Re: IS this for real?!

From: Bill Unruh (unruh_at_string.physics.ubc.ca)
Date: 08/18/04 

Date: 18 Aug 2004 19:24:29 GMT

Mok-Kong Shen <mok-kong.shen@t-online.de> writes:

]Douglas A. Gwyn wrote:

]> Mok-Kong Shen wrote:
]>
]>> I do have some knowledge of Fourier transform but, having
]>> no advanced knowledge in physics, a connection between
]>> the uncertainty principle and Fourier transform is unknown
]>> to me till now. Is there a good reference where the uncertainty
]>> priciple is derived in some details from the theory of
]>> Fourier transform? Thanks in advance.
]>
]> Although it's not done with full mathematical rigor,
]> section 24 of the 4th edition of Paul Dirac's "The
]> Principles of Quantum Mechanics" shows the connection.

]I didn't access that book but instead asked someone who
]studied physics. According to him: If one starts from the
]Schroedinger equation or the Dirac equation and employs
]Fourier transform (as a mathematical tool), one could get
]to the uncertainty principle.

]However, this is in my view quite different from what you
]wrote:

] the Heisenberg uncertainty principle, which is a
] direct consequence of a standard fact about Fourier
] transforms.

The Heisenberg uncertainty principle is a statement about operators on a
positive norm Hilbert space. As such it is an Mathematical proof. If the
Hilbert space is the set of functions with the standard L2 norm, and the
operators are the usual multiplication by x and derivative by x, then the
Heisenberg uncertainty relation is also a standard fact about Fourier
Transoforms. Ie, there is a mathematical relation between the two.

Now, the interpretation of this mathematical fact as a physical statement
depends on the physical interpretation of that Hilbert space, and those
operators. Heisenberg's Uncertainty relation was originally a mathematical
statement which was then interpreted.

]which would mean that the (pure) mathematics (the foundation)
]of the theory of Fourier transform by itself would imply
]the validity of the uncertainty principle (i.e. without

It does, if you take the uncertainty relation as its mathematical
statement.

]assuming the Schroedinger/Direc equation). Thus what you
]previously wrote is misleading in my humble view. Cf. a
]follow-up of Andrew Swallow, who also express to be unable
]to understand the quote of you.

]M. K. Shen

Relevant Pages

  • Re: ply What does it take to get a handle on math?
    ... Apart from basic maths and significant ... But from the viewpoint of physics ... it wouldn't be able to represent the uncertainty principle ... Very similar to the Fourier transform is the expansion of a function ...
    (sci.math)
  • Re: IS this for real?!
    ... >> Although it's not done with full mathematical rigor, ... > of the theory of Fourier transform by itself would imply ... > the validity of the uncertainty principle (i.e. without ... > assuming the Schroedinger/Direc equation). ...
    (sci.crypt)
  • Re: IS this for real?!
    ... >> of the theory of Fourier transform by itself would imply ... >> the validity of the uncertainty principle (i.e. without ... in Hilbert space are of positive length. ... consequence of a standard fact of [the mathematical theory ...
    (sci.crypt)
  • Re: uncertainty, wave function
    ... Can one express the uncertainty principle as a purely mathematical ... Any wave mechanics has uncertainty relations. ... space to momentum space in Cartesian coordinates with a Fourier transform. ... I was reading about a particle trapped in a box, so we look at it in ...
    (sci.physics)
  • Re: IS this for real?!
    ... >> the uncertainty principle and Fourier transform is unknown ... Is there a good reference where the uncertainty ... > Principles of Quantum Mechanics" shows the connection. ... to the uncertainty principle. ...
    (sci.crypt)
Quantcast