Re: need translation into english

From: John T Lowry (jlowry100_at_earthlink.net)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 10:52:02 GMT

"Paul" <paulw@mmail.ath.cx> wrote in message
news:3ba4d769.0408292111.3105f20a@posting.google.com...
> Hi
>
> Can someone translate the following statement into english:
>
> "If the eigenvalues of A (operator) are non-degenerate, the normalised
> eigenvectors |a_n> are unique to within a phase factor, ie the kets
> |a_n> and
> e^(i*theta) * |a_n>, where theta is any real number yield the same
> physical result."
>
> Assume I know everything mathematically there is to know about
> eigenvalues, eigenvectors, etc, but why emphasize "unique with a phase
> factor"? What's the physical significance?
>
> Thanks.

Since it's only inner products (probabilities) and expectation values of
observables which have physical significance, quantum states are
represented by RAYS in Hilbert space, not vectors. And it's a complex
vector space (over the field of complex numbers). Hence two vectors
differing only by factor exp(iA), A real, refer to the same physical
situation. Almost any good modern QM textbook will tell you about this.

John Lowry
Flight Physics

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