Dear Michael,
I am not sure whether I understand your question correctly,
so I can agree conditionally with part of it, but not with
all what you have written. Actually, I do not object the
statements from the latter part, but I do not understand it
well.
You wrote:
>It's my understanding that the quantum wavefunction
>corresponds, one to one, with the state vector in Hilbert
>space.
Please do not think that I am a maniac of precision, but I
would restate it as follows. For one wave function and one
vector the statement of one-to-one correspondence is not
clear to me.
"The set of wave functions corresponds, one to one, with the
state vectors (vectors of norm 1) of a Hilbert space."
This sentence is a tautology (necessarily true) from the
point of view of the Hilbert space formalism, or somebody
who accepts it. More specifically, wave functions can be
identified with elements of the specific representation of
the general Hilbert space as a Hilbert space of square-
integrable complex functions, distinguished by the condition
that their norms are equal one. We can have representations
using functions of position or functions of momentum giving
different Hilbert spaces.
But not everybody has to accept the Hilbert space formalism.
DIGGRESSION: The problem within this formalism (I wrote
about it before) is that there are infinitely many fifferent
wave functions, or if you prefere vectors, which describe
the same state. They differ by a complex factor of modulus
1, so typically (especially in practical applications) this
violation of one-to-one correspondence between the states
and vectors is ignored.
CONTINUATION: My first answer was conditional. Those who
accept Hilbert space formalism, would agree. But what about
those who do not?
I am not sure what assumptions you have made about "quantum
wavefunctions". If you assume that :
a) wave functions have to be square integrable complex
functions,
b) for a function to be a wave function is not necessary to
be a state of any specific physical system,
then the correspondence is clearly one-to-one.
However, if these assumptions are not made, two questions
may arise.
Question 1:
- Can every quantum state be represented this way?
For instance, those who think interms of wave functions
without the context of Hilbert space could argue that free
particle should be described by a function with infinite
norm (or no norm).
But there is additional problem. If you assume that physical
system can have only states described by wave functions, you
eliminate possibility of the state which is not "pure state"
(in terms of quantum logic which I presented briefly, you
assume that systems are purely quantum systems, where in
reality in most of cases we deal with systems which are only
partially quantum systems. Yes mixtures (as opposed to
superpositions) are incorporated into the picture in an
articial way which from the mathematical point of view is
questionable.
Question 2:
- Is every wave function a state of some physical system?
I do not think it is possible to prove or disprove this
statement. It is a matter of assumption. (I may be wrong in
this point. I can imagine that some smart reasoning could be
used to find example of a wave function for which there is
no possible physical model. But I do not remember meeting
with any such convincing reasoning)
If you ask for my own view, as I wrote before, both the
description refering to wave functions as phenomenological
tools without any specific mathematical theoretical
formalism and the Hilbert space formalism are for me too
narrow to give correct description of the physical reality.
Thus, you ask: do you agree that A is the same as B. My
answer (yes, under some conditions...) does not endorse
either of the approaches. In my opinion only approach of the
level of generality provided by quantum logic gives us full
picture.
It is more difficult to me to respond to your second
question, or second part of your question. It could be a
result of my limited knowledge, or a matter of very concise
form of your question, but I cannot understand it well.
You wrote:
>The spatial representation of the wavefunction, Psi(x), for
>example, is identical with the inner product, in Hilbert
>space, of the position eigenvector for x with the
>state vector, a. Or the momentum representation of the
>wavefunction, Psi(p), is identical to the inner product of
>the momentum eigenvector, p, with the state vector, a.
What I understand (not necessarily correctly) you are
writing that for given state "a" (represented as a vector in
Hilbert space) we can produce a wave function in position
space (or wave function as a function of the variable x
representing position) as the result of Hilbert space inner
product of the vector "a" and "the position eigenvector for
x".
The last part I do not understand, possibly because it is
without any context. When you write "the position
eigenvector for x" you apparently assume that there is one
particular eigenvector for observable x. But there is no
such unique eigenvector. Thus, you must have some way to
identify which one you want to use. I do not know which
eigenvector you wanted to choose, so it is difficult for me
to say definitely "I do not agree." I can only say that I do
not understand, and I am afraid there is something important
missing in your construction.
Additional problem is that the Hilbert space does not have
to be Hilbert space of complex functions defined for
positions, or worse it does not have to be Hilbert space of
functions at all.
But, it odoes not make sense to multiply possible
difficulties, when I should answer simply that I do not
understand the question.
Also, unfortunately, I do not have any of my books related
to quantum mechanics with me here in Japan. So, I can not
look it up in Merzbacher.
Sorry for not being able to give you more definite answer.
Regards,
Marcin
---- Original message ----
>Date: Tue, 06 Jun 2006 11:43:24 -0600
>From: Michael Devereux <dbar_x@cybermesa.com>
>Subject: [Fis] State Vector versus Wavefunction
>To: FIS Mailing List <fis@listas.unizar.es>
>
>Dear Marcin,
>
>Its my understanding that the quantum wavefunction
corresponds, one to
>one, with the state vector in Hilbert space. The spatial
representation
>of the wavefunction, Psi(x), for example, is identical with
the inner
>product, in Hilbert space, of the position eigenvector for
x with the
>state vector, a. Or the momentum representation of the
wavefunction,
>Psi(p), is identical to the inner product of the momentum
eigenvector,
>p, with the state vector, a. And so on. My authority for
this is
>Merzbachers textbook. Would you disagree?
>
>Cordially,
>
>Michael Devereux
>
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Received on Wed Jun 7 11:45:17 2006
