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Schrödinger Theory

The Schrödinger theory of quantum mechanics extends the de Broglie concept of matter waves by providing a formal method of treating the dynamics of physical particles in terms of associated waves. One expects the behavior of this wavefunction, generally called tex2html_wrap_inline1306 , to be governed by a wave equation, which can be written

equation144

where the first term of the left represents the particle's kinetic energy, the second the particle's potential energy, and H is called the Hamiltonian of the system. Making the assertion that p and H are associated with differential operators,

eqnarray151

this becomes

equation156

which is known as the time-dependent Schrödinger equation. For the specific case of a hydrogenic atom, the electron moves in a simple Coulomb potential, and hence the Schrödinger equation is

equation167

The solution proceeds by the method of separation of variables. First one writes the wavefunction as a product of a space component and a time component, for which the solution for the time part is easy and yields

equation177

Here E is the constant of the separation and is equal to the energy of the electron. The remaining equation for the spatial component is

equation182

and is called the time-independent Schrödinger equation. Due to the spherical symmetry of the potential, this equation is best solved in spherical polar coordinates, and hence one separates the spatial wavefunction as

equation191

The equations are more difficult but possible to solve and yield

eqnarray193

where L is an associated Laguerre polynomial, and for convenience the product of the angular solutions are written together in terms of a single function, the spherical harmonic Y. With foresight the separation constants tex2html_wrap_inline1320 and and l(l+1) were used. The meaning of the numbers n, l, and tex2html_wrap_inline1328 will now be discussed.

The physics of the Schrödinger theory relies on the interpretation of the wave function in terms of probabilities. Specifically, the absolute square of the wavefunction, tex2html_wrap_inline1330 , is interpreted as the probability density for finding the associated particle in the vicinity of tex2html_wrap_inline1332 at time t. For this to make physical sense, the wavefunction needs to be a well-behaved function of tex2html_wrap_inline1332 and t; that is, tex2html_wrap_inline1306 should be a finite, single-valued, and continuous function. In order to satisfy these conditions, the separation constants that appear while solving the Schrödinger equation can only take on certain discrete values. The upshot is, with the solution written as it is here, that the numbers n, l, and tex2html_wrap_inline1328 , called quantum numbers of the electron, can only take on particular integer values, and each of these corresponds to the quantization of some physical quantity. The allowed values of the energy turn out to be exactly as predicted by the Bohr theory,

equation209

The quantum number n is therefore called the principle quantum number. To understand the significance of l and tex2html_wrap_inline1328 , one needs to consider the orbital angular momentum of the electron. This is defined as tex2html_wrap_inline1354 , or as an operator, tex2html_wrap_inline1356 . With proper coordinate transformations, one can write the operators tex2html_wrap_inline1358 and the z-component of angular momentum tex2html_wrap_inline1362 in spherical coordinates as

eqnarray222

It can be shown that when these operators act on the solution tex2html_wrap_inline1306 , the result is

eqnarray236

It can also be shown that this means that an electron in a particular state has orbital angular momentum of constant magnitude tex2html_wrap_inline1366 and constant projection onto the z-axis of tex2html_wrap_inline1370 . Since the electron obeys the time-independent Schrödinger equation tex2html_wrap_inline1372 , and hence has constant energy, one says that the wavefunction tex2html_wrap_inline1306 is a simultaneous eigenstate of the operators H, tex2html_wrap_inline1358 , and tex2html_wrap_inline1362 . Table 1 summarizes this information and gives the allowed values for each quantum number. It is worth repeating that these numbers can have only these specific values because of the demand that tex2html_wrap_inline1306 be a well-behaved function.

   table242
Table 1: Some quantum numbers for the electron in the hydrogen atom.

It is common to identify a state by its principle quantum number n and a letter which corresponds to its orbital angular momentum quantum number l, as shown in table 2. This is called spectroscopic notation. The first four designated letters are of historical origin. They stand for sharp, primary, diffuse, and fundamental, and refer to the nature of the spectroscopic lines when these states were first studied.

   table252
Table 2: Spectroscopic notation.

Figure 1 shows radial probability distributions for some different states, labelled by spectroscopic notation. The radial probability density tex2html_wrap_inline1414 is defined such that

equation262

is the probability of finding the electron with radial coordinate between r and r+dr. The functions are normalized so that the total probability of finding the electron at some location is unity. It is interesting to note that each state has n-l-1 nodes, or points where the probability goes to zero. This is sometimes called the radial node quantum number and appears in other aspects of quantum theory. It is also interesting that for each n, the state with l=n-1 has maximum probability of being found at tex2html_wrap_inline1426 , the radius of the orbit predicted by Bohr theory. This indicates that the Bohr model, though known to be incorrect, is at least similar to physical reality in some respects, and it is often helpful to use the Bohr model when trying to visualize certain effects, for example the spin-orbit effect, to be discussed in the next section. The angular probability distributions will not be explored heregif, except to say that they have the property that if the solutions with all possible values of l and tex2html_wrap_inline1328 for a particular n are summed together, the result is a distribution with spherical symmetry, a feature which helps to greatly simplify applications to multi-electron atoms.

   figure268
Figure 1: Radial probability distribution for an electron in some low-energy levels of hydrogen. The abscissa is the radius in units of tex2html_wrap_inline1188 .




next up previous
Next: The Spin-Orbit Effect Up: Everything You Always Wanted Previous: The Bohr Model

Randy Telfer
Mon Oct 26 15:21:07 EST 1998