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By Harald Friedrich

This verified textual content and reference includes a sophisticated presentation of quantum mechanics tailored to the necessities of contemporary atomic physics. It contains issues of significant present curiosity akin to semiclassical concept, chaos and Bose-Einstein condensation in atomic gases. The 3rd version of Theoretical Atomic Physics extends the profitable moment version with an in depth remedy of the wave movement of atoms close to the anticlassical or severe quantum regime, and it additionally includes an creation to a couple elements of atom optics that are suitable for present and destiny experiments concerning ultra-cold atoms. a number of difficulties are incorporated including whole options. since it has extra emphasis on conception, this publication allows the reader to understand the elemental assumptions underlying average theoretical constructs and to embark on self reliant study projects.

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This can be clearly seen in Fig. 5, in which the radial wave functions have been renormalized such that their norm becomes inversely proportional to their separation in energy at large quantum numbers: n3 φn,l (r) . 139) 2R In this normalization the heights of the inner maxima are independent of n for large n, and the wave functions for a given l converge to a well defined (E=0) with infinitely many nodes in the limit n → ∞. 75) at energy E = 0 and has the explicit form √ r 8r (E=0) . 4). 141) Jν (x) = ν!

86) for positive energies E = ¯ h2 k2 /(2µ) . 3 Examples Square Well Potential In this case we have V (r) = −V0 0 for r < r0 , for r ≥ r0 . 122) If the effective potential Veff consists only of V (r) with no centrifugal potential and no Coulomb contribution, then for negative energies −V0 < E < 0 the odinger equation in the inner region is solution φr≤r0 of the Schr¨ φr≤r0 (r) = sin Kr . 123) The wave number K in the inner region depends on the energy E = −¯ h2 κ2 /(2µ) and the potential parameter K0 = K= 2µV0 /¯ h2 (see Fig.

F. 150)) 2µ sin(kr + δbg ) , r → ∞ . 222) π¯ h2 k δbg is a background phase shift, which originates mainly from the diagonal po¯ 2 k 2 /(2µ). 3 in Sect. 2). 223) and form the scalar product with the bra φ0 |, then we obtain an explicit expression for the coefficient A, A= φ0 |V2,1 |φreg . 224) The matrix element in the denominator is the double integral ˆ 1,2 |φ0 φ0 |V2,1 GV ∞ ∞ dr = 0 dr φ∗0 (r)V2,1 (r)G(r, r )V1,2 (r )φ0 (r ) . 221) and the corresponding irregular solution which behaves like a cosine asymptotically, φirr (r) = 2µ cos(kr + δbg ) , π¯ h2 k r→∞.

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