sec7 - VII. Central Potentials Before going any further...

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Unformatted text preview: VII. Central Potentials Before going any further with angular momentum, it is best to begin using the relations we already have so that we can get some idea what they are good for. Perhaps the best application of the angular momentum eigenfunctions we dealt with in the previous section comes when one deals with a spherically symmetric (or central) potential. In this case, the potential energy is only a function of the distance r to the origin, and the knowledge we already have will tell us a great deal about the eigenstates of the system irrespective of the particular potential . a. Spherical Polar Coordinates Since we are dealing with a potential that is a function only of the distance r to the origin, it is by far preferable to work in a set of coordinates where r is one of the basic variables, rather than some function of z y x , , .To this end, we need to convert our equations to spherical polar coordinates - , , , , r z y x . In most math textbooks, is defined to be the angle relative to the z axis while the vast majority of quantum mechanics texts use in this capacity. We will use the latter definition, but be careful that any equations taken from other sources use this same convention! Here are some useful relations in spherical polar coordinates: cos sin sin sin cos r z r y r x x r z y j i e k j i e k j i e cos sin sin cos sin cos cos cos sin sin sin cos +- - + + + r + + = + + sin sin 1 sin 1 1 1 sin 1 2 2 2 2 2 2 2 2 r r r r r r r r r r e e e b. Central Potentials For an arbitrary potential ) ( r V , we can write ( 29 r V m H 2 2 2 + - = & At this point, we convert the equations to natural units by choosing our unit of length and unit of mass so that 1 = = m & . Note that this leaves us one free standard unit (time, or, equivalently, energy). It is convenient to fix this dimension based on the problem at hand; for example, in a harmonic oscillator, it is useful to choose the energy so that 1 = & , while for the Coulomb interaction it is useful to choose the unit of electron charge to be that of one electron. These units are merely out of convenience and in the end, once we have calculated an observable (such as the position) we will need to convert the result to a set of standard units (such as meters). The main benefit at the moment is that it removes the relatively unimportant factors of & and m from our equation, so that in natural units: ( 29 ( 29 r V r r r r r V H ) sin sin 1 sin 1 ( 1 2 2 2 2 2 2 2 + + + - = + - = where the second equality just reinforces the gory details wrapped up in the Laplacian operator. c. Orbital Angular Momentum Operators In order to see what angular momentum has to do with this, we need to express the angular momentum operators in spherical polar...
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sec7 - VII. Central Potentials Before going any further...

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