Harmonic oscillator ladder operators

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... 

Taking as a particular case the harmonic oscillator ladder operators [20]... 

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... 

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... 

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... 

Occupation Number Representation of the Harmonic Oscillator. The Hamiltonian H for the harmonic oscillator, Eq. (3.4.1), can be rewritten in terms of ladder operators a + and a, which resemble the angular momentum ladder operators [6]. Substituing Eq. (3.4.2) into Eq. (3.4.1), H can be rewritten in terms of the momentum operator p (in the x direction) and the position operator x ... 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonic oscillator, 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. 

For obvious reasons, and a are known as step-up and step-down operators, respectively. They are also called ladder operators since they take us up and down the ladder of harmonic-oscillator eigenvalues. In the context of radiation theory, and a are called creation and annihilation operators, respectively, since their action is to create or annihilate a quantum of energy. 

Using ladder operators to evaluate matrix elements, calculate the average potential and kinetic energies for a harmonic oscillator in its nth quantum state. 

The two first equations follow from the commutation relations of the angular momentum components, Jf F = X, Y, Z, along space fixed axes. The latter two are not as obvious as they may appear at first glance. Since Jx and Jy do not commute with h we cannot factorize the vector space to treat // and Jz in a separated basis of vectors, (In, />, such as it is usually assumed when discussing the two-dimensional harmonic oscillator. Jy + iJx cannot be used as ladder operators of Jz, and similarly it may be shown that the usual ladder operators85 for H are inapplicable as well, since they do not commute with/2. 

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

The case of the parabolic barrier can be solved in a similar fashion however, the algebraic procedure becomes cumbersome due to the fact that the corresponding ladder operators are not adjoint to each other. A better approach is to use the concept of the Bargmann-Segal space, whereby avoiding long algebraic derivations [32]. We exemplify this method in the case of the harmonic oscillator. Let us consider the time-dependent Hamiltonian ... 

We now need a systematic way to evaluate matrix elements like — Rgf v y. This is provided by the second quantization formulation [5] of the one-dimensional harmonic oscillator problem, which parallels in some ways the ladder operator treatment of angular momentum. The harmonic oscillator Hamiltonian is... 

In addition, Dirac noted that Eq. (3.48) is identical to the classical expression for the energy of a harmonic oscillator with unit mass (Eq. 2.28). The first term in the braces corresponds formally to the kinetic energy of the oscillator the second, to the potential energy. It follows that if we replace Pj and Qj by momentum and position operators Pj and Q, respectively, the eigenstates of the Schrodinger equation for electromagnetic radiation will be the same as those for harmonic oscillators. In particular, each oscillation mode will have a ladder of states with wavefunctions r y and energies... 

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