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 e-book seemed in 2000, and earned the eu educational software program Award in 2001 for oustanding innovation in its box. whereas issues in ebook One mostly involved quantum mechanics in a single- and two-dimensions, e-book units out to give third-dimensional platforms, the hydrogen atom, debris with spin, and relativistic particles. It additionally includes a easy direction on quantum info idea, introducing themes like quantum teleportation, the EPR paradox, and quantum pcs. jointly the 2 volumes represent an entire path in quantum mechanics that locations an emphasis on rules and ideas, with a good to average quantity of mathematical rigor. The reader is predicted to be conversant in calculus and effortless linear algebra. to any extent further mathematical ideas can be illustrated within the textual content.
Th CD-ROM features a huge variety of Quick-Time videos offered in a multimedia-like setting. the flicks 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 is not any more challenging than utilizing the net.
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Additional resources for Advanced visual quantum mechanics
The expression for L3 in spherical coordinates is particularly simple. Just insert the third Cartesian component of eϑ and eϕ (see Eq. 90): ˆ 3 = −i L ∂ . 8. ˆ 3 cannot have half-integer eigenvalues m. The Now we can see that L ˆ 3 consists of continuous functions. As domain of the diﬀerential operator L ˆ 3 must therefore a function of the azimuthal angle ϕ, any eigenfunction of L be a periodic function: φ(r, ϑ, ϕ + 2π) = φ(r, ϑ, ϕ). 95) so that the ϕ-dependence of φ must be described by exp(imϕ), which is periodic with period 2π if and only if m is an integer.
From this, the Hamiltonian operator of the rigid rotator is obtained by replacing L2 with the quantum mechanical angularmomentum operator. 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, .
62): λ λ λ J3 ψ+ = J3 J+ ψm = (J+ J3 + [J3 , J+ ]) ψm = (J+ J3 + J+ ) ψm λ λ = (J+ m + J+ ) ψm = (m + 1) J+ ψm = (m + 1) ψ+ . 67) The vector ψ+ is still an eigenvector of J 2 belonging to the same eigenvalue λ, because J+ commutes with J 2 : λ λ λ λ = J+ J 2 ψm = J+ λ ψm = λ J+ ψ m = λ ψ+ . 68) λ . Either this vector is An analogous observation holds for the state J− ψm the zero vector, or it is a simultaneous eigenstate with eigenvalues m − 1 for λ , we J3 and λ for J 2 . 64) = λ − m2 ∓ m. 5. THE POSSIBLE EIGENVALUES OF ANGULAR-MOMENTUM OPERATORS λ From J± ψm 2 23 ≥ 0 it follows immediately that λ ≥ m2 ± m = m(m ± 1).