[orgo814]

"An unnormalized wave function for an electron in a carbon nanotube of length L is sin (2pix/L). Normalize this wave function"

Below is the solution to the problem. I am having trouble understanding some of these steps (The symbol between the 2 and x is supposed to be a pi symbol).

Ψ(x) = Nsin(2πx/L)

1 = (N^2) ∫ [sin(2πx/L)]^2 dx (I understand this)

1 = (N^2) ∫ ½ − ½ cos(4πx/L) dx (I understand this since sin^2(x) = 1/2-1/2 cos(2x)

1 = (N^2)[(x/2) - (L/8π)sin(4πx/L)] (I do not understand this step at all. I do not understand where the x/2 and 8/L came from and how we are now back at sin instead of cos)

where the normalization boundaries are x=0 --> x=L, hence the integration becomes:

1 = (N^2)(L/2) (I don't understand where the L/2 came from. I'm having trouble understanding the math to get it)

N = (2/L)^1/2

Finally, the normalized wavefunction is:

Ψ(x) = [(2/L)^1/2]sin(2πx/L) (no idea where the sin2pix/L came from..)

Please help for I am very lost and confused on these steps.

[orgo814]

Edit: I do not understand where the x/2 and 8/L came from and how we are now back at sin instead of cos); 8/L is supposed to be L/8pi

[Borek]

1 = (N^2) ∫ ½ − ½ cos(4πx/L) dx (I understand this since sin^2(x) = 1/2-1/2 cos(2x)

1 = (N^2)[(x/2) - (L/8π)sin(4πx/L)] (I do not understand this step at all. I do not understand where the x/2 and 8/L came from and how we are now back at sin instead of cos)

But you know what integration is?

[tex]\int \frac 1 2 dx= \frac x 2 + C[/tex]

[orgo814]

So that explains the x/2. Where does the L/8pi come from and why does it change back to sin

[Corribus]

butlerw2, this is really just an exercise in Introductory Calculus. The sin comes from the fact that the integral of a cos(x) function is proportional to sin(x). The L comes from the fact that one of the limits of integration is a factor of the variable L. I suggest you find a good Calculus textbook and look up how to do a definite integration. 

http://en.wikipedia.org/wiki/Definite_integral