How to Normalize the Wave Function ψ(x,t) = Ae^(-bx)e^(-iwt)?

[Gumbercules]

Homework Statement

I am unfamiliar with LaTeX (is there a tutorial around, or should I just wing it and risk posting a potential mess?). my problem is that I need to normalize a wave function:
psi(x,t) = Ae^(-bx)e^(-iwt). there are no constraints given.

Homework Equations

integral of psi*psi = 1, limits of integration are negative infinity to positive infinity

The Attempt at a Solution

I know that if I use the complex conjugates (psi*psi) the exponential expressions will cancel and I will be left integrating a constant. I get A²x |_-\infty^\infty = 1, which doesn't make much sense. I know this question is relatively easy and that I'm probably missing something simple, but I've been away from this material for a few years and getting back into it is a little rough.
Any help is appreciated!

 

[Pengwuino]

The imaginary exponential goes away. Remember conjugating simply changes i->-i, it doesn't do anything to real functions. Your e^{ - bx} remains.

 

[Gumbercules]

Yep, that's what I was forgetting. Thank you!

 

[turin]

Well, now, hold on. The function still cannot be normalized over the space x in (-∞,+∞) without some other ingredients. Perhaps you meant e^(-b|x|)?

 

[Gumbercules]

sorry, yes, I'm not quite sure how to insert the math symbols.
I checked the problem in the text again and I think the integral I looked up in the table is wrong as I neglected the absolute value. On top of that, the limits of integration were wrong as well. (must've been in a hurry, I guess) My math is very much out of practice, but I could use some help solving the integral

 

[turin]

I did not insert any "math symbols". I used shift+backslash to obtain the vertical bar for the absolute value. Also, you could have used "abs(x)". Don't let the notation get in the way of the meaning, but do try to keep it concise.

For the integral, for what values of x is the function different. Then, what transformation can you apply, for those values of x, to make the function the same. This will allow you to do the integral. Hint: the transformation is pretty simple. Think about what the absolute value operation does; the transformation is the opposite of that.

 

[Gumbercules]

My apologies, I am somewhat fuzzy on my notions of transforms (I did Laplace and Fourier transforms, but that was several years ago with no practice since). I am thinking that I need a negative sign in the exponent to undo what the absolute value does, but I don't see how that will help without changing the limits of integration. My apologies if this is too remedial, I don't want to waste anyone's time. Should I post this in the Calculus forum?

 

[turin]

You have the right idea. Split the integral into two ranges of x: one in which x is unaffected by the absolute value (call it X1), and the other in which x is changed by the absolute value (call it X2). You can do the integral over X1 no problem. For the integral over X2, you can do a change of variables (which also changes the limits of integration) ;) After you do this whole manipulation, it will seem quite simple. However, this is a nice example of the general approach of splitting up an integral and applying independent transformations to the integration variables.

 

[Gumbercules]

I've got it now, thank you very much, Turin!