Al Marino said:
I'm not clear about your reference to Hilbert spaces other than they can include discontinuous functions (besides, it hurts too much to think of them)
A [pure] 'quantum state' (of a 'quantum system') is a
vector in a
Hilbert space.
A "Hilbert space" is essentially a
vector space with an
inner product ... and for the finite-dimensional case this description is complete – there is no more to be said. For the infinite-dimensional case we need to add two more properties to the list, namely, "completeness" (as part of the formal definition) and "separability" (as a further condition) [note: for the finite-dimensional case both of these conditions hold by 'default']. In practice, however, there is no need at all to delve into the nuances of these two mathematical properties, because we can
simply define an infinite-dimensional separable Hilbert space as follows:
It has a countably-infinite orthonormal basis {
e1,
e2,...} and it consists of all of the "objects" (i.e. "vectors")
∑
nα
nen (where each α
nЄ
C)
such that
∑
n|α
n|
2 < ∞ .
Formally, this is quite simple ... is it not?
-------
... And here is a concrete example:
Consider the vector space of square-integrable functions on the real interval (0,L). Then, the functions
φ
n(x) = √(2/L) sin k
nx , n = 1,2,3,... ,
where k
n = nπ/L ,
form a countably-infinite orthonormal basis with respect to the inner product defined by
(f,g) ≡ ∫
0L f*(x)g(x) dx .
Any square-integrable function f on (0,L) can be written as
f = ∑
na
nφ
n ,
where a
nЄ
C, and
∑
n|a
n|
2 < ∞ .
[note: in the above, any pair of functions f and g such that
∫
0L |f(x) - g(x)|
2 dx = 0
are considered to be
one and the
same 'function'
]
-------
... I hope that the above remarks have
not been cause for aggravation.
___________
It seems reasonable that any formulation must include the measuring apparatus
... include the measuring apparatus
HOW?
In (what I am calling) the 'standard formulation' of QM, the
measuring apparatus does
not appear in the mathematical formalism on equal "footing" with the
quantum system. As mentioned above, a [pure] 'state' of the latter is a
vector in a Hilbert space ... whereas, a 'state' for the
measuring apparatus has NO representation whatsoever in the mathematical formalism! The apparatus enters into the formalism as something which, so to speak, "partitions" the Hilbert space (of the 'quantum system') into mutually orthogonal subspaces (... and this is a
fixed "partition", according to the
type of measurement which the apparatus performs), and it is only (any) one of these subspaces
into which the "state reduction" can occur.
___________
"Standard formulation" implies something other formulation; what would that (they) be?
When I use the term 'standard formulation', I mean the usual undergraduate-textbook type of presentation along with the associated elements of interpretation. These are essentially of the 'von Neumann' genre. Thus, the (implied) "something other" 'formulations' would then include all manners of approach which deviate from the said 'standard'. Most of these "other" 'formulations'
do not modify the Schrödinger equation, and they are more commonly referred to as "interpretations". Regarding those approaches which
do modify the Schrödinger equation, I am largely unfamiliar with them, but I have seen such descriptive terms as "stochastic extension of the Schrödinger equation" and "nonlinear state space".
___________
<To put it much more simply ... in the 'standard formulation' of Quantum Mechanics, the Dynamical Postulate (involving the Schrödinger equation) and the Projection Postulate (giving the rule for "state reduction") appear as two equally necessary, yet formally independent, axioms.>
Does this imply I can say (to paraphrase) "LET THERE BE RESOLUTION !" <g>
In my view, relative to the 'standard formulation', the answer is a definitive "YES!".
However, it must be emphasized that, to many practicing physicists the "RESOLUTION" is thought to be quite a
modest one. For example:
reilly said:
And, for another time, there's the approach that says. QM allows us to compute probabilities, state descriptions and the like. In short, the wave function collapse is simply the collapse from uncertainly to certainty...
Kemble's Quantum Mechanics ... demonstrates the reasonableness of the knowledge interp. of QM
And from another thread in this forum (
"Schrödinger's Cat", post #9):
reilly said:
This knowledge based approach, that |PSI|*|PSI| represents a normal probability density which reflects the state of our knowledge. That is, the wave function does not have a direct physical reality. Any wave function collapse is then associated with the change in our knowledge.
You may wish to note the rebuttals there in posts
#10,
#11, and
#13.
In my view, however,
none of those rebuttals are successful (... and indeed, post
#13, in particular, is not even
directed) at
refuting the contention that
a "knowledge-based approach" to the 'collapse' phenomenon can be consistent with the QM formalism .
Indeed, the above contention
must be true since "Bohm's interpretation" is an
instance of it.
On the other hand, the validity of this contention
does not negate that there must be
direct "physical reality" of some kind ascribable to the wavefunction. In Bohm's interpretation this can be seen most
explicitly, for, there, the wavefunction itself is what gives rise to the so-called "quantum potential".
___________
Perhaps you can point me in the right direction to learn more about the "measurement problem" and the experimental attempts to resolve or get around it.
As possibilities come to mind, I will let you know. It would, however, be helpful if you could indicate the level of mathematics which you are comfortable with. ... For example, how much of the above with regard to the "Hilbert space" were you able to follow?
By the way, why do you say
"experimental attempts to resolve or get around it" ? – it is a problem which concerns the
theory.
That's an active area of research.
