Regarding Kittel's solid state physics 167 page

[benz994]

Homework Statement

In Kittel's 'Introduction to solid state physics' (8th ed.), on page 167, it says "The wave functions at the Brillouin zone boundary ##k=\pi/a## are ##\sqrt{2} cos (\pi x/a)## and ##\sqrt{2} sin(\pi x/a)##, normalized over unit length of line."
Here I cannot understand what is the meaning of "normalized over unit length of line".
Does that mean that, when I integrate the square of the wave function from 0 to 1, the result should be 1? But
$$ \int_0^1 2cos^2 \frac{\pi x}{a} dx = \int_0^1 (1+cos \frac{2\pi x}{a})dx $$
$$ = 1+\frac{a}{2\pi} sin\frac{2\pi}{a} \neq 1. $$
Please help me find out what is wrong in my reasoning.
Thanks.

 

[voko]

Is ##a## given, or is it something you can specify at will? In the latter case, can you find ##a## that would satisfy the requirement?

 

[benz994]

Thank you for the responce.
I think ##a## is a given quantity. ##a## is the size of the unit cell in one-dimensional lattice.
The above equation is satisfied only when ##a=1##.

 

[voko]

The above equation is satisfied only when ##a=1##.

No, not only. The sine is a periodic function.

 

[HalfLostAndSearching]

I would like to clarify the concept of normalization in quantum mechanics and its relevance to the wave functions mentioned in Kittel's solid state physics book.

Normalization is a fundamental concept in quantum mechanics that ensures the probability of finding a particle in a given state is equal to 1. In other words, the wave function must be normalized in order to accurately describe the probability distribution of a particle in a particular state.

In the context of Kittel's book, the wave functions mentioned at the Brillouin zone boundary are normalized over a unit length of line. This means that when the wave function is integrated over a unit length of the line, the result should be equal to 1. Therefore, your understanding that the integral of the square of the wave function from 0 to 1 should be equal to 1 is correct.

However, your calculation is incorrect. The correct integral should be:
$$ \int_0^1 2cos^2 \frac{\pi x}{a} dx = \frac{1}{2} + \frac{a}{4\pi} sin \frac{2\pi}{a} = 1$$

The mistake in your calculation is that you have integrated the term ##cos \frac{2\pi x}{a}## incorrectly. It should be integrated as follows:
$$ \int_0^1 cos \frac{2\pi x}{a} dx = \frac{a}{2\pi} sin\frac{2\pi}{a} $$

Once you correct your calculation, you will see that the integral is indeed equal to 1, as expected. This confirms that the wave functions mentioned in Kittel's book are normalized over a unit length of line.

I hope this clarification helps to address your confusion. It is important to have a solid understanding of normalization in quantum mechanics, as it is a crucial concept in accurately describing the behavior of particles at the atomic and subatomic level.