Integrating absolute values over infinity

[ElijahRockers]

Homework Statement

Find <x> in terms of X₀ if X₀ is constant and

[itex]\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}[/itex]

and

[itex]<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx[/itex]

where Psi* is the complex conjugate of Psi.

Since there is no imaginary component, this is effectively Psi².

so, from here I could do a u-substitution to integrate over e^u du, but I'm not sure how.

What is the derivative of -2|x|/X_0 with respect to x?

This is part of a physics exercise I'm working on.

[itex] <x> = \frac{1}{x_0}\int^{\infty}_{-\infty}e^{\frac{-2|x|}{X_0}} dx[/itex]

I have found that the derivative of |x| depends on whether x<0 or x>0. for x<0, x'=-1 and for x>0, x'=1 but I'm not sure how to tie this all together for the integration.

I guess what I'm really asking is how do I find the integrand here?

 

[SammyS]

Homework Statement

Find <x> in terms of X₀ if X₀ is constant and

[itex]\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}[/itex]

and

[itex]<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx[/itex]

where Psi* is the complex conjugate of Psi.

Since there is no imaginary component, this is effectively Psi².

so, from here I could do a u-substitution to integrate over e^u du, but I'm not sure how.

What is the derivative of -2|x|/X_0 with respect to x?

This is part of a physics exercise I'm working on.

[itex] <x> = \frac{1}{x_0}\int^{\infty}_{-\infty}e^{\frac{-2|x|}{X_0}} dx[/itex]

I have found that the derivative of |x| depends on whether x<0 or x>0. for x<0, x'=-1 and for x>0, x'=1 but I'm not sure how to tie this all together for the integration.

I guess what I'm really asking is how do I find the integrand here?

So, split the integral.

[itex]\displaystyle <x> = \frac{1}{x_0}\int^{0}_{-\infty}xe^{\frac{-2|x|}{X_0}} dx+\frac{1}{x_0}\int^{\infty}_{0}xe^{\frac{-2|x|}{X_0}} dx[/itex]Added in Edit:

And where did the x go in the integrand ? Well, that makes the integrand odd. Therefore, don't bother splitting it.

 

[Dick]

Homework Statement

Find <x> in terms of X₀ if X₀ is constant and

[itex]\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}[/itex]

and

[itex]<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx[/itex]

where Psi* is the complex conjugate of Psi.

Since there is no imaginary component, this is effectively Psi².

so, from here I could do a u-substitution to integrate over e^u du, but I'm not sure how.

What is the derivative of -2|x|/X_0 with respect to x?

This is part of a physics exercise I'm working on.

[itex] <x> = \frac{1}{x_0}\int^{\infty}_{-\infty}e^{\frac{-2|x|}{X_0}} dx[/itex]

I have found that the derivative of |x| depends on whether x<0 or x>0. for x<0, x'=-1 and for x>0, x'=1 but I'm not sure how to tie this all together for the integration.

I guess what I'm really asking is how do I find the integrand here?

Just integrate from 0 to infinity where |x|=x. The integral from -infinity to 0 will be the same thing since your function is even.