Power of Two-by-two Matrices
Harmonic oscillators and two-by-two matrices are the two basic building
blocks of modern physics. It is not uncommon to reduce complicated
problems into a coupled-oscillator problem. Then what do we do?
Decouple the oscillators. If you think you can diagonalize the system
by making a simple rotation, you are wrong.
Let us go back to the first edition of Goldstein's book on classical mechanics. It is a beautiful book. Goldstein diagonalizes two coupled oscillators by rotating the coordinate system. In addition, however, he makes a scale adjustments when the masses are not equal. Goldstein used the word "congruent transformation" in his book. Do you know what congruent transformation means?
This problem was later carefully examined by Aravind in his article in the American Journal of Physics [see P. K. Aravind, Am. J. Phys. 57, 309 (1989)]. See also Han, Kim, and Noz, Am. J. Phys. 61, 61 (1999). We can explain what Aravind did in the following way.
Let us start with the Hamiltonian for two coupled oscillators. The kinetic energy can be described by an ellipse on the left-hand side of the following figure. The coupled potential energy can be illustrated as a tilted ellipse as shown on the right side.
The diagonalization means aligning the major and minor axes to the coordinate axes. The kinetic-energy part of the Hamiltonian is initially diagonal We can diagonalize the potential energy by rotating the corresponding ellipse, but this rotation will made the kinetic energy tilted? Thus, it is not possible to diagonalize the two ellipses simultaneously.
|In order to solve this problem, let us consider the squeeze transformation like this figure. One axis is expanded while the other is contracted. In this way, we can transform the diagonal kinetic-energy ellipse into a circle. This transformation can be achieved by a two-by-two matrix. We can safely use the word "squeeze" for this scale transformation. Under this squeeze transformation, the potential-energy becomes another ellipse.|
Since the kinetic-energy circle is invariant under rotations, we can rotate the potential-energy ellipse to a diagonal form as shown below.
The combination of rotation and squeeze is called a "symplectic transformation. The word "symplectic" is relatively new in physics. Eugene Wigner once asked me what the symplectic group is, while he had been working on this group since 1937. The Wigner rotation is a product of the symplectic group. The word "symplectic" was invented by Hermann Weyl. Wigner perhaps did not read Weyl's book on group theory and quantum mechanics, because he did not like Weyl.
Paul A. M. Dirac did not know he was doing the symplectic group while
he was writing his paper on the O(3,2) deSitter goup for coupled oscillators
[J. Math. Phys. 4, 901 (1963)]. Earlier in 1949, Dirac published a paper
in the Rev. Mod. Phys. 21, 392 (1949), where he introduced his light-cone
coorinate system. He translates the Lorentz boost into a transformation of
a square into a rectangle. This is a
physics of two-by-two matrices.
If you did not know what the symplectic group is, it is quite OK. Remember
this. You cannot do physics without two-by-two matrices. They do rotations,
but they can also squeeze the world. In physics, we need squeeze transformations.
Finally, what does Einstein say about the squeeze transformation?
Click here if you are interested.
If you did not know what the symplectic group is, it is quite OK. Remember this. You cannot do physics without two-by-two matrices. They do rotations, but they can also squeeze the world. In physics, we need squeeze transformations.
Finally, what does Einstein say about the squeeze transformation? Click here if you are interested.
Y. S. Kim (2007.3.10)
PS. For two-by-two matrices in optics, you may click here.