Notes for Desai Chapter 26
Posted by peeterjoot on December 9, 2010
Chapter 26 notes for .
To verify equations 26.3-5 in the text it’s worth noting that
the transformed coordinates are
This allows us to read off the rotation matrix. Without all the messy trig, we can also derive this matrix with geometric algebra.
Here we use the Pauli-matrix like identities
and also note that commutes with the bivector for the plane . We can also read off the rotation matrix from this.
Recall that in the problems of Chapter 5, one representation of spin one matrices were calculated . Since the choice of the basis vectors was arbitrary in that exersize, we ended up with a different representation. For as found in (26.20) and (26.23) we can also verify easily that we have eigenvalues . We can also show that our spin kets in this non-diagonal representation have the following column matrix representations:
Verifying the commutator relations.
Given the (summation convention) matrix representation for the spin one operators
let’s demonstrate the commutator relation of (26.25).
Now we can employ the summation rule for sums products of antisymmetic tensors over one free index (4.179)
Continuing we get
General infinitesimal rotation.
Equation (26.26) has for an infinitesimal rotation counterclockwise around the unit axis of rotation vector
Let’s derive this using the geometric algebra rotation expression for the same
We note that and thus the exponential commutes with , and the projection component in the normal direction. Similarily anticommutes with . This leaves us with
For , this is
Position and angular momentum commutator.
Equation (26.71) is
Let’s derive this. Recall that we have for the position-momentum commutator
and for each of the angular momentum operator components we have
The commutator of interest is thus
A note on the angular momentum operator exponential sandwiches.
In (26.73-74) we have
so from the first two terms of (10.99)
we get the desired result.
Trace relation to the determinant.
Going from (26.90) to (26.91) we appear to have a mystery identity
According to wikipedia, under derivative of a determinant, , this is good for small , and related to something called the Jacobi identity. Someday I should really get around to studying determinants in depth, and will take this one for granted for now.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.
 Peeter Joot. Notes and problems for Desai Chapter V. [online]. http://sites.google.com/site/peeterjoot/math2010/desaiCh5.pdf.
 Wikipedia. Determinant — wikipedia, the free encyclopedia [online]. 2010. [Online; accessed 10-December-2010]. http://en.wikipedia.org/w/index.php?title=Determinant&oldid=400983667.