Normalize the eigenfunction of the momentum operator

[LCSphysicist]

Homework Statement

I will post a print

Relevant Equations

The momentum operator identity

I am just solving the equation $$\frac{h}{2\pi i}\frac{\partial F}{\partial x} = pF$$, finding $$F = e^{\frac{ipx2\pi }{h}}C_{1}$$, and$$ \int_{-\infty }^{\infty }C_{1}^2 = 1$$, which gives me $$C_{1} = \frac{1}{(2\pi)^{1/2} }$$, so i am getting the answer without the h- in the denominator

Pretty sure the error is in the third line. But can't say where there

 

[vela]

Doesn't $\int_{-\infty }^{\infty }C_{1}^2\,dx$ diverge?

 

[LCSphysicist]

Doesn't $\int_{-\infty }^{\infty }C_{1}^2\,dx$ diverge?

Yeh but we use dirac notation here just to normalize it (i am still trying to get how this work)

 

[vela]

Yeh but we use dirac notation here just to normalize it (i am still trying to get how this work)

The point is that your third line is non-sensical. It's clear that
$$\int_{-\infty}^\infty C_1^2\,dx \ne 1$$ for any value of $C_1$, so how you ended up concluding on the following line that $C_1 = 1/\sqrt{2\pi}$ is a mystery.

Do you know the integral representation of the Dirac delta function?

 

[vanhees71]

Where is this from? Quantum mechanics is usually presented with some (healthy) mathematical sloppiness in physics textbooks, but obviously you have a book where the sloppiness is used in a way where it becomes unhealthy.

There are many ways out of this. One is to do quantization of a particle in a finite volume (or here in the 1D case finite line). In order to have a proper momentum operator you have to use periodic boundary conditions, i.e., consider a particle at the interval $(-L,L)$ an impose the periodic boundary conditions $\psi(-L)=\psi(L)$ on the wave functions. The momentum operator is given as in infinite space by $\hat{p}=-\mathrm{i} \hbar \partial_x$.

Now solve the problem first for this space and then take the limit $L \rightarrow \infty$ in the proper way such as in this limit
$$\int_{\mathbb{R}} \mathrm{d} x e_p^*(x) e_{p'}(x)=\delta(p-p').$$