Undergrad About normalization of periodic wave function

[KFC]

Hi all,
I am reading something on wave function in quantum mechanics. I am thinking a situation if we have particles distributed over a periodic potential such that the wave function is periodic as well. For example, it could be a superposition of a series of equal-amplitude plane waves with different wave number (some positive and some negative) so to give a form of $f(x+2\pi)=f(x)$. In this case, I wonder how do we normalize the wave function. I try the following but it almost give something close to zero because the integral gives something very large

$f [\int_{-\infty}^{+\infty}|f|^2dx]^{-1}$

But since it is periodic, do you think I should normalize the wave function with the normalization factor computed in one period as follows:

$\int_{-\pi}^{+\pi}|f|^2dx$

 

[vanhees71]

That's very simple to answer. Since $\int_{\mathbb{R}} \mathrm{d} x |f(x)|^2$ doesn't exist in this case, it is not a wave function that describes a physical state, and thus you never ever need to consider it let alone normalize it.

If you have in mind the momentum eigenstates, you should realize that these are not wave functions but generalized functions which allow you transform from the position representation to momentum representation and vice versa. Here you normalize them "to a $\delta$ distribution". The momentum eigenstates are given by the equation
$$\hat{p} u_p(x)=-\mathrm{i} \partial_x u_p(x)=p u_p(x) \; \Rightarrow\; u_p(x)=N_p \exp(\mathrm{i} x p).$$
To "normalize" these functions conveniently you use
$$\int_{\mathbb{R}} \mathrm{d} x u_{p}^*(x) u_{p'}(x)=N_p^* N_{p'} \int_{\mathbb{R}} \mathrm{d} x \exp[\mathrm{i} x(p-p')=2 \pi \delta(p-p') |N_p|^2 \stackrel{!}{=} \delta(p-p') \;\Rightarrow \; N_p=\frac{1}{\sqrt{2 \pi}},$$
up to an irrelevant phase factor. So for convenience one uses
$$u_p(x)=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x).$$
Then the momentum-space wave function is given by the Fourier transformation of the position-space wave function, i.e.,
$$\tilde{\psi}(p)=\int_{\mathbb{R}} \mathrm{d} u_p^*(x) \psi(x),$$
which is inverted by
$$\psi(x)=\int_{\mathbb{R}} \mathrm{d}p u_p(x) \tilde{\psi}(p).$$

 

[KFC]

Thanks for your reply. I am still reading your reply but I am still confusing on some parts. Since you mention the momentum space, I wonder if the following is physically possible or not. Taking crystal as example, in the text they always start the discussion with periodic lattice in position space so the k space is also periodic. So if k space is periodic, is it possible to input some wave in some form onto the crystal such that the wave in k space is periodic. If that's possible, how do we normalize the wave in k space? It is confusing me. I am always thinking a picture that in k space, we may see a Gaussian in every single recipical lattice site but such Gaussian is repeating from and to infinity so they don't add up to a finite value. In your example, you consider the delta function and derive the normalization factor, but that's still for plane wave. What I am thinking is something periodic in k space but not a plane wave.

 

[vanhees71]

Sorry, I misunderstood your question. It's not about periodic wave functions but particles in a periodic potential as models of crystals. This is a bit more complicated. So have a look in some solid-state physics book (like Ashcroft&Mermin) on Bloch states.

 

[TheConsciousCat]

Sorry, I misunderstood your question. It's not about periodic wave functions but particles in a periodic potential as models of crystals. This is a bit more complicated. So have a look in some solid-state physics book (like Ashcroft&Mermin) on Bloch states.

So are Bloch states actual wavefunctions ? because after all they can only be normalized in an individual unit cell, which means that depending on the unit cell you choose the electron will always be in there.