By Paolo Pasini; Slobodan Žumer; Claudio Zannoni

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4-69), and hence, considering Eq. (3-83), the eigenvalues of L z are Lz = lim; m = 0, ± 1, ... , ± I (4-75) Thus, the angular momentum is a vector whose length equals IiJ[I(/ + 1)] This vector is oriented so that the component in a chosen direction, is an integral multiple of Ii. The behaviour of the vector of the angular momentum, as follows from Eqs. (4-70), (4-71), (4-74) and (4-75), can be understood as its spatial quantization. e. Eq. (4-72), with the eigenfunctions for the hydrogen atom (see Table 3-1) shows that the angular parts (spherical harmonics) are identical for the two cases.

The rigid rotator. : a .. 34 a system consisting of two point particles of mass m1 and m2 , which are held a constant distance apart by a massless bond. This system rotates around the (0) axis passing through the centre of mass of the system and lying perpendicular to the projection plane. Here, translational motion of the rotator is not considered and therefore the centre of mass of the rotator is considered to be at rest, fixed in the origin of the coordinate system. In classical mechanics the following relation [cf.

4-15) and multiply it by function f, after integration we obtain *_ o - Sf(J)*f* d. SIfl2 d. (4-18) From comparison of Eqs. (4-17) and (4-18) and from the condition of hermicity of operator (4-14) it can be seen that 0=0* (4-19) Thus, it obviously holds that: Theorem 1. The eigenvalues of Hermitian operators are real numbers. 2 Axiomatic foundation of quantum mechanics Scientific disciplines, deductive by character, depend on axioms or postulates which are considered to be fundamental and non-deducible.

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