# Classical and Quantum Mechanical Aspects of Heavy Ion by H.L. Harney, P. Braun-Munzinger, C.K. Gelbke

By H.L. Harney, P. Braun-Munzinger, C.K. Gelbke

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4 gives the radial probability density for the 1s ground state of the H atom and the average value of the distance of the electron from the nucleus hri1s . It is seen that, while the probability density P1s ðrÞ has a maximum at r ¼ a0 (the Bohr radius), hri1s ¼ 1:5a0 (the vertical bar in the figure). 5 Real Form of the Atomic Orbitals In valence theory, it is customary to use the real form for the angular part of the AOs, which is the same for all orbitals, even those that are not 44 ATOMIC ORBITALS hydrogen-like.

Multiplying a matrix A by a complex number c implies multiplication of all elements of A by that number: cA ¼ B Bij ¼ cAij ð2:5Þ The product, rows by columns, of two (or more) matrices A by B is possible if the matrices are conformable (the number of columns of A equals the number of rows of B): AB ¼ C mÂn nÂp mÂp Cij ¼ n X Aia Baj ; a¼1 Dij ¼ p n X X ABC ¼ D ; mÂn nÂp pÂq mÂq ð2:6Þ Aia Bab Cbj a¼1 b¼1 Matrix multiplication is usually not commutative, the quantity ½A; B ¼ AB À BA ð2:7Þ being the commutator of A and B.

The first step is the separation of the partial differential equation in the angular variables u and w into two differential equations, one for each variable, by posing Yðu; wÞ ¼ QðuÞFðwÞ ð3:20Þ This implies a second separation constant, which will for convenience be called m2. Of the two resultant total differential equations d2 F þ m2 F ¼ 0 dw2 ð3:21Þ 1 d d m2 sin u þlÀ QðuÞ ¼ 0 sin u du du sin2 u ð3:22Þ and the first is the well-known equation of harmonic motion (Atkin, 1959), whose regular solutions in complex form are4 1 Fm ðwÞ ¼ pﬃﬃﬃﬃﬃﬃ expði mwÞ 2p m ¼ 0; Æ1; Æ2; .