Particle in a ring

In this system, the electron can move only along the circumference and therefore polar coordinates can be used to advantage, as shown in Fig. 1.5.3  

The constant A can again be determined with the normalization condition  

The allowed energy values for this system can be determined using eq. (1.5.47)  

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. 

If we apply the free-electron model to the six n electrons of benzene, we see that the Vd, in, and // i orbitals are filled with electrons, while in and Vy—2 and all the higher levels are vacant. To excite an electron from in (or //-1) to in (or ir-2), we need an energy of 

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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. 

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. 

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

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. 

Consider two related systems—a particle in a ring of constant potential and another just like it except for a very thin, infinitely high barrier inserted at = 0. When this barrier is inserted,... 

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

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... 

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

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... 

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

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