Eigenvalue total angular momentum

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,... 

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

Thus () is an eigenvalue of Lz with eigenvalue The angle-dependent part of the wave equation is seen to contain wave functions which are eigenfunctions of both the total angular momentum as well as the component of angular momentum along the polar axis. 

From (27) and (29) it follows that every component of the total angular momentum operator J = L + S and J2 commute with the Dirac Hamiltonian. The eigenvalues of J2 and Jz are j(j + 1 )h2 and rrijh respectively and they can be defined simultaneously with the energy eigenvalues E. 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

For systems of two or more particles orbital and spin angular momenta are added separately by the rules of vector addition. Integral projections of the shorter vector on the direction of the longer vector, are added to the long vector, to give the possible eigenvalues of total angular momentum. 

From this equation, at various specific values of the ranks K, k, K, k, K", k", we can work out a set of equations for sums of a definite parity of ranks of the total angular momentum. The values of the ranks are selected to yield operators with known eigenvalues on the left side. For complete scalars K" = k" = 0, we get... 

The eigenvalues and eigenfunctions of the orbital angular momentum operators can also be derived solely on the basis their commutation relations. This derivability is particularly attractive because the spin operators and the total angular momentum obey the same commutation relations. 

We recall that the square of the total angular momentum J1 commutes with all the components of / and hence with the rotation operator R(rotation operator is applied to an angular momentum eigenfunction j, m), the result is also an eigenfunction of J2 with the same eigenvalue j(j + 1) ... 

Before going into a detailed description, a mathematical definition of the problem is useful. If we have conserved quantities, like the total angular momentum J and its projection J, the eigenvalues of the linearized motion matrix M has now the following eigenvalue structure. Let us have, as usual, n degrees of freedom and k < n —2 conserved quantities. Then we have... 

One eigenvalue pair = 1, in the J,%j plane, corresponding to the conservation of total angular momentum, along the p.o. 

We sometimes abbreviate the description of jm) by calling it an eigenstate of J with eigenvalue /. Atomic states are characterised by the total angular momentum quantum number j. 

The parameter k describes the angular momentum. It is the eigenvalue of the operator —K defined in (107). The quantum number k determines the total angular momentum... 

Despite their formidable appearances, these expressions can be computed numerically with minimum effort. We discuss the essential role of this operator not only for molecules close to the normal limit, but also for symmetric molecules, even if very local ones. Finally, the J or < o<3) operator gives the eigenvalue of the square of the total angular momentum operator. 

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