Potentials Poschl-Teller

The Poschl-Teller potential can be put into a correspondence with the algebra U(2). The procedure is identical to that discussed in Section 2.8. Consider the chain (2.31)... 

This is identical to Eq. (6.45) with A = D and X - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras 0(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case. 

Having established the correspondence between the Poschl-Teller potential and the algebra U(2), one can proceed to a quantization of bending vibrations along the lines of Section 4.2. We emphasize once more that the quantization scheme of bending vibrations in U(2) is rather different from that in U(4) and implies a complete separation between rotations and vibrations. If this separation applies, one can quantize each bending oscillator i by means of an algebra U,(2) as in Eq. (6.6). The Poschl-Teller Hamiltonian... 

This Hamiltonian is identical to that of stretching vibration [Eq. (6.7)]. The only difference is that the coefficients A, in front of C, are related to the parameters of the potential, D and a, in a way that is different for Morse and Poschl-Teller potentials. The energy eigenvalues of uncoupled Poschl-Teller oscillators are, however, still given by... 

Frank, A., and Wolf, K. B. (1985), Lie Algebras for Systems with Mixed Spectra. I. The Scattering Poschl-Teller Potential, J. Math. Phys. 26, 973. 

The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Poschl-Teller potential in quantum mechanics. 

As an example to illustrate this method, we study the case of a single particle in the presence of the Poschl-Teller potential [181]... 

Figure 33. Poschl-Teller potential Panels (a) and (b) show the scaled correlation length /1 and the scaled radial mean distance R/L as a function of the potential strength X for different system sizes with L = 3,6,9,..., 30. The grid spacing is kept fixed AL = 0.03, so the smallest system has Nl = 100 and the largest has Nl = 1000 points. Panels (c) and (d) illustrate the scaling of aE, and R with the system size L for different values of X = 1(0), 1-02,1.04,1.06,1.08,1.10. All numerical values are in atomic units.

Two points of interest in our algebraic analysis are that (1) this function is symmetrical under 0- -6 (as expected in a bending motion), and (2) it can be shown that for one-dimensional problems, the Poschl-Teller and Morse potentials are isospectral (i.e., they have the same bound-state spectrum) [25]. Figure 4 shows the typical behavior of the Poschl-Teller potential function. 

We now show that the algebraic realization of the one-dimensional Morse potential can be adopted as a starting point for recovering this same problem in a conventional wave-mechanics formulation. This will be useful for several reasons (1) The connection between algebraic and conventional coordinate spaces is a rigorous one, which can be depicted explicitly, however, only in very simple cases, such as in the present one-dimensional situation (2) for traditional spectroscopy it can be useful to know that boson operators have a well-defined differential operator counterpart, which will be appreciated particularly in the study of transition operators and related quantities and (3) the one-dimensional Morse potential is not the unique outcome of the dynamical symmetry based on U(2). As already mentioned, the Poschl-Teller potential, being isospectral with the Morse potential in the bound-state portion of the spectrum, can be also described in an algebraic fashion. This is particularly apparent after a detailed study of the differential version of these two anharmonic potential models. Here we limit ourselves to a brief description. A more complete analysis can be found elsewhere [25]. As a... 

A final comment about wavefunctions. We already have analytical expressions for wavefunctions of the harmonic, Morse, and Poschl-Teller potentials in the one-dimensional case. They can be obtained in terms of the single coordinate in a conventional differential approach. Instead, in the algebraic framework (or in a second-quantization scheme, more generally speaking), wavefunctions are expressed in terms of boson annihilation-creation operators acting on the Fock vacuum state, as shown in Eq. (2.29). In the specific case of U(2), for example, the basis state N,m) can be written (created) as... 

To begin with, we recall that in certain cases, the algebraic model has been already put in a one-to-one correspondence with a specific potential function for the usual space coordinates. We have already studied dynamic symmetries providing exact solutions for the one-, two-, and three-dimensional truncated harmonic oscillators, the Morse and Poschl-Teller potential functions. When we consider more complicated algebraic expansions in terms of Casimir operators, or when we deal with coupled... 

This is the precise analytical form of the one-dimensional Poschl-Teller potential, suited for bending vibrations. The corresponding n m Fermi interactions is thus given by... 

Other potentials, such as the Poschl-Teller potential and related forms, have been suggested, and one should not... 

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