procedure is continued similarly for q,, •••, q_N_, and we see a recursion process is established, so that after (JV- 1) steps we have:

where

Relation 6.28 coincides with the quantum mechanical transition amplitude and the phase of the exponential factor is the classical action along the classical path, with the condition that q(0) = q₀ and q(T) = q.

Gaussian Functional Feynman Path Integrals

We consider the following generalized Feynman path integral: where, the Lagrangian:

is quadratic both in the set of generalized coordinates q as well as generalized velocities q and f (f') is the driving force. Such an integral in 6.30 is therefore Gaussian and so may be exactly solvable analytically with

being, respectively, the starting and ending coordinates of the path. All the matrices in 6.31 are realvalued with к and b being respectively symmetric and anti-symmetric matrices.

We write the Euler-Lagrange equation described by the relation in 3.30 of chapter 3 with the help of 6.31 in the matrix form:

The solution of this equation gives the classical path q_d with the property of extremizing the action S. We observe two non-quadratic terms in the Lagrangian described in 6.31. So, it is necessary to transform the Lagrangian to quadratic form for the integral in 6.30 to be easily evaluated. We consider the quasi-classical approximation via the method of stationary phase method as mentioned earlier. For this, we assume that only those paths quite near to the classical path q_cl(f) are important. So, from the quasi- classical approximation, we select a fluctuating path q'(f') and interpreted as the quantum fluctuations around the classical path q_C|(f'):

So, the classical action:

Then considering 6.34 we have

where, the Gaussian integral around the classical path is the reduced propagator:

The interpretation of the expression in 6.36 is that the quantum transition amplitude of a particle can be separated into two factors:

• The first factor _exp|—s(q_d(f'),f,t₀)j *^s written in terms of the action of the classical path q_d.

Indeed, the first factor contains often most of the useful information contained in the propagator.

• The second factor F(T) is the path integral over fluctuations q'(f') from the classical path q_d. This second factor is independent of the initial q(f₀) and final points q(f).

So, the quasi-classical limit should then correspond to the possibility to reduce the path integral to a Gaussian integral. Hence our problem is reduced to determining the classical action, once relation 6.36 is shown to be true for the system. This follows that the normalization F(T) of the propagator becomes the only challenging task.

Setting f (f') = 0 in equation 6.36 and considering q₀ = q, we then integrate both sides of the resultant equation by q and this yields the reduced propagator:

where,

and

The reduced propagator 6.37 or 6.38 describes a cyclical path from q and back to q with period T and frequency со and permits us to Fourier series expand the generalized coordinates q via the fundamental harmonic frequency equal to _ :

T

with the frequency

The fluctuation vector satisfies cyclical boundary conditions and is expressed:

The Fourier coefficients a„ will be observed to serve as a discrete set of integration variables with each path having its own coefficients a„. The expansion of у in terms of the Fourier coefficients a„ will result in the diagonalizing of the integrand of the action which is Lagrangian and path integral over all у equal integral overa,,.

From

then substituting 6.41 into 6.38 and considering the Hermitian matrix denoted by

then we have the following quadratic action functional as a result of the diagonalizing procedure:

We consider extremizing the action function and we find the Fourier coefficient corresponding to the classical path:

Substituting into 6.41 and letting then

From equations 6.48 to 6.50 then the action functional 6.47 for the classical path:

This permits us to elegantly compute F,(T):

We find now F,(T) by time-slicing the time axis: with

and

From 6.53 to 6.55 we compute the path integral in 6.39: where, the integral over all Fourier coefficients a„:

and the factor J J JA„ is the Jacobian transformation for the manifold integral.

П

For a free particle then and

So, the following quadratic action functional: and for fixed y^{, v|} then 6.56 becomes:

From 6.46 and 6.58 then

So, equation 6.56 is satisfied when the Jacobian J takes the value

So,

and

with

We compute the product in 6.66 supposing the matrix m to be positive definite. So, v is a symmetric and positive definite matrix:

From here, considering 6.46 then the following Hermitian matrix: and consequently, where,

From 6.67 to 6.70 it is obvious that Then,

Here,

and differentiating this by k' then Substituting this into 6.69 then Considering 6.68 and 6.74 then

This product can be simplified via the formula:

and consequently,

Considering 6.76, then and consequently,

This is indeed the transition amplitude describing a cyclical path from q and back to q with period T and scaled frequency w.

This problem may be applied to the case of a harmonic oscillator of frequency w and driving force y(t) under chapter 5 where we set

Charged Particle in a Magnetic Field

The problem of a charged particle in a magnetic field was first solved by L.D. Landau in 1930 in the Schrodinger theory [15,16]. We consider the case of the magnetic field with the magnetic field induction В pointing along the z -axis direction:

The Lagrangian for such a charged particle e in the uniform magnetic field and subjected to the force

m

Compare equation 6.84 with 6.31 and apply these equations to 6.70 then The expression , can be obtained:

Here, we introduce the field-dependent magnetic frequency:

with (o_c, being the Landau frequency or cyclotron frequency defined as

Here, c is the speed of light. To calculate the reduced propagator F(T), we substitute 6.86 into 6.80 and consequently 6.81:

From the Lagrangian in 6.84 we can construct a Feynman path integral that will be Gaussian and exactly solvable analytically with

being respectively, the starting and ending position vectors. We write the Euler-Lagrange equation described by the relation in 3.30 of chapter 3 via 6.84:

The solution of this equation gives the classical path p_d with the property of extremizing the action S: Here, 9(s - t) is the Heaviside step function and the antisymmetric matrices:

and

with

From here, then substituting the classical path r_ci (f) in equation 6.92 into 6.35 then we have the action functional:

where the orthogonal part of the action: and

Hie magnetic interaction of a particle of charge e for the 3D case has the transition amplitude:

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