Generating a Hilbert space representation of a wavefunction

[vanhees71]

Only two vectors can be orthogonal or non-orthogonal, but not a single vector. A single function (which is just a representation of the vector in the position representation) can also consequently not be orthogonal or non-orthogonal.

The functions
$$u_k(x)\exp(\mathrm{i} k x)/\sqrt{2 \pi},$$
where $k \in \mathbb{Z}$, form a complete orthonormal set, since
$$\langle k'|k \rangle=\int_0^{2 \pi} \mathrm{d} x u_{k'}^*(x) u_k(x)=\frac{1}{2 \pi} \int_0^{2 \pi} \mathrm{d} x \exp[\mathrm{i} x(k-k')]=\delta_{k k'}.$$

 

[SeM]

I am not sure we are talking about the same here. In MQM the non-orthogonality of a function and its complex conjugate is clearly described by that integral above.

 

[vanhees71]

Ok, I give up. You should first read a good textbook on quantum theory or functional analysis.

 

[SeM]

Mark, the hat function approach seems similar to the Delta function approach advised by Bill. I take the Dirac function approach further for the moment,

Thanks

 

[vanhees71]

I've no clue what MQM means. Quantum theory is quantum theory. There's only one such theory, and it is based on clear definitions, and you should use these clear definitions. It doesn't make sense to say "one vector is orthogonal"! First of all one must learn proper and concise definitions (and, of course, understand them)!

 

[SeM]

Vanhees, it's Molecular Quantum Mechanics

Mark, I read your suggestion with care now, and I noticed I already did what you wrote, however, I called it the hat function a "testing function", as derived from one of Bills links. The testing function is 1, and the integral is for 1 * Psi over the interval 0 to 2pi, which makes it a tempered distribution, allowed for Fourier Transform.

Whether it makes any physical sense is not the scope of this paper, however, I have done a Hilbert transform of it, to see its behaviour as a signal in a physical world.

Bill, I will look into the Dirac function alternative.

Thanks all

 

[SeM]

I've no clue what MQM means. Quantum theory is quantum theory. There's only one such theory, and it is based on clear definitions, and you should use these clear definitions. It doesn't make sense to say "one vector is orthogonal"! First of all one must learn proper and concise definitions (and, of course, understand them)!

Vanhees:

I didnt say a vector is orthogonal, I said the wavefunction is non-ortogonal to its complex conjugate, which is the reason for the overlap integral being a non-zero value. However, the very function is square integrable over a finite region, and it is normalizable.

 

[vanhees71]

Please read the passage you quoted again carefully word by word! Nowhere it's said a wave function is orthogonal or non-orthogonal to its complex conjugate. This wouldn't make any sense! It doesn't matter whether you work in molecular quantum mechanics or any other application, the math of QM is unique.

 

[PeterDonis]

Thread closed for moderation.