Quantum Numbers Selection Rules Population Numbers

Quantum Numbers. The Bohr model needs only one quantum number, n, to describe all the energy levels of the electron. This can explain only the coarse features of an atom with a single electron.

Quantum mechanics describes the electron as a three dimensional wave, which only gives the probability of finding the electron in a certain place. Quantum mechanics has accurately predicted all the energy levels of hydrogen atoms. The energy levels of heavier atoms and molecules can also be computed; however, such calculations are very complicated. Also the existence of quantum numbers can be understood from the quantum mechanical point of view.

The quantum mechanical description involves four quantum numbers, one of which is our n, the principal quantum number. The principal quantum number describes the quantized energy levels of the electron. The classical interpretation of discrete energy levels allows only certain orbits given by (5.6). The orbital angular momentum of the electron is also quantized. This is described by the angular momentum quantum number l. The angular momentum corresponding to a quantum number l is l = y l(l+1) n.

The classical analogy would be to allow some elliptic orbits. The quantum number l can take only the values l = 0,1,... ,n - 1.

For historical reasons, these are often denoted by the letters s, p, d, f, g, h, i, j.

Although l determines the magnitude of the angular momentum, it does not give its direction. In a magnetic field this direction is important, since the orbiting electron also generates a tiny magnetic field. In any experiment, only one component of the angular momentum can be measured at a time. In a given direction z (e.g. in the direction of the applied magnetic field), the projection of the angular momentum can have only the values

Lz = ml n, where ml is the magnetic quantum number mi = 0, ±1, ±2,... ,±l.

The magnetic quantum number is responsible for the splitting of spectral lines in strong magnetic fields, known as the Zeeman effect. For example, if l = 1, ml can have 2l +1 = 3 different values. Thus, the line arising from the transition l = 1 ^ l = 0 will split into three components in a magnetic field (Fig. 5.7).

The fourth quantum number is the spin describing the intrinsic angular momentum of the electron. The spin of the electron is s = y s(s +1) n,

5.4 Quantum Numbers, Selection Rules, Population Numbers

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AE AE AE AE

AE AE

Fig. 5.7. The Zeeman effect. In strong magnetic fields each energy level of a hydrogen atom splits into (2l +1) separate levels, which correspond to different values of the magnetic quantum number ml = l, l — 1, ..., -l. The energy differences of the successive levels have the same constant value AE. For example the p state (l = 1) splits into three and the d state (l = 2) into five sublevels. The selection rules require that in electric dipole transitions Aml equals 0 or ±1, and only nine different transitions between p and d states are possible. Moreover, the transitions with the same Aml have the same energy difference. Thus the spectrum has only three separate lines where the spin quantum number is s = 1. In a given direction z, the spin is

Sz = ms h, where ms can have one of the two values: 1

All particles have a spin quantum number. Particles with an integral spin are called bosons (photon, mesons); particles with a half-integral spin are fermions (proton, neutron, electron, neutrino etc.).

Classically, spin can be interpreted as the rotation of a particle; this analogy, however, should not be taken too literally.

The total angular momentum J of an electron is the sum of its orbital and spin angular momentum:

Depending on the mutual orientation of the vectors L and S the quantum number j of total angular momentum can have one of two possible values,

(except if l = 0, when j = 1). The z component of the total angular momentum can have the values mj = 0, ±1, ±2,... ± j .

Spin also gives rise to the fine structure of spectral lines. Lines appear as close pairs or doublets.

Selection Rules. The state of an electron cannot change arbitrarily; transitions are restricted by selection rules, which follow from certain conservation laws. The selection rules express how the quantum numbers must change in a transition. Most probable are the electric dipole transitions, which make the atom behave like an oscillating dipole. The conservation laws require that in a transition we have

In terms of the total angular momentum the selection rules are

The probabilities of all other transitions are much smaller, and they are calledforbidden transitions; examples are magnetic dipole transitions and all quadrupole and higher multipole transitions.

Spectral lines originating in forbidden transitions are called forbidden lines. The probability of such a transition is so low that under normal circumstances, the transition cannot take place before collisions force the electron to change state. Forbidden lines are possible only if the gas is extremely rarified (like in auroras and planetary nebulae).

Spins parallel

Proton

Electron

Spins antiparallel

Fig. 5.8. The origin of the hydrogen 21 cm line. The spins of the electron and the proton may be either parallel or opposite. The energy of the former state is slightly larger. The wavelength of a photon corresponding to a transition between these states is 21 cm

The spins of an electron and nucleus of a hydrogen atom can be either parallel or antiparallel (Fig. 5.8). The energy of the former state is 0.0000059 eV higher. But the selection rules make an electric dipole transition between these states impossible. The transition, which is a magnetic dipole transition, has a very low probability, A = 2.8 x 10—15 s-1. This means that the average lifetime of the higher state is T = 1/A = 11 x 106 years. Usually collisions change the state of the electron well before this period of time has elapsed. But in interstellar space the density of hydrogen is so low and the total amount of hydrogen so great that a considerable number of these transitions can take place.

The wavelength of the radiation emitted by this transition is 21 cm, which is in the radio band of the spectrum. Extinction at radio wavelengths is very small, and we can observe more distant objects than by using optical wavelengths. The 21 cm radiation has been of crucial importance for surveys of interstellar hydrogen.

Population Numbers. The population number ni of an energy state i means the number of atoms in that state per unit volume. In thermal equilibrium, the population numbers obey the Boltzmann distribution:

where T is the temperature, k is the Boltzmann constant, AE = Ei — E0 = hv is the energy difference between the excited and ground state, and gi is the statistical weight of the level i (it is the number of different states with the same energy Ei). The subscript 0 always refers to the ground state. Often the population numbers differ from the values given by (5.11), but still we can define an excitation temperature Texc in such a way that (5.11) gives correct population numbers, when T is replaced by Texc. The excitation temperature may be different for different energy levels.

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