Solving Integration of Wave Equation for x|Psi|^2

[diewlasing]

where the wave equation is Psi_n = sqrt(2/a)sin(n*pi*x/a). When you do the integral of -inf to +inf of x|Psi|^2, the CRC handbook works it out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2).

And I know the solution works out to be a/2 somehow but I don't know how to get it. I worked it down to:

ax - 2sin(2ax) - 2/ax = 0. I don't know if this is the right track. Can someone shed some light on this?
Thanks in advance.

 

[kdv]

where the wave equation is Psi_n = sqrt(2/a)sin(n*pi*x/a). When you do the integral of -inf to +inf of x|Psi|^2, the CRC handbook works it out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2).

And I know the solution works out to be a/2 somehow but I don't know how to get it. I worked it down to:

ax - 2sin(2ax) - 2/ax = 0. I don't know if this is the right track. Can someone shed some light on this?



Thanks in advance.

Don't integrate from minus inifnity to plus infinity. The wavefunction you give is valid only inside the well. Outside the well, the wavefunction is zero. So integrate over the width of the well only.

 

[diewlasing]

right my fault, but the integral works out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2)

My question is how does that simplify to a/2?

 

[kdv]

right my fault, but the integral works out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2)

My question is how does that simplify to a/2?

This can't be right since it does not have the dimensions of a length. Even worse, all the terms don't have the same dimensions. So check your calculation carefully.

 

[Biest]

Where did all the factors of $$n\pi$$ go? Those will help when simplifying