By Bernd Thaller

Visual Quantum Mechanics is a scientific attempt to enquire and to coach quantum mechanics simply by computer-generated animations. even though it is self-contained, this e-book is a part of a two-volume set on visible Quantum Mechanics. the 1st booklet seemed in 2000, and earned the eu educational software program Award in 2001 for oustanding innovation in its box. whereas themes in publication One quite often involved quantum mechanics in a single- and two-dimensions, e-book units out to give third-dimensional structures, the hydrogen atom, debris with spin, and relativistic particles. It additionally includes a uncomplicated path on quantum info conception, introducing issues like quantum teleportation, the EPR paradox, and quantum desktops. jointly the 2 volumes represent an entire direction in quantum mechanics that areas an emphasis on rules and ideas, with a good to average quantity of mathematical rigor. The reader is anticipated to be conversant in calculus and common linear algebra. to any extent further mathematical suggestions can be illustrated within the textual content.

Th CD-ROM encompasses a huge variety of Quick-Time video clips provided in a multimedia-like setting. the films illustrate and upload colour to the text, and let the reader to view time-dependent examples with a degree of interactivity. The point-and-click interface isn't any tougher than utilizing the web.

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156) where Jν and Nν are the Bessel function and Neumann functions of order ν. ˆ is singular for > 0. 15 allows investigation of the dependence on . 2. Special Topic: Properties of the Riccati-Bessel functions The real-valued functions jˆ (z) and n ˆ (z) are for z > 0 solutions of the equation − ( + 1) d2 y(z) + y(z) − y(z) = 0. 158) 46 1. 159) jˆ (z) = −(−z) +1 jˆ0 (z) , z dz z 1 d 1 n ˆ 0 (z) . 160) z dz z Their limiting behavior for small z is given by 2 ! 161) (2 + 1)! (2 )! 162) z 1 + O(z 2 ) , as z → 0.

During one period, the trajectory goes twice through ϑ, hence we multiply the time by 2 to obtain the total time spent in dϑ during one period. The time needed to complete a period is T = 2π/L. Hence, the probability of ﬁnding the particle in dϑ is L dt(ϑ) L 2 dt(ϑ) = dϑ. 134) 2π π dϑ Hence, one needs to invert the function ϑ(t) and diﬀerentiate with respect to ϑ. It is suﬃcient to invert the function ϑ(t) on a part of the time interval during which the trajectory goes through the point ϑ under consideration.

In a classical picture, the angular momentum vector (if measured with respect to a certain direction) thus lies on certain cones. Eigenvalues of the orbital angular momentum: The eigenvalues of the operator L2 are precisely the numbers 2 ( + 1), where is a non-negative integer. For each , the operator L3 has the eigenvalues m, where m = − , − + 1, . . , . 7. In quantum mechanics, this picture should not be taken seriously because the same result would be obtained for the possible values of L1 and L2 (or the component of L in an arbitrary direction).