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an energy of +13.6 eV, or greater. The values of the energy differences between these states are unaffected by this shift in the zero point of the energy.

We can use equation (3.6) to derive the Balmer formula. First, we rewrite the equation as

where

The energy of an emitted or absorbed photon must equal the energy difference between the two states:

Taking the energies from equation (3.7b) gives

which looks very similar to the Balmer formula, except that the Balmer formula has a 2 instead of the m. This means that the Balmer series all have the second energy level as their lower level.

We can use equation (3.8) to divide the hydrogen spectrum into different series. A given series is characterized by having the same lower energy state. For example, the Balmer series consists of absorptions accompanying transitions from level 2 to any higher levels, and emissions accompanying transitions from higher levels down to level 2. The first Balmer transition (involving levels 2 and 3) has the smallest energy difference of the series. (Clearly the energy difference between levels 2 and 3 is less than the energy difference between levels 2 and 4, or between levels 2 and 5, and so on.) The Balmer series is important because the first few transitions fall in the visible part of the spectrum. The series with the lower energy level being level 1 is called the Lyman series. Even the lowest transition in the Lyman series is in the ultraviolet.

We have developed a labeling system for various transitions. First we give the chemical symbol for the element (e.g. H for hydrogen). Then we give the m for the lowest level that characterizes the series (1 for Lyman, 2 for Balmer, etc.). Finally, we give a Greek letter denoting the number of levels jumped. For example, if n = m + 1, we have an alpha (a) transition; if n = m + 2, we have a beta 0) transition. The first Balmer line is then designated H2a. (Note that for the Balmer series of hydrogen only, we sometimes drop the 2 and just say Ha, Hp, etc.)

3.3.2 Quantum mechanics

The Bohr model of the atom allowed physicists to understand the organization of energy levels. However, it was far from a complete theory. One shortcoming was that it did not explain why some spectral lines are stronger than others. More fundamentally, it was an ad hoc theory. Bohr had no explanation of why stationary states exist, or why angular momentum must be quantized in some particular way. These were just postulates. A much deeper understanding was needed.

An important step was made by Louis de Broglie, who proposed the revolutionary idea that if light could exhibit a wave-particle duality, then maybe all matter could. That is, an electron orbiting a nucleus has certain wavelike properties, and it is those properties that determine the states that are stable. One could think of the electron as having a certain wavelength. Stationary states could be those whose circumference contained an integral number of wavelengths, producing a pattern that reinforced during each orbit (like a standing wave). It was necessary to have expressions for the wavelength and frequency of a particle, and de Broglie noted that if the wavelength was taken as h/p (where p is the momentum of the particle) and the frequency as E/h, then the orbits allowed by the standing wave idea were the same as the orbits that Bohr found from his postulates (see Problem 3.8).

This is clearly a departure from our normal way of looking at matter around us, and we cannot go through all of the ramifications here. To this point, we have gone far enough to understand stellar spectra. The picture as presented by Bohr and de Broglie is quantum theory in its most naive form. It was realized that if particles behave, in some fashion, like waves then the description of particle motions (mechanics) must be changed from Newton's laws of motion to laws of motion involving waves. (Of course, in the limit of large objects, such as apples falling to Earth, these new laws of motion must reduce to

Newton's laws, because we know that Newton's laws work quite well for apples and planets.) Theories that describe the mechanics of waves are called wave mechanics or quantum mechanics. One such theory was presented in 1925 by the German physicist Erwin Schrodinger. In his theory the information about the motion of a particle is contained in a function, called a wave function. Schrodinger's interpretation of the wave function was that it is related to the probability of finding a particle in a particular place with a particular momentum. This replaced the absolute determinism of classical physics, with the statement that we can only predict where a particle is likely to be, but not exactly where it will be. However, we can predict the average positions and momenta of a large group of particles, and it is these average properties that we see (and measure) in our everyday world. Many physicists (including Einstein) were not comfortable with this probabilistic interpretation, but quantum theory has been very successful in predicting the outcome of a wide variety of experiments. We will pick up on some of the threads of the quantum revolution later in this book.

3.4 I Formation of spectral lines

Now that we have some idea of how atoms can emit or absorb radiation, we can return to stellar spectra. The first point to realize is that in a star we are not talking about the radiation from a single hydrogen atom, but from a large number of them. We see a strong Ha absorption line in stars because many photons are removed from the continuum by this process. It is clear, however, that having a lot of hydrogen does not assure us of a strong Ha absorption. In order for such absorption to take place, a significant number of atoms must be in level 2, ready to absorb a photon. If all the hydrogen is in level 1, you will not see the Balmer series, no matter how much hydrogen is present.

3.4.1 Excitation

In general, the strength of a particular transition (emission or absorption) will depend on the number of atoms in the initial state for that transi tion. The number of atoms per unit volume in a given state is called the population of that state. In this section we look at the factors that determine the populations of the various states. We refer to processes that can alter the populations as excitation processes. We have already seen one type of excitation process - the emission and absorption of photons. Electrons can jump to a higher level when a photon is absorbed or they can jump to a lower level when a photon is emitted.

Populations can also be changed by collisions with other atoms, as illustrated in Fig. 3.6. For example, atom 1 can be in state i. It then undergoes a collision with atom 2, and makes a transition to a higher state, j. In the process the kinetic energy of atom 2 is decreased by the difference between the energies of the two states in atom 1, Ej — Ei. The reverse process is also possible, with atom 2 gaining kinetic energy and atom 1 dropping from state j to state i.

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