Normalize the wave function and more Please help

[s_gunn]

normalize the wave function and more! Please help!

Homework Statement

i) Normalize the wave function

ii) Calculate <x>

iii) Calculate $$<x^{2}>$$

iv) What would happen if a < 0?

Homework Equations

$$\psi\left(x\right) = N\left(1+i\right)exp\left(-a|x|\right)$$, for -inf < x < inf and a > 0

The Attempt at a Solution

It would take ages for me to work out the latex for all my steps (I'm new to latex!) so I'll do what i need to and hope someone can help!

First:

$$^{inf}_{-inf}\int 2N^{2}e^{-2a|x|}dx$$
$$=N^{2}\left[-e^{-2a|x|}\right]^{inf}_{-inf}$$

so: $$\frac{-N^{2}}{a}=1$$
so: $$N=\sqrt{\frac{-1}{a}}$$

therefore:
$$\psi\left(x\right) = \sqrt{\frac{-1}{a}}\left(1+i\right)exp\left(-a|x|\right)$$


ii+iii) for the expectation values, I got both equalling zero

iv) and if a < 0, you'd get exponential growth as x approaches infinity (+ and -)

Is this right??!

I get so confused when the limits are infinity!

 

[flatmaster]

I think you lost a 2 when doing your initial normalization.

You should get an expectation value for position of zero. Think about physically why that is the case.

The expectation value of x^2 should not be zero. Check that integral.

Your completely correct on the a<0 regieme. The function would increase to infinity as x goes to infinity and you could not normalize the function. The solution becomes non-physical.

 

[nicksauce]

Your integration to normalize the WF looks wrong. You shouldn't get a negative answer when you're integrating a positive definite function. I'd try it again more carefully if I were you.

 

[flatmaster]

Also, it is difficult to integrate an absolute value. You need to split the integral into two regions. One where the Abs(x) = x and one where Abs(x) = -x

 

[s_gunn]

Ok! I redid the integration for the wave function and got $$N=\sqrt{\frac{a}{2}}$$ now, which looks a lot better and my expectation value of x is still zero so that's good too! My problem now is that when i try to calculate the expectation value of x^2, (using integration by parts) I just keep getting deeper and deeper into it!

$$<x^{2}>=^{inf}_{-inf}\int\psi^{*}x^{2}\psi\: dx$$
$$=\sqrt{\frac{a}{2}}\left(1-i\right)e^{-a|x|}x^{2}\sqrt{\frac{a}{2}}\left(1-i\right)e^{-a|x|}\: dx$$
$$=^{inf}_{-inf}\int ax^{2}e^{-2a|x|}$$
$$=^{0}_{-inf}\int ax^{2}e^{2ax}+\;^{inf}_{0}\int ax^{2}e^{-2ax}$$
$$=\left[\frac{x^{2}e^{2ax}}{2}\right]^{0}_{-inf}-\:^{0}_{-inf}\int \frac{e^{2ax}}{x}\:dx+\left[\frac{-x^{2}e^{-2ax}}{2}\right]^{inf}_{0}-\:^{inf}_{0}\int \frac{-e^{-2ax}}{x}\:dx$$

=...

so far I have carried on another 2 integrals and the denominator in each new integral has an increased power of x (this one is x, next x^2, then x^3)! I can see no end!

Have I made yet another mistake?!

 

[s_gunn]

I don't know how where i was going with that intgral! I started again from scratch following your advice and every thing seemed to fall into place itself so thanks to both of you for your help!