Free and Driven Harmonic Oscillator Classical Action and Transition Amplitude

Free Oscillator Classical Action

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We devote this example to a study of an important physical system called the one-dimensional harmonic oscillator which is some system having one degree of freedom and executing in its direction a harmonic oscillation. We examine a one-dimensional motion of a point mass that is attracted to a fixed center by a restoring force F₍₎: This restoring force is proportional to the displacement q from that center:

where m and m are the mass and the classical frequency respectively. This provides one of the fundamental problems of classical dynamics. This study is very important in physics as a large number of systems may always be analyzed in terms of normal modes of motion. In the limit of small oscillations, the normal modes are formally equivalent to harmonic oscillators. The harmonic oscillator has practical applications in a variety of domains of modern physics, such as molecular spectroscopy, solid state physics, nuclear structure, quantum field theory, quantum statistical mechanics and so on. The harmonic oscillator also provides the path in the study of the electromagnetic field. The vibrations of the electromagnetic field in a cavity provide an infinite number of stationary waves, i.e., normal modes of the cavity. We may expand the electromagnetic field in terms of these modes. If we consider Maxwell’s equations, it can be seen that each coefficient of expansion conforms to a differential equation. This equation is identical to that of a harmonic oscillator having angular frequency w associated with a normal mode. This follows that the vibrations of the electromagnetic field in a cavity may be analyzed into harmonic normal modes each of which energy levels is of the oscillator type. Quantizing the given oscillators associated with various normal modes of the cavity may quantize the field. So, a detail investigation of the problem of the harmonic oscillator in quantum mechanics is very important.

We examine the harmonic oscillator with the procedure in reference [1] considering the stationary phase method. Иге Lagrangian for this harmonic oscillator has the form:

The aim here is to compute the leading contribution to the path integral in the formal limit Pi -> 0. As seen above, this will correspond to deriving a general expression for the Gaussian, or quasi-classical approximation. So, the main contribution to the path integral for the transition amplitude will come from paths at the vicinity of the classical path as earlier indicated. This implies those paths with an action within about Pi of the action of the classical path where the least action principle is satisfied for

From here, we arrive at the Euler-Lagrange equation for the classical path:

Consider a particle of mass m executing a harmonic oscillation in the direction of the q-axis with a cyclical frequency a). From the restoring force 5.106 we find the potential energy V(q):

then from equation 5.109 follows with solution in the form:

Here, a is the initial phase. Letting Then and

The classical action for the harmonic oscillator can be found: or

The second term in 5.117 vanishes due to the equation of motion in 5.109. From

then this permit to find for X and У in 5.115:

Letting,

So, the following solution only makes sense

provided

This implies, if

So, if T is not an integral multiple of half the period r,

then there exists a unique classical path that starts from the initial point q„ and ends at q_b within the time interval T. From here and 5.117, then follows the classical action of a harmonic oscillator that is, of course, force free:

Driven or Forced Harmonic Oscillator Classical Action

This involves the problem of interaction of a particle with a harmonic oscillator where now the potential energy:

Here, the oscillator of mass m and frequency a> is subjected to a time-dependent force y{t) that changes the state of the self-consistent system. This force y{t) may be due to an external field, e.g. an electric field coupling via dipole interaction to a charged particle. Considering dissipative quantum mechanics, the harmonic oscillator could represent a degree of freedom of the environment under the influence of a force exerted by the system. This type of problem may imitate the interaction of an electron with phonons. Here, we have the electronic as well as the phonon subsystems.

Substitute 5.126 into 5.109 then the classical path has to satisfy the equation of motion:

The homogenous solution q_H(t) for y{t) = 0 is already written in 5.121 and the non-homogeneous NH(f) solution for y(t) Ф 0 can be obtained by writing the full solution in the form: