Orbital angular momentum classical

The Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

The quantum-mechanical operators for the components of the orbital angular momentum are obtained by replacing px, Py, Pz in the classical expressions (5.2) by their corresponding quantum operators. 

The analysis is performed for the calculations with rrot=0 K for the CH3C1 reactant, so that the angular momentum distribution for the complex P(j) is the distribution of orbital angular momentum for complex formation P(i). This latter distribution is given in ref. 37. Jm , the quantum number for j, varies from 282 for Enl = 0.5 kcal/mol to 357 for rel = 3.0 kcal/mol. The term k iEJ) in equation 24 is written as k (.EJ)=k Ejyf E), where k EJ) is the classical RRKM rate constant with the CH3C1 intramolecular modes inactive and / ( ) is treated as a fitting factor. 

Figure 4. The average change in the squared value of the classical orbital angular momentum ((A/. )) and the standard deviation of the A/. distribution

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

The unpaired pn electron has an orbital angular momentum L with projections A = 1 on the body-fixed 2-axis which — in a classical sense — describe rotation of the electron in opposite directions about the internuclear axis. Furthermore, it possesses a spin S with components = 1/2. Coupling of A and leads to two (2 1 = 2) manifolds with A = 1/2 and 3/2 which are represented by 2nx/2 and 2n3/2, respectively. The splitting is in the range of several hundredths cm-1. 

The orbital angular momentum quantum number / has other strange characteristics. Notice that / = 0 is allowed. However, for a classical orbit, circular or elliptical, we have L = mv R (Equation 5.19), which cannot be zero. The vector L points in the... 

PROBLEM 2.11.2. Evaluate the speed of the orbital motion of an electron that has orbital angular momentum m,vr0 = h, if the mass is concentrated at the electron radius r0 estimated above either from electron-electron scattering (1(P16 m) or from its "classical radius" (2.892 x 10-15 m) or from the Compton wav elength Ac (2.426 x 10 12m). 

The integrals in Eqs. (149) and (150) can be evaluated as follows. We slightly deviate from a strictly classical analysis by quantizing the orbital angular momentum I and the diatom rotational angular momentum j. That is, we assign... 

The classical expression for the magnitude of the angular momentum of a rotating particle having mass m and describing a circular path of radius r with speed v is mvr. Since it is also necessary to characterize the plane and the direction of the rotation, the orbital angular momentum is a vector equal to the cross product of the position vector r and the linear momentum p ... 

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