Operator angular momentum squared

By combining with Equations 3-22, 3-23, and 3-24, the total angular momentum squared operator is obtained, and, to no surprise, it is proportional to the legendrian. 

The spherical harmonic wavefunctions are eigenlimctions only of the z angular momentum operator and the overall angular momentum squared operator. 

Confirm that the Y21 is not an eigenfunction of the x or y angular momentum operators but is an eigenfunction of the z angular momentum and overall angular momentum squared operators. 

Since an electron has an intrinsic spin, there must be a corresponding operator for the overall intrinsic spin angular momentum squared,. It is expected that the intrinsic spin eigenfunctions, Xsm, are analogous to the spatial spherical harmonic wavefunctions, Yi (6, ). The operators S, and 5 will be the only operators for which the intrinsic spin functions are eigenfunctions just like YUd. < >) are only eigenfunctions of i) and operators. 

The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

This new operator is referred to as the square of the total angular momentum operator. 

Again, the square of the total rotational angular momentum operator appears in Hj-ot... 

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

The coefficient of 1/r2 now takes the place of the operator for the square of the orbital angular momentum in the nonrelativistic Schro-dinger equation. This correspondence can be made more explicit by introducing the Johnson operator... 

Accordingly, the quantum-mechanical Hamiltonian operator H for this system is proportional to the square of the angular momentum operator U-... 

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

If in the non-BO calculation one chooses a basis set of eigenfunctions of the operator representing the square of the total angular momentum and the... 

In the absence of an electric field, the non-BO Hamiltonian commutes with the square of the angular momentum operator, [H, P] = 0, and so the eigenfunctions of the Hamiltonian also have to be eigenfunctions of J. This condition is met, for example, by functions such as... 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

Physically, it means that it is possible to know simultaneously the square of the intensity of the spin angular momentum and its component along z. Since the spin wavefunctions are not eigenfunctions of the operators S or /, it is impossible to... 

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