Re: Tweaking junctions

```On Mon, Nov 1, 2010 at 7:24 AM, Moritz Lenz <mor...@faui2k3.org> wrote:
> On 10/22/2010 06:16 AM, Damian Conway wrote:
>> That is, a C<\$value> is an eigenstate of a C<\$junction> if-and-only-if:
>>
>>     \$value !~~ Junction  &&  \$value ~~ \$junction
>
> In general this definition makes it impossible to return a list of
> eigenstates from the junction. Just think of junctions containing Code
> objects. Or anything more complicated than the built-in value types.```
```
[Originally sent to Moritz alone because of "Reply" not sending to the list]

Is it too late in this discussion to point out that, in non-perl
usage, eigenstates are associated with the operator, not with the
value fed into the operator?

In linear algebra, eigenvectors and eigenvalues are non-trivial
solutions to the equation Ax=λx, where x is a vector in a vector
space, A is a operator (a function from a vector space to itself) and
λ is a member of the field the vector space is defined over.  For a
given operator A, only certain values of λ allow that equation to be
solved, and those values are called the "eigenvalues" for A.  Also,
for a given operator A, only certain vectors x will solve the
equation, and those vectors are called "eigenvectors".  It should also
be clear that different values of λ work with different sets of
vectors x  (the solutions to Ax = ax and Ax=bx are different if a !=
b), so it's typical to talk about the eigenvectors of A associated
with a given eigenvalue λ.

Since A is linear, if Ax=λx  and Ay=λy, then A(ax) = a(Ax) = a(λx) =
λ(ax) and A(x+y)=Ax+Ay=λx+λy=λ(x+y), so fir a given eigenvalue λ,
there are typically multitudes of eigenvectors which form a vector
space of their own.  Eigenvectors for different eigenvalues are
orthogonal, and any eigenvector can be scaled to be a unit
eigenvector.  If an operator has a full set of eigenvalues, one can
pick a set of unit eigenvectors to act as a natural orthonormal basis
for the operator.  If operator A has three eigenvalues a, b, c, and
three unit eigenvectors x, y, z, such that Ax=ax, Ay=by, and Az=cz,
then if w = dx+ey+fz, Aw = a(dx)+b(ey)+c(fz), which is really easy to
compute.

In quantum mechanics, especially the Heisenberg matrix formulation
(but by analogy, also every other formulation, including wave
mechanics), quantum states are represented by vectors in a complex
vector space, and vectors which differ by a real-valued scaling factor
are generally considered equivalent.  Transformations (i.e., anything
which modifies the quantum state of the system, including but not
limited to the passage of time) are represented by (unitary) operators
on the state space.  (Unitary in this case means that the norm of Ax
is the same as the norm of x, for all x.)  The standard notation is a
bit odd, where the 'ket' |x> represents a system in state x (and
therefore |x+y> a state in a superposition of x and y), The 'bra' <x|
is the complement of the ket |x>, and can be multiplied by a ket to
get a braket <x|y> which represents the probability that a system in
state y is also in state x.  |x> is, naturally, usually normalized
such that <x|x> = 1.  Operators act on kets and return kets, so A|x>
is the braket notation way of writing the linear algebra Ax.
Naturally, that means that <y|A|x> is the probability that a system
that starts in state x will be in state y after the transform A.

Since A is a linear operator, it has eigenvalues and eigenvectors.  In
the quantum mechanical world, where vectors represent states, the
eigenvectors are called "eigenstates".  Eigenstates |i>, |j> of an
operator A have the property that <i|A|i> = <j|A|j> = 1, but <i|A|j> =
0 (informally, if you start in an eigenstate of A, then the transform
leaves you unchanged).  However, A|i+j> = |ai+bj>, so A can change the
nature of a superposition of states.  <i|i+j> = 1/2, <j|i+j> = 1/2,
but <i|A|i+j> = a/(a+b), <j|A|i+j> = b/(a+b).

Schrodinger's Wave Equation, in matrix notation, is of the form Hx=Ex,
where H is the "Hamiltonian operator" of the system, and E is the
energy of the system, so the only allowed solutions of the wave
equations are for energy levels E which are eigenvalues of H,and for
quantum states which are eigenstates of H.  Similar equations exist
for virtually every "observable", so the only allowable momenta are
the eigenvalues or eigenstates of the momentum operator, the only
allowable positions are the eigenvalues or eigenstates of the position
operator, etc.

So asking for the eigenstates of a quantum superposition is asking the
wrong object for the property.
```