By Zhuangqi Cao, Cheng Yin
Advances in One-Dimensional Wave Mechanics presents a finished description of the movement of microscopic debris in one-dimensional, arbitrary-shaped potentials according to the analogy among Quantum Mechanics and Electromagnetism. using a deeper figuring out of the wave nature of topic, this e-book introduces the concept that of the scattered sub-waves and a sequence of latest analytical effects utilizing the Analytical move Matrix (ATM) strategy. This paintings might be necessary for graduate scholars majoring in physics, mostly in easy quantum idea, in addition to for tutorial researchers exploring electromagnetism, particle physics, and wave mechanics and for specialists within the box of optical waveguide and built-in optics.
Prof. Zhuangqi Cao is a Professor of Physics at Shanghai Jiao Tong college, China.
Dr. Cheng Yin is a instructor at Jiangsu Key Laboratory of strength Transmission and Distribution gear expertise, Hohai college, China.
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Extra info for Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View
7), there is 00 0 S1 ¼ À 0 S0 p ðxÞ , 0 ¼ À 2pðxÞ 2S0 ð3:11Þ 30 3 Semiclassical Approximation and Eqs. 10) become h|p0 | < < p2 and h|p0 /p| < < |p|, respectively. 13) also requires that the potential should vary slowly; there is λ dV ðxÞ ð3:15Þ 2½E À V ðxÞ dx << 1: If Eq. 15) is satisfied, and we ignore all terms of Oðhn Þ ðn ! 2Þ in Eq. 4), we can rewrite down the first-order WKB wave function Eq. 2) as A i ψ ðxÞ ¼ pﬃﬃﬃ exp h p Z x ! 16) holds only when the classically allowed region is involved, that is, the potential strength should be smaller than the particle energy V(x) < E .
On the other side, the new condition Eq. 26) gives QðxÞ / n n 1 À xnÀ2 , 4 4 ð3:28Þ which equals zero for the special case n ¼ 4, while the first-order WKB wave function is always exact. For the rest of the cases, the new condition and the old one gives the same result. It should be noted here that Eq. 26) is an important physical potential for it can be used in the description of various physical phenomena, such as the Coulomb potentials, centrifugal or monopole–dipole potentials, and van der Waals potentials .
Am. J. Phys. 74, 572 (2006) 20. W. Chen, T. Hong, H. Lin, Semiclassical quantization rule for the bound-state spectrum in quantum dots: scattering phase approximation [J]. Phys. Rev. A 68, 205104 (2003) 21. A. Einstein, Zur Quantentheorie der Strahlung [J]. Ver. Deut. Phys. Ges. 19, 82 (1917) 22. L. Brillouin, Remarques sur la me´canique ondulatoire [J]. J. Phys. Radium 7, 353 (1926) 23. J. Keller, Corrected Bohr-Sommerfeld quantization conditions for nonseparable systems [J]. Ann. Phys. 4, 180 (1958) Chapter 4 Exact Quantization Condition via Analytical Transfer Matrix Method Abstract The transfer matrix and the layer segment method are applied to study the energy eigenvalue spectrum of an arbitrary potential well, and a general quantization condition without approximation is presented.
Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View by Zhuangqi Cao, Cheng Yin