Modern Physics Unit QM4 Angular Momentum and the Hydrogen Atom

Overview: This unit is devoted to the quantum mechanical description of the hydrogen atom. (In a sense this is the only consistent picture, since in classical physics the accelerating electron orbiting the proton would emit elecromagnetic radiation, lose energy, and move closer and closer to the nucleus.) Before treating this problem, however, we have to review the classical theory of angular momentum covered in unit CM2 and then introduce the quantum description of angular momentum which is essential to treating the 3-dimensional quantum mechanical problem of a particle in a central force.

Prerequisite: QM3, CM2

R. Eisberg and R. Resnick Quantum Mechanics of Atoms, Solids, Nuclei and Particles (2nd Ed.) , Chapt. 7 - One-Electron Atoms

COMMENT:

The simplest microscopic physical system of interacting particles is the hydrogen atom. It was in trying to understand the spectrum of hydrogen that Bohr came up with his model of quantized energy levels. We will now see how the Schrödinger equation also proves itself in this context. (Actually, the most convincing proof of the Schrödinger equation's superiority is in the calculation of transition strengths, which is discussed in Sec. 8.7, but we will not cover that.)

To treat real physical systems we will have to deal with three dimensional problems. When we have a central force problem, as for the hydrogen aton, there are very powerful techniques which exploit the spherical symmetry of the system.

In following the mathematical derivations here it is important to keep in mind what it is we are trying to do. We cannot find analytical (as opposed to numerical) solutions of a partial differential equation in three variables unless we can find a way to reduce it to ordinary differential equations.

After completing this unit you should understand:

1. The decomposition of the kinetic energy into a radial part and an angular momentum part.
2. How the equation is separated into a radial equation and an angular momentum part.
3. The quantization of angular momentum. The meanings of l and m_l , and the possible values for these quantities.
• Degeneracy: how different wavefunctions can have the same energy eigenvalues.
• The qualitative descriptions of the different states.
• Problems:

Chapt. 7: Questions 2,7,9,10,18 and Problems 1,3,4,5,6.

What are the lowest four energy levels of hydrogen? Describe the quantum numbers of all the states at these levels. How many states in total are there for each of these levels?