Operators, angular momenta

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,... 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. 

In exereise 7 above you determined whether or not many of the angular momentum operators eommute. Now, examine the operators below along with an appropriate given funetion. Determine if the given funetion is simultaneously an eigenfunetion of both operators. Is this what you expeeted ... 

The components of the quantum mechanical angular momentum operators along the three principal axes are ... 

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... 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

Again, the rotational kinetic energy, which is the full rotational Hamiltonian, can be written in terms of the total rotational angular momentum operator J2 and the component of angular momentum along the axis with the unique principal moment of inertia ... 

An example will help illustrate these points. The px, Py and pz orbitals are eigenfunctions of the angular momentum operator with eigenvalues equal to L(L+1) h ... 

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

The above results apply to any angular momentum operators. The essential findings ean be summarized as follows ... 

A. The Hamiltonian May Commute With Angular Momentum Operators... 

There are cases in which the angular momentum operators themselves appear in the Hamiltonian. For electrons moving around a single nucleus, the total kinetic energy operator T has the form ... 

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

The corresponding total angular momentum operator can be obtained as... 

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

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... 

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