Operators ladder

For the following basis of funetions (T 2p p and F2p ), construet the matrix representation of the Lx operator (use the ladder operator representation of Lx). Verify that... 

There are two procedures available for solving this differential equation. The older procedure is the Frobenius or series solution method. The solution of equation (4.17) by this method is presented in Appendix G. In this chapter we use the more modem ladder operator procedure. Both methods give exactly the same results. 

We now solve the Schrodinger eigenvalue equation for the harmonic oscillator by the so-called factoring method using ladder operators. We introduce the two ladder operators d and a by the definitions... 

Application of equation (3.33) reveals that the operator is the adjoint of a, which explains the notation. Since the operator d is not equal to its adjoint d neither d nor d is hermitian. (We follow here the common practice of using a lower case letter for the harmonic-oscillator ladder operators rather than our usual convention of using capital letters for operators.) We readily observe that... 

We have already introduced the use of ladder operators in Chapter 4 to find the eigenvalues for the harmonic oscillator. We employ the same technique here to obtain the eigenvalues of and Jz. The requisite ladder operators and J-are defined by the relations... 

Equation (6.24) may be solved by the Frobenius or series solution method as presented in Appendix G. However, in this chapter we employ the newer procedure using ladder operators. 

We now solve equation (6.24) by means of ladder operators, analogous to the method used in Chapter 4 for the harmonic oscillator and in Chapter 5 for the angular momentum. We define the operators Ax and Bx as... 

Equations (7.16) illustrate the behavior of and S- as ladder operators. The operator S+ raises the state /3) to state ja), but cannot raise a) any further, while lowers a) to /3), but cannot lower y3). From equations (7.7) and (7.16), we obtain the additional relations... 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonie oseillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

The reason for rejecting the solution s = —(/ + 1) is actually more complicated for states with 1 = 0. I. N. Levine (1991) Quantum Chemistry, 4th edition (Prentice-HaU, Englewood Cliffs, NJ), p. 124, summarizes the arguments with references to more detailed discussions. The complications here strengthen the reasons for preferring the ladder operator technique used in the main text. 

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-... 

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

Berrondo, M Palma, A., and L6pez-Bonilla, J. L. (1987), Matrix Elements for the Morse Potential Using Ladder Operators, Int lJ. Quant. Chem. 31,243. 

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