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College Physics for AP® Courses 2e

30.8 Quantum Numbers and Rules

College Physics for AP® Courses 2e30.8 Quantum Numbers and Rules

Learning Objectives

By the end of this section, you will be able to:

  • Define quantum number.
  • Calculate angle of 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 En1/n2En1/n2, where n=1, 2, 3, ...n=1, 2, 3, .... We define nn to be the principal quantum number that labels the basic states of a system. The lowest-energy state has n=1n=1, the first excited state has n=2n=2, and so on. Thus the allowed values for the principal quantum number are

n=1, 2, 3, ....n=1, 2, 3, ....

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 nn, 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 LL can have only the values

L=ll+1hl=0, 1, 2, ...,n1,L=ll+1hl=0, 1, 2, ...,n1,

where ll is defined to be the angular momentum quantum number. The rule for ll in atoms is given in the parentheses. Given nn, the value of ll can be any integer from zero up to n1n1. For example, if n=4n=4, then ll can be 0, 1, 2, or 3.

Note that for n=1n=1, ll can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not h/2π h/2π 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 30.46. 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 n=2n=2, so that ll can be either 0 or 1, according to the rule in L=ll+1hL=ll+1h . Similarly, for n=3n=3, ll can be 0, 1, or 2. It is often most convenient to state the value of ll, a simple integer, rather than calculating the value of LL from L=ll+1hL=ll+1h. For example, for l=2l=2, we see that

L=22+1h=6h=0.390h=2.58×1034 Js.L=22+1h=6h=0.390h=2.58×1034 Js.

It is much simpler to state l=2l=2.

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 zz-axis, can have only certain values of LzLz. 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 LzLz are

Lz=mlhml=l,l+1, ...,1, 0, 1, ...l1,l,Lz=mlhml=l,l+1, ...,1, 0, 1, ...l1,l,

where LzLz is the zz-component of the angular momentum and mlml is the angular momentum projection quantum number. The rule in parentheses for the values of mlml is that it can range from ll to ll in steps of one. For example, if l=2l=2, then mlml can have the five values –2, –1, 0, 1, and 2. Each mlml 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 zz-component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in Figure 30.52.

The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.
Figure 30.52 The component of a given angular momentum along the zz-axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for l=1l=1, for which ml=1, 0, and +1ml=1, 0, and +1. The direction of LL is quantized in the sense that it can have only certain angles relative to the zz-axis.

Example 30.3

What Are the Allowed Directions?

Calculate the angles that the angular momentum vector LL can make with the zz-axis for l=1l=1, as illustrated in Figure 30.52.


Figure 30.52 represents the vectors LL and LzLz as usual, with arrows proportional to their magnitudes and pointing in the correct directions. LL and LzLz form a right triangle, with LL being the hypotenuse and LzLz the adjacent side. This means that the ratio of LzLz to LL is the cosine of the angle of interest. We can find LL and LzLz using L=ll+1hL=ll+1h and Lz=mhLz=mh.


We are given l=1l=1, so that mlml can be +1, 0, or −1. Thus LL has the value given by L=ll+1hL=ll+1h.


LzLz can have three values, given by Lz=mlhLz=mlh.

Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1 Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1

As can be seen in Figure 30.52, cos θ= L z /L, cos θ= L z /L, and so for ml=+1ml=+1, we have




Similarly, for ml=0ml=0, we find cosθ2=0cosθ2=0; thus,


And for ml=1ml=1,


so that



The angles are consistent with the figure. Only the angle relative to the zz-axis is quantized. LL can point in any direction as long as it makes the proper angle with the zz-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 ( θ 1 θ 1 in the example) is for the maximum value of ml=0ml=0, namely ml=lml=l. For that smallest angle,


which approaches 1 as ll becomes very large. If cosθ=1cosθ=1, then θ=θ=. Furthermore, for large ll, there are many values of mlml, so that all angles become possible as ll gets very large.

Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction

There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum, ss, of an electron is given by

S=ss+1h(s=1/2 for electrons),S=ss+1h(s=1/2 for electrons),

where ss is defined to be the spin quantum number. This is very similar to the quantization of LL given in L=ll+1hL=ll+1h, except that the only value allowed for ss for electrons is 1/2.

The direction of intrinsic spin is quantized, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the zz-axis, can have only the values

S z = m s h m s = 1 2 , + 1 2 S z = m s h m s = 1 2 , + 1 2

for electrons. SzSz is the zz-component of spin angular momentum and msms is the spin projection quantum number. For electrons, ss can only be 1/2, and msms can be either +1/2 or –1/2. Spin projection ms=+1/2ms=+1/2 is referred to as spin up, whereas ms=1/2ms=1/2 is called spin down. These are illustrated in Figure 30.51.

Intrinsic Spin

In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles ss is half-integral, whereas for others ss is integral—there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have s=1/2s=1/2, whereas photons have s=1s=1, and other particles called pions have s=0s=0, and so on.

To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form n, l,ml,msn, l,ml,ms —see Table 30.1 For electrons in atoms, the principal quantum number can have the values n=1, 2, 3, ...n=1, 2, 3, .... Once nn is known, the values of the angular momentum quantum number are limited to l=1, 2, 3, ...,n1l=1, 2, 3, ...,n1. For a given value of ll, the angular momentum projection quantum number can have only the values ml=l,l+1, ...,1, 0, 1, ...,l1,lml=l,l+1, ...,1, 0, 1, ...,l1,l. Electron spin is independent of n, l,n, l, and mlml, always having s=1/2s=1/2. The spin projection quantum number can have two values, ms=1/2 or 1/2ms=1/2 or 1/2.

Name Symbol Allowed values
Principal quantum number n n 1, 2, 3, ... 1, 2, 3, ...
Angular momentum l l 0, 1, 2, ... n 1 0, 1, 2, ... n 1
Angular momentum projection m l m l l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l ) l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l )
Spin1 s s 1/2 ( electrons ) 1/2 ( electrons )
Spin projection m s m s 1/2, + 1/2 1/2, + 1/2
Table 30.1 Atomic Quantum Numbers

Figure 30.53 shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.

The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.
Figure 30.53 Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as n,l,mln,l,ml. The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.

We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.


  • 1The spin quantum number s is usually not stated, since it is always 1/2 for electrons
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