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 ebook 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 issues in ebook One frequently involved quantum mechanics in a single- and two-dimensions, e-book units out to offer 3-dimensional platforms, the hydrogen atom, debris with spin, and relativistic particles. It additionally incorporates a easy direction on quantum details conception, introducing subject matters like quantum teleportation, the EPR paradox, and quantum desktops. jointly the 2 volumes represent a whole path in quantum mechanics that locations an emphasis on principles and ideas, with a good to reasonable quantity of mathematical rigor. The reader is predicted to be acquainted with calculus and undemanding linear algebra. any longer mathematical strategies should be illustrated within the textual content.
Th CD-ROM incorporates a huge variety of Quick-Time video clips offered in a multimedia-like setting. the flicks illustrate and upload colour to the text, and permit the reader to view time-dependent examples with a degree of interactivity. The point-and-click interface isn't any more challenging than utilizing the web.
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Example text
Hence, the degree of degeneracy of the eigenvalue E ;nr is at least 2 + 1. Experimentally, the states belonging to different eigenvalues of L3 can only be distinguished in the presence of magnetic fields. Hence, m is often called the magnetic quantum number. 191) r where f ;nr is the nr -th eigenfunction of the radial Schr¨ odinger operator 2 ( + 1) d2 + + V (r). 192) 2 2m dr r2 The corresponding eigenvalues E ;nr do not depend on the quantum number m. Hence, the multiplicity of each eigenvalue of H is at least 2 + 1.
Note that r, the radius of the sphere, is treated as a fixed parameter. 91) for the angular momentum in spherical coordinates, we arrive at ˆ = 1 L ˆ 2. 121) 2 2m r 2I 36 1. SPHERICAL SYMMETRY ˆ 2 has a discrete spectrum of eigenvalues, therefore the same The operator L is true for the energy of the rotator. Eigenvalues of the rigid rotator: A particle with mass m on a sphere with radius r can only have the energies 2 E = ( + 1), = 0, 1, 2, 3, . . 123) ψ ,m (ϑ, ϕ) = Y m (ϑ, ϕ), m = − , − + 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. n ˆ (z) = 2 ! In scattering theory, one often defines the Riccati-Hankel functions ˆ ± (z) = n ˆ (z) ± iˆ j (z) = e±i(z− π/2) 1 + O(1/z) , as z → ∞. 163) h jˆ (z) = Further details about the Riccati-Bessel functions can be found in the book [1], where the notation jˆ(z) = zj(z), and n ˆ (z) = zy(z) is used. 3. Special Topic: Expanding the plane wave The plane waves exp(ik·x) are important solutions of the free-particle Schr¨ odinger equation, despite the fact that they are not square-integrable.