The Hydrogen Atom A Central Force Problem

It would have been easier (and shorter ) to use the formulas for the energy and momenta to determine the values of these three quantized observables. But it is important to understand that these differential equations actually do work when the wavefunctions are operated on by them. The above example shows that all of the operators do yield the appropriate values of the observables. 

There are a few other analytically solvable systems, but most are variations on the themes presented here and in the last chapter. For now, we will halt our treatment of model systems and move on to a system that is more obviously relevant chemically. But before we do, it is important to reemphasize a few conclusions about the systems we have treated so far. (1) In all of our model systems, the total energy (kinetic -I- potential) is quantized. This is a result of the postulates of quantum mechanics. (2) In some of the systems, other observables are also quantized and have analytic expressions for their quantized values (like momentum). Whether other observables have analytic expressions for their quantized values depends on the system. Average values, rather than quantized values, may be all that can be determined. (3) All of these model systems have approximate analogs in reality, so that the conclusions obtained from the analysis of these systems can be applied approximately to known chemical systems (much in the same way ideal gas laws are applied to the behavior of real gases). (4) Classical mechanics was unable to rationalize these observations of atomic and molecular systems. It is this last point that makes quantum mechanics worth understanding in order to understand chemistry. 

Consider the atomic scale, however. The uncertainty principle suggests that specifying a certain position of an electron is incompatible with other observables that we use to describe the state of the electron, like momentum and energy. Maybe we can t fix the electron to a certain radius. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

Quantum Mechanics Model Systems and the Hydrogen Atom 

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