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David Macmanus
science forum beginner
Joined: 20 May 2005
Posts: 16
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 Posted: Fri May 20, 2005 8:56 am Post subject:
Hilbert space and Uncertainty Principle
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Is there any connection between the need for infinite dimensional spaces
in quantum mechanics and the Uncertainty Principle (UP)?
In classical mechanics objects have a known position and momentum and
(generally) 3D vectors suffice to describe them. In QM, on the other
hand, we need an infinite number of state vectors to describe either
position or momentum - and this seems connected to the UP. Is there
indeed a connection?
Thanks,
David.
--
Posted via Mailgate.ORG Server - http://www.Mailgate.ORG |
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Arnold Neumaier
science forum Guru
Joined: 24 Mar 2005
Posts: 379
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| Posted: Fri May 20, 2005 5:03 pm Post subject:
Re: Hilbert space and Uncertainty Principle
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David Macmanus wrote:
| Quote: | Is there any connection between the need for infinite dimensional spaces
in quantum mechanics and the Uncertainty Principle (UP)?
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No. There is already an uncertainty relation for any pair of
noncommuting variables, in particular between two spins
in nonparallel directions, and the state space is there
only 2-dimensional.
Arnold Neumaier |
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Hans Aberg
science forum beginner
Joined: 08 May 2005
Posts: 27
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| Posted: Fri May 20, 2005 5:03 pm Post subject:
Re: Hilbert space and Uncertainty Principle
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In article <a61183d39f486de9991e520bcbe8068b_35661@mygate.mailgate.org>,
"David Macmanus" <macmanus@tripos.com> wrote:
| Quote: | Is there any connection between the need for infinite dimensional spaces
in quantum mechanics and the Uncertainty Principle (UP)?
In classical mechanics objects have a known position and momentum and
(generally) 3D vectors suffice to describe them. In QM, on the other
hand, we need an infinite number of state vectors to describe either
position or momentum - and this seems connected to the UP. Is there
indeed a connection?
|
The uncertainty pricniple follows from the fact that the P and Q operators
(when taken in the same direction) do not commute, and does not as such
have anything to do on what they operate on. Of course, what they operate
on determines what you physically measure. So the different parts are tied
together that way.
--
Hans Aberg |
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John T Lowry
science forum beginner
Joined: 21 May 2005
Posts: 7
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| Posted: Sat May 21, 2005 8:43 am Post subject:
Re: Hilbert space and Uncertainty Principle
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"David Macmanus" <macmanus@tripos.com> wrote in message
news:a61183d39f486de9991e520bcbe8068b_35661@mygate.mailgate.org...
| Quote: | Is there any connection between the need for infinite dimensional
spaces
in quantum mechanics and the Uncertainty Principle (UP)?
In classical mechanics objects have a known position and momentum and
(generally) 3D vectors suffice to describe them. In QM, on the other
hand, we need an infinite number of state vectors to describe either
position or momentum - and this seems connected to the UP. Is there
indeed a connection?
Thanks,
David.
I don't think so. Spin variables are encompassed by finite dimensional |
spaces and yet have uncertainty relations among them.
John Lowry
Flight Physics |
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Guest
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| Posted: Sat May 21, 2005 8:43 am Post subject:
Re: Hilbert space and Uncertainty Principle
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David Macmanus wrote:
| Quote: | Is there any connection between the need for infinite dimensional
spaces
in quantum mechanics and the Uncertainty Principle (UP)?
|
Clearly so. The algebra given by the properties:
ab - ba = c
ac - ca = 0
bc - cb = 0
has no non-trivial finite dimensional matrix representation. That's
obvious by the fact that for finite dimensional matrices the trace
operator Tr() satisfies the properties:
Tr(kA) = k Tr(A)
Tr(AB) = Tr(BA)
Tr(A+B) = Tr(A) + Tr(B).
Therefore
Tr(ab - ba) = Tr(ab) - Tr(ab) = 0 = Tr(c).
Since c commutes with everything, a general representation for the
algebra will have a block diagonal form with c constant on each block.
The representation is irreducible if there is only one block, in which
case c is a multiple of the identity which, since Tr(c) = 0, must be 0.
If the representation is reducible, it has two or more blocks. The
trace operator can be confined to each block (i.e., the trace just
takes the sum of the diagonal elements within the block). Then all the
above holds for each block of the representation, so that c = 0 on each
block and therefore c = 0 overall.
Heisenberg requires a non-zero c. Therefore, the matrix representation
is infinite dimensional. The simplest one is given by vectors:
(z0 z1 z2 ... zN 0 0 0 ...)
that have a finite number of non-zero entries. Then
a = 0 0 0 0 ...
1 0 0 0 ...
0 1 0 0 ...
0 0 1 0 ...
...
b = 0 -k 0 0 ...
0 0 -2k 0 ...
0 0 0 -3k ...
0 0 0 0 ...
...
c = k 0 0 0 ...
0 k 0 0 ...
0 0 k 0 ...
0 0 0 k ....
...
This is realized by the correspondence
(z0 z1 z2 ... zN 0 0 ...) = z0 + z1 x + z2 x^2 + ... + zN x^N
with
a = x, b = -k d/dx, c = k. |
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Eckard Blumschein
science forum Guru Wannabe
Joined: 28 Apr 2005
Posts: 128
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| Posted: Wed May 25, 2005 4:21 pm Post subject:
Re: Hilbert space and Uncertainty Principle
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On 5/21/2005 6:48 PM, Baugh wrote:
[snip]
| Quote: | Some recent work has
been done considering non-unitary time evolution (non-hermitian
hamiltonians).
