[Q]Momentum eigenstate normailization

[good_phy]

Hi

I use liboff quantum mechanics textbook fourth edition.

5.25 fomula of 122 page is $\frac{1}{\sqrt{2\pi}}e^{ikx}$

I thought it is nomalized, but i don't know exactly why $\sqrt{2\pi}$ is denominator.

I think it seemed to be linked Dirac-delta function [itex] \int_{\infty}^{\infty}\frac{1}{2\pi}e^{i(k-k^{'})x} = \delta(k-k^')[/atex] but i have no idea what is going on exactly.

Please Help me

Thank for reading this question.

 

[CompuChip]

I think it is because of his definition of Fourier transform. Different people use different conventions. In the end, they are all equivalent of course and should give the same results.

See, for example this link ("Other conventions" section).

 

[good_phy]

Thank you so much that i found Fourier transform is included general innerproduct

in some mathematical space as l Fourier transform is just analogous to vector

innerproduct between momentum eigenfunction and statefunction. thank you.

but i have one questiong. what is hilbert space in which general vectorproduct is

defined?

 

[George Jones]

but i have one questiong. what is hilbert space in which general vectorproduct is defined?

What do you mean by general vector product?

 

[good_phy]

It means innerproduct such as $$<\varphi_{k}|\varphi{_{k^'}}>$$ i described

product of this form as more general form of 'vector innerproduct in normal vector space'