By Zhuangqi Cao, Cheng Yin

ISBN-10: 3642408907

ISBN-13: 9783642408908

ISBN-10: 3642408915

ISBN-13: 9783642408915

*Advances in One-Dimensional Wave Mechanics* presents a finished description of the movement of microscopic debris in one-dimensional, arbitrary-shaped potentials in line with the analogy among Quantum Mechanics and Electromagnetism. using a deeper knowing of the wave nature of topic, this booklet introduces the idea that of the scattered sub-waves and a sequence of recent analytical effects utilizing the Analytical move Matrix (ATM) approach. This paintings could be priceless for graduate scholars majoring in physics, in general in simple quantum conception, in addition to for educational 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 energy Transmission and Distribution gear know-how, Hohai collage, China.

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In the following chapter of this book, we will introduce the concept of the scattered subwaves, which takes all the multiple reflection effect into consideration. 2 33 Semiclassical Limit A few more words is necessary here to explain the semiclassical limit h ! 0, in which the WKB approximation is expected to work well. Relatively, we refer to the anticlassical limit as h ! ∞, where the quantum effect becomes obvious. In Eq. 15) we derived a rough criterion of the semiclassical limit by requiring that the absolute value of the second term on the left-hand side of Eq.

Consider the double-well potential as plotted in Fig. 1 Double-Well Potentials 49 V(x) Fig. 1 The main waves and the subwaves in a double-well potential V0 V3 V1 0 d1 V2 d1+d2 x where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 À Áﬃ < κj ¼ 2μ E À V j =h qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : α ¼ 2μÀV À EÁ=h j j ðj ¼ 1, 2Þ ðj ¼ 0, 3Þ : ð4:3Þ Applying the continuity conditions of the ψ(x) and ψ 0 (x) at boundaries of x ¼ 0, x ¼ h1, and x ¼ h1 + h2, we can obtain the following expression, that is, the energy eigenvalue equation: Á Â Â À κ1 À κ2 exp i2ðκ1 h1 þ κ2 h2 À φ10 À φ23 Þ þ exp i2 κ1 h1 À φ10 κ1 þ κ2 þ κ2 À κ1 exp½i2ðκ2 h2 À φ23 Þ¼ 1, κ2 þ κ1 ð4:4Þ where 8 0 1 > > α0 > > φ10 ¼ [email protected] A > > > κ1 < 0 1: > > > > > φ32 ¼ [email protected]α3 A > > : κ2 ð4:5Þ It should be noted that 2φ10 and 2φ32 in Eq.

N. Bhor, On the theory of atomic constitution [J]. Philos. Mag. 26, 471 (1913) 3. N. Bhor, On the theory of atomic constitution [J]. Philos. Mag. 26, 857 (1913) ¨ ber die Ausbreitung der Wellen in der drahtlosen Telegraphie [J]. Ann. der 4. A. Sommerfeld, U Physik 50, 385 (1916) 5. J. Zeng, Quantum Mechanics, vol. I [M] (Science Press, Beijing, 2007) 6. G. Wentzel, A generalisation of the quantum constraints for the purposes of the wave mechanics [J]. Z. Physik 38, 518 (1926) 7. A. Kramers, Wave mechanics and half-integral quantization [J].

### Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View by Zhuangqi Cao, Cheng Yin

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