Quantum
Mechanics of the Crap Shoot
The purpose of this is to show some of the
fundamentals of QM for a beginner in quantum theory or for someone planning to
get into it soon. Like QM, the crap shoot (rolling dice) is a game of chance,
so it is a handy vehicle for explaining the basics. When I studied QM, it was
not always clear to me what was math and what was physics. This is all math in this note. Also I found that sometimes the symbolism
could be a roadblock to understanding some simple ideas. I hope this little
exercise helps you.
I will stick to real numbers here. In QM
there are a lot of complex numbers.
Your mother will think that getting involved
with a crap shoot is not very discreet, but tell her that this is a discrete
case. You can have continuous eigenvalues in physics,
but not in the crap game.
A reminder about vectors in space: We have
unit vectors (or basis vectors) i, j, and
k which could also be labeled (1,0,0), (0,1,0), and (0,0,1). Then any
vector is a linear combination of these. If A = 3i + 2j -k,
note that the dot product i×A = 3, the
component of A in the i
direction. A QM vector has these two properties also. To ease into the QM
symbols, that dot product is written <iï A> = 3.
The unit vectors i, j, k are called
eigenvectors in QM.
In QM we have an abstract vector
"space." The vector A above could be
regarded as simply a list of numbers, (3, 2, -1), and that is the way to think
of vectors in QM, nothing to do with directions or the space we live in. The
state vector in QM is a list of "amplitudes:" the square roots of the
probabilities of all the possible outcomes of a measurement. When you flip a
coin, the amplitude for getting heads is 0.51/2.
We use Dirac's bra
- ket (bracket) notation; <yç is called
a bra vector and ïy> a ket vector. In the
language of matrix algebra, bras and kets are row and
column vectors. Each component of one is the complex conjugate of the
corresponding component of the other, but for the crap game we can use all real
numbers, so they are the same. More on this later.
In the crap game, a pair of dice is thrown,
and the outcome could be 2, 3, 4, …,12. There is one
way to get a 2, two ways to get 3, and the number of ways increases by one
until we have six ways to get 7. Then we decline by one each step: five ways to
get 8, etc. until we end with one way to get a 12. If you add these up you get
36 ways. (It is no coincidence that this is 62; look it up in a
statistics book, or figure it out.). So the probability of throwing a 7 is 6
out of 36 or 1/6, and the amplitude for getting a 7 is the square root of 1/6.
The state vector ïy>:
While the dice are flying through the air, we assume the above probabilities
are in effect, and each component of the state vector is an amplitude, the
square root of the probability of getting a 2, 3, …, 12. So
<yï = (1/6,
Eigenvalues and eigenvectors:
The eigenvalues are the possible results of a
"measurement": 2, 3, 4,…,12. For each eigenvalue
there is an eigenvector which is a unit vector for that value. This set of
vectors defines an eleven dimensional "space." The eigenvalues are the eleven numbers, and like the vector in
3 space, which can be written as a linear combination
of 3 unit vectors, any vector in our eleven space can be written as a linear
combination of the eigenvectors.
People usually name the eigenvectors after
their eigenvalues. Thus
<2ï = (1,0,0,0,0,0,0,0,0,0,0)
<3ï = (0,1,0,0,0,0,0,0,0,0,0)
etc., but you do not always need to write out the list of components like this.
The corresponding ket vectors ï2> and ï3> are
column vectors in matrix algebra. If you know about that, fine. If not, at the
bottom of this document I have some stuff on that.
Dot products (inner products):
In QM the dot product of the two vectors in the previous paragraph is written
<2ï3>. It is the sum of the products of corresponding components, and
obviously for eigenvectors the result is zero if they are different, one if
they are the same. Note that <yïy> = 1, because it is the sum of the probabilities
of all the possibilities. Note also that (for example)
<5ï y > = the component of ï y > in the 5 "direction." So if you
square it, you find the probability of getting a 5. In QM, the component could
be complex, so square the absolute value: Probability of getting n is
ï
<nï y >ï 2.
Operators:
An operator does something to a vector or to another operator. For example in
our crap game we might need an operator to tell us the number that has come up
(an eigenvalue). Let ïn> represent an eigenvector,
where n = integers 2 to 12, and let D be a dice-value operator that multiplies
the eigenvector (a unit vector) by n.
