Re: Hilbert space and Uncertainty Principle

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? Please assume it to deviate from
zero just for positive values of time.

> 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

Eckard

.

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