Remember the commutative property that we all
learned in Algebra I? It's
ab = ba
where (in Algebra I at least) a and b are real numbers. That property bugged me when I started
taking algebra. Maybe the associative and distributive properties (of
real numbers) also bothered me, but according to my memory of it, I was
especially perplexed and put out with the commutative idea because I'd already
learned that numbers could be multiplied in forward or backward order. Why did
the textbook make such a big deal out of it? The answer has to do with
the emphasis on set theory in the New Math of the sixties and early seventies.
What I didn't know is there are things
besides plain old real numbers that are noncommutative under multiplication. I wish someone had told me that way back then.
(Yeh, don't we all wish someone had told us something way back when?)
For instance, two ARRAYS of real numbers multiplied together don't in general
have the commutative or reversal-of-order property. An array of numbers is written as a
rectangular or square table called a matrix. Actually, an array of any sort of
mathematical animal such as variables, operators, or functions is a
matrix.
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Aside:
In some dialects, matrix is pronounced “mah-trix,” as opposed to the
English-speaking version, “may-trix.” When I moved into a University-owned
apartment with a fellow graduate physics student in Columbia, South Carolina, at the turn of the 21st century, he
asked me if I wanted to watch a movie he’d rented. He was from Kazakhstan, and spoke decent English, but I could not understand what movie he was
talking about. The nearest I could guess,
he was saying “The Mattress.” He must have finally gotten the box with the DVD (or VHS tape)
in it and showed me: It was The Matrix.
------------------
A matrix is usually designated with a capital letter rather than a lower case letter. So let's let A and B be 2 x 2 matrices of consecutive real numbers, like so:
A matrix is usually designated with a capital letter rather than a lower case letter. So let's let A and B be 2 x 2 matrices of consecutive real numbers, like so:
1 2 5 6
A = and B =
A = and B =
3
4
7 8 .
Multiplying
A times B gives (row elements times column elements, then add the result,
and repeat till finished)
1 2 5
6 1*5+2*7 1*6+2*8
x =
3 4
7 8 3*5+4*7 3*6+4*8
19 22
=
43 50 .
19 22
=
43 50 .
(The
multiplications shown here are done first, then the addition.) And here’s the “backwards” multiplication of
these two matrices:
5 6
1 2 5*1+6*3 5*2+6*4
x =
7 8
3 4 7*1+8*3 7*2+8*4
23 34
=
31 46 .
23 34
=
31 46 .
So,
we have AB ≠ BA. A and B don’t commute, or,
writing it differently,
AB – BA ≠ 0.
We
can give a name to the difference in AB
and BA, and also get rid of the pesky
“not-equal” sign. We could just use a name or symbol such as C, but the
difference in two matrices multiplied forwards and backwards has important
properties in quantum mechanics, so we’ll use a different symbology, called “the
commutator"
AB – BA = [A, B].
In
our consecutive numbers case [A, B]
is equal to
19 22
23 34 -4
-12
- =
- =
43 50
31 46 12 4
.
This
little example turned out to be more interesting than I thought it would be: I discovered that the commutator of any
consecutive numbers put into 2 X 2 matrices is equal to this -4, -12; 4, 12
matrix. For instance, let
57 58 61 62
A = and B
=
59
60
63 64 .
Then
-4 -12
[A, B] =
12 4
.
This
happens because the consecutive numbers are of the form n+1, n+2, …, n+8 where
n can be any integer. The n’s can be put in their own matrix which is merely
added to the basic 1, 2, 3, 4, etc, matrices, like so
n+1 n+2 n n
1 2
= +
n+3 n+4 n n 3 4
.
The
“n” matrices cancel each other when the commutator is calculated, and so do
cross-terms that have a factor of n in them, leaving only the commutator of the two matrices with
1, 2, …, 8 in them. Very cool!
I’m
writing about matrices because matrix multiplication of operators and wave
functions (state vectors) is the bread and butter of quantum mechanics. For instance, certain quantum mechanical matrices
that commute, meaning AB = BA, or that “anti-commute,” meaning AB = -BA, are
important for predicting the results of physical measurements. Matrices that don’t commute are responsible
for the theory behind the Heisenberg Uncertainty Principle--well, that's also important for physical measurements! I could just say "measurements" instead of "physical measurements." Or in general: experiments (as opposed to just thought experiments).
Matrices used in quantum mechanics can be transposed
(rows and columns interchanged) and have their complex conjugate taken (replace i by -i in a complex number). Sometimes they can be inverted (have their reciprocal taken). When a
matrix is equal to its transpose, it’s called symmetric. When it’s equal to the negative of its
transpose, it’s called anti-symmetric or skew-symmetric. The most important matrices for quantum
mechanics are the ones that equal their own complex-conjugate transpose (Hermitian
matrices), and ones that equal their own complex-conjugate inverse
(unitary matrices).
Question: is the above "consecutive number commutator" matrix,
-4 -12
12 4 ,
symmetric or maybe anti-symmetric? It has a common factor of 4 in all its elements that can be taken out, giving
-1 -3
4 times
3 1 ,
which really looks (even more) symmetric. But if you take its transpose (interchange the -3 with the 3) you don't get the same matrix. To get the same matrix, you'd have to have both 3's be positive or both be negative. It's also not anti-symmetric, which you can check for yourself (adding the transpose to the original would give the zero matrix if the original matrix were anti-symmetric).
Okay
that’s enough self-directed math review for today.
