The Particle in a Ring

The diagrams summarizing relations between energies, y, and number of states are collected in Fig. 15-3. 

Chapter 15 Molecular Orbital Theory of Periodic Systems 

Cyclic structures with varying potentials exist (e.g., benzene), and such systems retain the degeneracy and real or complex orbital features. Also, j continues to have meaning (restricted) with respect to wavelength and number of nodes. However, angular momentum and energy are no longer simply related to j. 

We next examine benzene—a cyclic system having a nonuniform potential. We consider only the n electrons as described by the simple Hiickel method, since the features we wish to point out are already present at that elementary level. A formula is presented (without derivation) in Chapter 8 [Eq. (8-52)] for the MOs of molecules like benzene. This gives, for the yth MO of benzene. 

In these formulas, j is restricted to the integer values ranging from 0 to 5, giving six MOs. If one tries other integer values of j in formulas (15-4) and (15-5), one simply reproduces members of the above set. Indeed, any six sequential integer values for j produces the same set of solutions as does the sequence 0-5. In particular, the set -2, -1,0,1,2, 3 is perfectly acceptable and provides a match with the conventions of solid-state physics and chemistry. 

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In other words, only the ground state is non-degenerate, while all the excited states are doubly degenerate. The quantum mechanical results of the particle-in-a-ring problem are summarized in Fig. 1.5.4. 

In Section 1.5.1, it was mentioned that the energy of the lowest state of a particle confined in a one-dimensional box is not zero and this residual energy is a consequence of the Uncertainty Principle. Yet the ground state energy of the particle-in-a-ring problem is zero. Does this mean the present result is in violation of the Uncertainty Principle The answer is clearly no, and the reason is as follows. In a one-dimensional box, variable x starts from 0 and ends at a, the length of the box. Hence Ax can at most be a. On the other hand, in a ring, cyclic variable does not lie within a finite domain. In such a situation, the uncertainty in position cannot be estimated. 

Section 2-6 The Particle in a Ring of Constant Potential TABLE 2-1 51... 

The solution of Eq. (4-36) is similar to that of the particle in a ring problem of Section 2-6. The normalized solutions are... 

Let us consider the operations associated with the n-fold proper axis oriented along the z axis and ask what will become of the function /=exp(n ) as it is rotated clockwise about this axis by Inin radians. (We entertain the idea that exp(i< ) might be a convenient basis for a representation since such functions were found to be eigenfunctions for the particle-in-a-ring problem in Chapter 2.) Since the clockwise direction is opposite to the normal direction of the coordinate, the effect of the rotation is to put /((f>) where f ) = Rfexp i), is given by exp(/< )/exp[/(< — 27t/ )], or... 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

This gives the magnitude of the angular momentum vector. The 0 dependence of the functions in Table 3.1 is exactly the same as for the particle on a ring, and m indeed has the same significance it gives the component of angular momentum about the z axis ... 

We now consider, following Flygare, the model system of a particle in a ring, with a magnetic field defining the z axis perpendicular to the plane of the ring (see figure 8.10). This model system has obvious similarities to real molecular systems... 

This is for the three-dimensional case. In the two-dimensional case illustrated in the figure, the average number of particles in a ring of width da is plnada. 

Use the simple approach presented in Problem 2-3 to demonstrate that A = 1 / for the trigonometric particle-in-a-ring eigenfunctions and 1 for the exponential eigenfunctions. 

For a single particle-in-a-ring system having energy 9h /Sjv I we can say that the angular momentum, when measured, will equal... 

Calculate the average angular momentum, Lz, for a particle in a ring of constant potential having wavefunction... 

Figure 15-3 (a) The energy for a particle in a ring has parabolic dependence on y, but exists only... 

Just as the particle in a box gives us some imderstanding of the distribution and energies of n electrons in linear conjugated systems, the particle on a ring is a useful model for the distribution of k electrons around a cyclic conjugated system. 

The particle in a two-dimensional well is a useful model for the motion of electrons around the indole ring (3), the conjugated cycle found in the side chain of tryptophan. We may regard indole as a rectangle with sides of length 280 pm and 450 pm, with 10 electrons in the conjugated % system. As in Case study 9.1, we assume that in the... 

Figure 3 Perspective views of a 60 atom Fe ring recorded at tunnelling current of 1 nA and tip bias voltages of (A) 10 mV and (B) -10 mV. The quantum interference patterns inside the ring change with energy. The energy dependence of the lowest density of states at the centre of the ring is iiiustrated by the d//d /spectra in (C). The sharp peaks in the spectra indicate sharp resonances in the iowest density of states. These data match theoreticai resuits based upon the particle-in-a-box model very closely. Reprinted from Physica DB3 Crommie MF, Lutz CP, Eigier DM and Heller EJ, Quantum corrals, pp 98-108, 1993 with kind permission of Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.

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