Forced Harmonic Oscillator Research Articles

Wave-Interference Effects for a Forced Harmonic Oscillator

Wave-interference effects for transitions between unperturbed states of a forced harmonic oscillator are explicitly demonstrated by comparing the exact quantum-mechanical solution for this problem with a solution which assumes transitions to the different states are independent and therefore neglects wave-interference effects.

Vibrational Energy Transfer in High-Energy Collisions

Time-dependent wavefunctions are used to evaluate the exact transition probabilities for a forced harmonic oscillator. The forcing is represented by a time-dependent potential, where this potential has a linear dependence on the oscillator coordinate. The results are compared with available numerical solutions for a harmonic oscillator forced with a potential which has an exponential dependence on the oscillator coordinate. The comparison is made for the collision of an N2 molecule with another particle, and it is found that although the results for the two cases are similar, the linear potential gives higher values for the multiquantum transitions. It is then shown that time-dependent wavefunctions which contain the corresponding classical motion as a parameter provide a good set of functions for a perturbation calculation. The energy transfer to these oscillating wavefunctions is always identical to the energy transfer to the classical oscillator. Thus the perturbation value of the energy transfer represents the difference between the classical and the quantum-mechanical result. It is shown that this value, which is zero for the linear forcing potential, is very small for higher-order potentials, even at high velocity of impact. This demonstrates that classical calculations can be used to obtain the energy transfer in molecular collisions at high temperature.

A generalization of the phase relations in a forced harmonic oscillator

When a simple harmonic oscillator is subjected to a sinusoidally varying external force whose frequency is different from the natural frequency of the oscillator, there is a simple and well-known phase relation between the purely forced response and the forcing function. They will have the same or opposite signs (except when both functions vanish) according as the forcing frequency is less or greater than the natural frequency. It is shown here that when the sinusoidal forcing function is replaced by any member of a wide class of periodic functions, the relation between the signs of the forcing function and the forced response continues to hold when the forcing frequency is greater than the natural frequency, but no longer holds in the other case.

Operator Calculus of Quantized Operator

The notational ambiguities in Feynman's calculus are all remedied here by setting a more natural foundation of the ordered exponential operators, which will be called briefly “expansional” operators in this paper. The essential point to be stressed is than the pure exponential operator is merely a special case of the more wider class of operators, i.e., expansional operators, and the latter type of operators generally appears in quantum mechanics. A clear-cut distinction between between these two types of operators is all important. The so-called disentangling process is a device to decompose any one expansional operator into a product of some simple exponential operators. In the first place the general view-points concerning the expansional operators are presented, in which a rigorous organization of disentangling procedure is accomplished. Next are given some examples treating a generalized forced harmonic oscillator, where the transformation operator is completely disentangled and its representative yields automatically the classical action in its compact form, which process constitutes the original intention of this work. No accounts are given here concerning the quantized field theory.

Reactors in hydroelectric stations

The advantages and limitations in the use of current limiting reactors in hydroelectric stations are not so well recognized or understood as they are in steam turbine stations, but it is now coming to be realized that their use may be quite as justifiable and necessary in the former as in the latter. The two beneficial results of reactance, protection and localization, are distinct; the former being associated with the square, and the latter with the first power of the reactance. Of two detrimental results of reactance, or limitations to its use, one, that of voltage drop, is well understood but the other, the reduction of synchronous stability is not so well understood. The installation of reactance usually results in a decrease in the stability of synchronism. Such stability may be expressed in terms of the angular phase displacement between two groups, due to a sudden load disturbance. Practically, complete instability or asynchronism occurs in a large station when the phase angle exceeds 90 deg. The asynchronizing effect of a sudden load change is approximately twice that of a gradual change of the same magnitude. The origination of a power surge by sudden loss of load and the accompanying hunting oscillation is described. Increased reactance increases the amplitude of both the power surge and the phase-angle oscillation and also increases the period. A certain amount of reactance will result in complete asynchronism immediately following the load disturbance, and a smaller amount may cause troublesome and persistent hunting by forced harmonic oscillation of the turbine governors in combination with certain hydraulic conditions, such as long penstocks. It is shown that these phenomena form a practical limitation to the use of reactance, in that the reactances necessary to create the conditions described are of the same magnitude as those often indicated for protective and localizing purposes. The actual occurrence of a surge is described.