Learning Objectives
By the end of this section, you will be able to do the following:
- Define quantum number
- Calculate the angle of an angular momentum vector with an axis
- Define spin quantum number
Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.
The energy states of bound systems are quantized, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as where We define to be the principal quantum number that labels the basic states of a system. The lowest-energy state has the first excited state has and so on. Thus, the allowed values for the principal quantum number are
This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of as can other characteristics, such as the orbital radii of the hydrogen atom.
The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum can have only the values
where is defined to be the angular momentum quantum number. The rule for in atoms is given in the parentheses. Given the value of can be any integer from zero up to For example, if then can be 0, 1, 2, or 3.
Note that for can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in Figure 13.48. The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has so that can be either 0 or 1, according to the rule in Similarly, for can be 0, 1, or 2. It is often most convenient to state the value of a simple integer, rather than calculating the value of from For example, for we see that
It is much simpler to state
As recognized in the Zeeman effect, the direction of angular momentum is quantized. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the axis, can have only certain values of The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of are
where is the component of the angular momentum and is the angular momentum projection quantum number. The rule in parentheses for the values of is that it can range from to in steps of one. For example, if then can have the five values –2, –1, 0, 1, and 2. Each corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in Figure 13.55.
Example 13.3 What Are the Allowed Directions?
Calculate the angles that the angular momentum vector can make with the axis for as illustrated in Figure 13.55.
Strategy
Figure 13.55 represents the vectors and as usual, with arrows proportional to their magnitudes and pointing in the correct directions. and form a right triangle, with being the hypotenuse and the adjacent side. This means that the ratio of to is the cosine of the angle of interest. We can find and using and
Solution
We are given so that can be +1, 0, or −1. Thus, has the value given by
can have three values, given by
As can be seen in Figure 13.55, and so for we have
Thus,
Similarly, for we find ; thus,
And for
so that
Discussion
The angles are consistent with the figure. Only the angle relative to the axis is quantized. can point in any direction as long as it makes the proper angle with the axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle in the example) is for the maximum value of namely For that smallest angle
which approaches 1 as becomes very large. If then Furthermore, for large there are many values of so that all angles become possible as gets very large.