Thus Dï n> = nï n>.
If you have had a course in linear algebra (matrices and all that), you will
recognize that ïDï could be regarded as a square matrix with the eigenvalues 2 through 12 on the diagonal (from upper left)
and zeros elsewhere. ï n> could be regarded as a unit column vector.
(If you have not studied the matrix methods
of linear algebra, do it before a course on QM, unless your course is strictly
wave mechanics. Check out the nature of the course. There is a minimal
discussion of matrices at the bottom of this page.)
You will find that <nï is a row
vector, and S ï n><nï = I, the identity matrix, which has all ones on the
diagonal and zeros elsewhere.
Even without knowing about matrices, you can
see why S ï n><nï acts as an identity operator:
Recall that <nï y > is the component of ï y > in the
n "direction," and by the same reasoning, <f ï n> is
the component of <f ï . Now a vector is simply a list of its components
(amplitudes), and if we let S mean to combine them as a vector sum over all n, not
add components as scalars,
then S <nï y > = ï y > and S <f ï n> = <fï .
Hence S <f ï n><nï y > = <fïy >.
So S ú n><nï = 1
Where the heck did f come from? It's any vector. You might have
unbalanced dice, thus changing the odds, hence changing the state vector. I did
not use y for both because that would be a special case.
Expectation value:
Or average value of a large number
of identical measurements. For any observable there is an operator (hermitian operator in QM), the D operator in our game, such
that when it operates on an eigenvector, the result is the appropriate eigenvalue. The expectation value of the measurement, the
dice throw in this case, is calculated using D and ïy >. In
the crap game the expectation value for throwing the dice is 7, because 2 and
12 have the same probability, and the average of 2 and 12 is 7; the same can be
said of 3 and 11; etc. Now let's see how to do it with D and ïy >. We
need the sum of each dice value times its probability: The beast that does the
job is
<yïDïy >. Here it is:
(1/36)(2+6+12+20+30+42+40+36+30+22+12) = 7 . We symbolize the expectation of D
as <D>, (It really is the expectation of the eigenvalues
of D, but I am using the conventional terminology.) Anyway, <D> = <y ï Dï y > is a
useful thing in QM.
Degeneracy:
Include a coin with the dice, and
then with each number you have a head or a tail. Call heads + and tails - or up
and down (like electron spin). Then for each eigenvector there are two
different states. This is degeneracy. You can remove the degeneracy by
expanding to 22 "dimensions." The new eigenvectors would be labeled ï2,+>, ï2,-> , etc.
Your mother thinks the game itself is
degenerate.
Continuous eigenvectors and eigenvalues:
There are cases in which things do
not need to take quantum jumps- there can be continuous change, so we need
infinitely many infinitesimal steps. I discuss this in another document. If you
know why <xïy > = y(x), for example, do not go
there.
Matrix Multiplication
Addition and subtraction are the way you
would expect. Let's multiply. A row vector on the left times a column vector on
the right is the sum of the corresponding products: the first in the row times
the first in the column + the 2nd times the 2nd, etc. The
result is a single number called the dot product or scalar product or inner
product. Example:
For matrix multiplication, follow the same
row times column pattern as with the vectors above.
That is, row 1 on the left is multiplied by column 1 on the right in the manner
above to give you a single number which is the row 1, column 1 entry in the
product matrix. Continue this row-column game: the ij
entry in the product matrix is the ith row
of the first times the jth column of the 2nd.
With matrices, the pattern is always row-column. (In the Excel spreadsheet, b3
is column b, row 3. Go figure.) Example:
From the above discussion you know that if A and B are matrices, in order for the product AB to exist,
the number of rows of A must equal the number of columns of B. You can also
conclude, if you think about it, that if A is a column vector with n components
and B is a row vector with n components then BA is a single number, but AB is
an n by n matrix! In QM symbols, <BïA> = a number and ïA><Bï = a matrix.
Example:
Now leave this crap (the game,
that is), and go to the main quantum page, or send hate mail or comments,
questions: fredrick.gram at tri-c.edu (but remove “at”
and spaces and insert @) but if you are very
far along in QM, I probably won't be of any help.
You probably want to hit your browser's back button.
My main pages:
Mechanics
Fluids, heat, electricity and magnetism
Vibrations and waves
Quantum
Index