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In quantum physics, energy is usually real because the hamiltonian is
hermitian. Did someone try and assume a real (non-hermitian) hamiltonian
but a hermitian spectrum of energy instead?
Regards,
E. Blumschein |
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Igor Khavkine
science forum Guru
Joined: 01 May 2005
Posts: 607
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| Posted: Thu May 26, 2005 7:07 pm Post subject:
Re: Hilbert space and Uncertainty Principle
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On 2005-05-25, Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
| Quote: | On 5/21/2005 6:48 PM, Baugh wrote:
[snip]
Some recent work has
been done considering non-unitary time evolution (non-hermitian
hamiltonians).
In quantum physics, energy is usually real because the hamiltonian is
hermitian. Did someone try and assume a real (non-hermitian) hamiltonian
but a hermitian spectrum of energy instead?
|
Eigenvectors of a non-hermitian operator with real eigenvalues are not
orthogonal. Thus projection onto eigenstates is not orthogonal. (Modulo
subtleties in infinite demensions.) That would be incompatible with the
inner product. The inner product can't be abandoned due to the
probabilistic interpretation and its role in the classical limit.
Igor |
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Eckard Blumschein
science forum Guru Wannabe
Joined: 28 Apr 2005
Posts: 128
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| Posted: Sat May 28, 2005 9:14 am Post subject:
Re: Hilbert space and Uncertainty Principle
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On 5/26/2005 9:07 PM, Igor Khavkine wrote:
| Quote: | Some recent work has
been done considering non-unitary time evolution (non-hermitian
hamiltonians).
In quantum physics, energy is usually real because the hamiltonian is
hermitian. Did someone try and assume a real (non-hermitian) hamiltonian
but a hermitian spectrum of energy instead?
Eigenvectors of a non-hermitian operator with real eigenvalues are not
orthogonal.
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Not orthogonal with respect to what? Did you consider that time
evolution may be extremely non-unitary? Please assume it to deviate from
zero just for positive values of time.
| Quote: | Thus projection onto eigenstates is not orthogonal. (Modulo
subtleties in infinite demensions.) That would be incompatible with the
inner product. The inner product can't be abandoned due to the
probabilistic interpretation and its role in the classical limit.
Igor
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Eckard |
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Igor Khavkine
science forum Guru
Joined: 01 May 2005
Posts: 607
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| Posted: Sun May 29, 2005 6:26 pm Post subject:
Re: Hilbert space and Uncertainty Principle
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On 2005-05-28, Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
| Quote: | On 5/26/2005 9:07 PM, Igor Khavkine wrote:
Some recent work has
been done considering non-unitary time evolution (non-hermitian
hamiltonians).
In quantum physics, energy is usually real because the hamiltonian is
hermitian. Did someone try and assume a real (non-hermitian) hamiltonian
but a hermitian spectrum of energy instead?
Eigenvectors of a non-hermitian operator with real eigenvalues are not
orthogonal.
Not orthogonal with respect to what? Did you consider that time
evolution may be extremely non-unitary?
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Orthogonal with respecto the the inner product which is presupposed to
be define for whatever Hilbert space of states we are working with.
Without an inner product, it is not even possible to define what is and
is not hermitian.
Unitary evolution is equivalent to a hermitian Hamiltonian. (That's for
finite dimensional spaces. I should refrain from making statements about
infinite dimensional spaces that I do not sufficiently understand, lest
I be on the receiving end of the wrath of more knowledgeable readers. 
But there is a close analogy.) If you abandon one, you abandon the
other. Vice versa, if you keep one, you must keep the other.
| Quote: | Please assume it to deviate from zero just for positive values of
time.
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I don't understand this statement.
Igor |
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Eckard Blumschein
science forum Guru Wannabe
Joined: 28 Apr 2005
Posts: 128
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| Posted: Tue May 31, 2005 6:37 am Post subject:
Re: Hilbert space and Uncertainty Principle
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On 5/29/2005 8:26 PM, Igor Khavkine wrote:
| Quote: | Some recent work has
been done considering non-unitary time evolution (non-hermitian
hamiltonians).
In quantum physics, energy is usually real because the hamiltonian is
hermitian. Did someone try and assume a real (non-hermitian) hamiltonian
but a hermitian spectrum of energy instead?
Eigenvectors of a non-hermitian operator with real eigenvalues are not
orthogonal.
Not orthogonal with respect to what? Did you consider that time
evolution may be extremely non-unitary?
Orthogonal with respecto the the inner product
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The inner product of mutually orthogonal vectors is zero. Perhaps you
refer to mutual orthogonality of two eigenvectors. Which ones should not
be orthogonal and why?
| Quote: | Unitary evolution is equivalent to a hermitian Hamiltonian.
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Yes, and non-unitary time evolution is equivalent to a non-hermitian
hamiltonian.
[snip]
| Quote: | But there is a close analogy.) If you abandon one, you abandon the
other. Vice versa, if you keep one, you must keep the other.
Please assume it to deviate from zero just for positive values of
time.
I don't understand this statement.
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While I did not understand what to abandon or keep, I will tell you more
in detail what I meant:
Since the very beginning, quantum physics starts with the assumption of
just positive and real frequencies implying zero values at negative
frequency. On the other hand, time is tacitly assumed to be positive as
well as negative.
The result of inverse Fourier transform is complex with hermitian
symmetry and to be made real by arbitrary multiplication with the
complex conjugate.
Real world is only known for past time. I called this time positive.
Therefore the missing future values are usually replaced by zeroes.
Accordingly in complex domain, there is hermitian symmetry with respect
to positive as well as negative values of frequency.
Eckard |
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