course index

Recall from last lecture:

• Under the rules of quantum physics, the energy of a photon is E = hf.
• Planck's constant, h = 6.63×10^-34Js
• Uncertainty principle: Dx Dp_x >= h/4p

Chapter 28: Atomic Physics

Throughout this chapter we will learn the Physics of the hydrogen atom. The hydrogen atom, with one proton and one electron, is the most simple atomic system possible, and many results are derivable for this system. Some of these results provide the most precise comparison between calculation and measurement ever achieved. Though derived for this special case, the results we obtain are more general and tell us about the structure of all atoms, and the organization of the periodic table of elements.

We begin with some history on the structure of atoms, then move to the approximate Bohr model and refinements. The structure of the periodic table is discussed, and some applications will be discussed at the end.

28.1 Early Models of the Atom

"Atom" is a Greek word meaning the smallest piece, a whit. The original idea of an atom is traced to the Greek philosopher Democritus who suggested that if you take something and cut or break it in half, again and again and again, eventually you will be left with a piece that you could no longer break in two. This remaining piece is an atom. In more modern usage, an atom is the smallest unit of an element that still retains the characteristics of that element.

We can't see atoms directly (or at least we couldn't until very recently), so we make a model that allows us to rationalize how atoms behave. The earliest model, due to Newton, was a hard sphere, kind of like billiard balls. Atoms will collide and elastically bounce off each other, conserving energy and momentum. This picture is used in formulating the kinetic theory of gases, which generally works quite well.

When people realized that atoms are connected with the electrical properties of materials, then charge had to be incorporated into the model of the atom. After J.J. Thompson demonstrated the existence of the electron, the "raisin pudding" model came into vogue. This model consists of a sphere of uniform positive charge, with negatively charged electrons stuck in it like raisins in pudding. The overall charge was zero, and electrons were incorporated, so everyone was happy.

In 1911, Rutherford and his students performed an experiment, scattering alpha particles (He nucleus) off a thin gold foil. They were very surprised when they observed some of the alpha particles scattering backward! An analogy for this experiment is to imagine hitting a golf ball through a tree. If the golf ball is up near the small branches and leaves, it will pass right through the tree, with only a small change from its original direction. If the golf ball hits the trunk of the tree, it will bounce off at a large angle, maybe even coming right back at the golfer. This is what Rutherford and his students observed. But the raisin pudding model of an atom doesn't have a "trunk", it doesn't have a dense core for something to bounce off of. So Rutherford proposed a model with a small, dense core of positive charge (the nucleus) surrounded by negatively charged electrons moving in orbits about the nucleus.

This model is closer to our present picture, but still has problems. First, it doesn't explain the spectral lines of atoms (and molecules). Second, electrons in circular motion are accelerated, and, according to the equations of electromagnetism, an accelerated charge will radiate electromagnetic energy, losing energy itself. The electrons orbiting the nucleus should eventually radiate away all their kinetic energy, and fall into the nucleus.

Both these problems are solved by the application of quantum theory and Schrödinger's wave equation.

28.2 Atomic Spectra

You have seen spectral lines in one of the laboratory experiments. Different types of atoms emit different spectral lines. In fact, the spectral lines can be used to fingerprint an atom. Even more common is to look at the absorption lines produced, for instance, when light from a blackbody (a star) passes through a gas in space. The atoms in the gas will selectively absorb certain frequencies, the same frequencies that are emitted when the atom is excited. In this way, astronomers can determine what kind of atoms are in space, in what abundances, and even tell how fast the gas is moving toward or away from us by the doppler shift of the lines.

The spectral lines of hydrogen are particularly interesting. Hydrogen emits in the visible at wavelengths of 656.3nm, 486.1nm, 434.1nm, and 410.2nm. In 1885, Johann Balmer realized that these wavelengths are described by the formula:

Inserting n=3 into the Balmer formula gives the wavelength 656.3nm. Inserting n=4 gives 486.1nm, and so on.

Another set of spectral lines called the Lyman series is given by a similar formula:

The element helium was discovered because astronomers observed absorption lines from the sun that didn't correspond to any known element. The name helium is derived from helios, greek for sun.

28.3 The Bohr Theory of Hydrogen

Niels Bohr was a famous Danish physicist -- "Copenhagen", a recent play, is concerned with a conversation between Bohr and Heisenberg (of the Heisenberg uncertainty principle) after World War II regarding the German program to build a nuclear bomb. Bohr's model of the hydrogen atom is able to explain the spectral lines, and many other features. The model is based on assumptions that mix together ideas from classical and quantum physics. The model is superseded by Schrödinger's wave equation, but Bohr's model leaves us with a nice picture of an atom that is not directly evident from the involved mathematics of the wave equation.

I quote the assumptions from the text:

1. The electron moves in circular orbits about the proton under the influence of the Coulomb force of attraction, as in Figure 28.5. In this case, the Coulomb force is the force that produces centripetal acceleration.
2. Only certain electron orbits are stable. These are orbits in which the hydrogen atom does not emit energy in the form of radiation. Hence, the total energy of the atom remains constant, and classical mechanics can be used to describe the electron's motion.
3. Radiation is emitted by the hydrogen atom when the electron "jumps" from a more energetic initial state to a lower state. The "jump" cannot be visualized or treated classically. In particular, the frequency, f, of the radiation emitted in the jump is related to the change in the atom's energy and is independent of the frequency of the electron's orbital motion. The frequency of the emitted radiation is
   E_i - E_f = hf
   where E_i is the energy of the initial state, E_f is the energy of the final state, h is Planck's constant, and E_i > E_f.
4. The size of the allowed electron orbits is determined by a condition imposed on the electron's orbital angular momentum: The allowed orbits are thos for which the electron's orbital angular momentum about the nucleus is an integral multiple of hbar = h/2p.
   m_e v r = n hbar     n = 1, 2, 3, ...

We can now use these assumptions to calculate the energy levels of the allowed orbits of the hydrogen atom.