Raising and lowering operators

To illustrate the use of raising and lowering operators to find the states that ean not be identified by inspeetion, let us again foeus on the p ease. Beginning with three of the states that are easy to reeognize, piapoot, piap.ia, and p.iapoot, we apply S. to obtain the Ms=0 funetions ... 

Construct the 3 triplet and 1 singlet wavefunetions for the Ei+ ls 2s configuration. Show that each state is a proper eigenfunction of S2 and Sz (use raising and lowering operators for S2)... 

The Raising and Lowering Operators Change the Jz Eigenvalue but not the J2 Eigenvalue When Aeting on j,m>... 

As stated above, the CG coefficients can be worked out for any particular case using the raising and lowering operator techniques demonstrated above. Alternatively, as also stated above, the CG coefficients are tabulated (see, for example, Zare s book on angular momentum the reference to which is given earlier in this Appendix) for several values of j, j, and J. 

Erection load The load produced in the mast and its supporting structure during the raising and lowering operation. 

We now apply the raising and lowering operators to find the eigenvalues of and Jz. Equation (5.17) tells us that for a given value of X, the parameter m has a maximum and a minimum value, the maximum value being positive and the minimum value being negative. For the special case in which X equals zero, the parameter m must, of course, be zero as well. 

Using the raising and lowering operators J+ and show that... 

The raising and lowering operators for spin angular momentum as defined by equations (5.18) are... 

Chapters 4, 5, and 6 discuss basic applications of importance to chemists. In all cases the eigenfunctions and eigenvalues are obtained by means of raising and lowering operators. There are several advantages to using this ladder operator technique over the older procedure of solving a second-order differ-... 

Our next goal is to transform this expression into one based on the total electron spin operator, S = si + s2. The first three terms can be simplified by making use of the identity (derived using raising and lowering operators) ... 

If we expand the hyperfine term of the spin Hamiltonian and write the operators in terms of raising and lowering operators ... 

Making use of the raising and lowering operators defined in Tables II and III, Hamiltonians (78) and (79) become finally... 

The standard procedure used to solve Eq. (29) is to transform the spin operators into fermionic operators [61]. Let us define the raising and lowering operators... 

If A] is phase-free, as discussed in Section III, and in Ref. 15, there are no longitudinal electric field components. This also occurs if A,-3"1 is zero [17]. The B(3) field is then a Fourier sum over modes with operators a qaq and is perpendicular to the plane defined by A and /1<2>. The four-dimensional dual to this term is defined on a time-like surface, following Crowell [17], which can be interpreted as E under dyad vector duality in three dimensions. The ( field vanishes because of the nonexistence of the raising and lowering operators l3 , . The BM is nonzero because of the occurrence of raising and lowering... 

The sign convention is that prescribed by Condon and Shortley (5) and is not that often seen in quantum-mechanics texts. It is required to give correct results when using the raising and lowering operators /+ and /. ... 

The raising and lowering operators were introduced by Dirac in his book. The Principles... 

The results of the current section, both the lowering operators and the classification, will come in handy in Section 8.4, where we classify the irreducible representations of so(4). One can apply the classification of the irreducible representations of the Lie algebra sm(2) to the study of intrinsic spin, as an alternative to our analysis of spin in Section 10.4. More generally, raising and lowering operators are widely useful in the study of Lie algebra representations. 

Like the raising and lowering operators, the Casimir operator does not correspond to any particular element of the Lie algebra 5m(2). However, for any vector space V, both squaring and addition are well defined in the algebra gt (V) of linear transformations. Given a representation, we can define the Casimir element of that representation. ... 

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