Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona

Angular momentum problems in Quantum Physics

Angular momentum forms an important part of any course on Quantum Physics. It establishes the angular nature of wavefunctions for 3-dimensional (3D) problems, with applications to real systems such as atoms (s-, p-, d-, f-states, etc), molecules, nuclei and solids. In addition to the intrinsic interest of the topic, in this course we are also using the 1D simple harmonic oscillator (SHO), closely followed by angular momentum in 3D, to introduce operator methods and algebra, and to make the transition to matrix mechanics. If you get confused about these different methods, pose questions in class: in general, we have a several to one relationship between the maths and the physics. Because of time pressures, we are illustrating the different mathematical methods on mostly different physical problems. This means that each method is in the form of an illustration, rather than an exhaustive treatment.

The course book, Gasiorowicz (third edition, G3), spends chapter 6 on operator methods for the SHO, chapter 7 on Angular Momentum per se, with web appendices on rotational invariance, followed by chapter 8 on the central force problem. This is a substantial change from the second edition (G2) and much closer to our course order. References to G2 are given here.

I am putting up some thoughts which may become pages for clarification, or which might lead to projects. Please suggest additions which may be of use to current and future students. Those of you who need page numbers in Liboff (L3 or L4), please see me in person.

Operator Methods (G3 p107-119)

  • Chapter 6 starts with operator methods for the SHO, and then goes on to time dependence of operations and questions of representations. Master the SHO material early on, in time for the mid-term exam and part of Problem set#3. We will come back to the other stuff later.
Angular Momentum (G3 p121-130 and web supplements 7A and 7B)
  • Chapter 7 starts with the eigenvalue equations for both Lz and L2 in terms of the spherical harmonics Ylm which are a function of the two angular variables, q (theta) and f (phi). You should have handouts showing the angular character for the few lowest l and m values, and be fully familiar with the conditions |m| ≤ l, etc, for l = 0,1,2,.. An arbitrary angular wavefunction can be expanded in terms of these eigenfunctions (see G3 p128-9).

  • To find the eigenfunction and show that the eigenvalue of Lz = mh is straightforward. To show that the eigenvalue of L2 is l(l+1) h2 requires the raising and lowering operators (L+ and L_ ), and some algebra (see G3 p122-3), which we could usefully make into a web page. Note how similar much of this material is to the corresponding treatment of the SHO.
This page uses the Symbol and MT Extra character set. Latest version: 21st February 2007.