Quaternion Numbers in Quantum Physics

Date: 04/11/2001 at 22:51:53
From: Reuben
Subject: Quaternion numbers in Physics

I am aware that the wave function is defined by a imaginary or complex 
number, but was wondering if there are any applications in quantum 
physics for quaternion numbers. I was also wondering how quaternion 
exponents and logarithms operate, e.g. i^k, j^i, etc. 

I hope and thank you if you can answer my two questions, being that I 
am unaware of anyone else who can answer these questions considering 
my extreme interest in mathematics and physics.

Date: 04/11/2001 at 23:52:01
From: Doctor Mitteldorf
Subject: Re: Quaternion numbers in Physics

Dear Reuben,

All the common elementary particles have a "spin" equal to half hbar.  
Spin 1/2 particles have just two states, usually called "spin up" and 
"spin down."  If you know whether the spin is "up" or "down," you know 
everything about the spin of the particles. But "up" and "down" can be 
measured along any axis, so there are three independent spin 
operators, usually written as Sx, Sy, and Sz.  

The Heisenberg Uncertainty Principle tells you that if you measure one 
of these, say Sx, you lose all information about the others (Sy and 
Sz). Of course, it's also possible to measure spin along some 
intermediate axis, at an angle in between x and z, say.

Measuring first Sx and then Sz in Quantum Physics is represented as 
the "product" Sz times Sx. I'm telling you all this because it turns 
out that the three spin operators have the same properties under 
multiplication that the quaternions have. For example, Sx * Sy = iSz, 
just as i*j=k. The spin operators, like the quaternions, anti-commute:  
Sx*Sy = - Sy*Sx.

The spin operators are often represented by two-by-two matrices, named 
for Wolfgang Pauli. These matrices can be multiplied using ordinary 
matrix algebra, and they have the same properties that the spin 
operators and the quaternions have. Conveniently, they also tell you 
the behavior of the wave function if you take a particle in one state 
(say Sz) and measure it with another operator (say Sx).

The Pauli matrices are

Sz = (1  0)
     (0 -1)

Sx = (0  1)
     (1  0)

Sy = (0 -i)
     (i  0)

1 = (1  0) 
    (0  1)

As for your second question: I haven't come across the combinations 
that you write, like i to the j power, but I think I can point you in 
a direction to get started understanding them. You may know that e^x 
can be represented by a Taylor series  1 + x + x^2/2 + x^3/6 + ...   
You can use this series to define e^i, where i is the square root of 
-1. Similarly, you can use it on matrices or quaternions. If you can 
multiply the object x, then you can find x^2, x^3, etc., and therefore 
you can define e^x. I leave it to you to calculate e to the power Sz.

Similarly, you can find the natural log ln(x) as a Taylor series, and 
use that series to define ln(i) or ln(Sz). I'll leave that to you as 
well.

Once you've done that, you can define i^j as e raised to the power 
(j*ln(i)). Try it! And please do write back with your results.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/