摘要:

The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ‘‘quantum bouncer’’ is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.

ISSN:0002-9505    

DOI:10.1119/1.16673

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.

ISSN:0002-9505    

DOI:10.1119/1.16674

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

As a concrete example of the idea of renormalization, quantum mechanical scattering of particles by a two‐dimensional delta‐function potential is considered. The renormalization of the scattering cross section is carried out exactly and analytically. The calculation, free from obscuring mathematical details required for realistic field theories, may aid in making the idea of renormalization more accessible.

ISSN:0002-9505    

DOI:10.1119/1.16675

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

A simple formula is presented for the change in energy of a quantum system in a box when the box is infinitesimally deformed. The result is applied to rederive a formula given by Bartonetal., [Am. J. Phys.58, 751–755 (1990)] for the difference in energy of a bound state calculated with and without box boundary conditions. The formula can also be used to discuss pressure in a quantum system. A generalization of the result has been used for calculating scattering amplitudes for waves scattered by a rough extended surface. Here, a similar manifestly reciprocal result is derived for scattering by bounded objects.

ISSN:0002-9505    

DOI:10.1119/1.16676

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

A pedagogical approach to the emission of electromagnetic radiation is outlined in which Maxwell’s equations are solved directly forEandBwithout making use of the vector and scalar potentials.

ISSN:0002-9505    

DOI:10.1119/1.16677

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

A long‐standing problem and major shortcoming of semiclassical (WKB) bound‐state wave functions is their divergence at the classical turning points. Presently available regularization schemes are accurate but rather complicated. It is shown how finite wave functions can be simply obtained by reorganizing the WKB perturbation expansion in powers of ℏ and retaining appropriate higher‐order terms. The wave functions obtained by this method are compared with exactly known ones for the harmonic‐oscillator and Morse potentials.

ISSN:0002-9505    

DOI:10.1119/1.16678

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16679

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16680

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16681

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16838

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP