Normalization of a wave function with cosine

In summary, the conversation discusses how to normalize the given wave function by squaring it and integrating. It is noted that the time dependent part drops out, but the resulting integral is still complicated. It is suggested to split the function into positive and negative parts, and to use the identity cos(pi*x)=(exp(i*pi*x)+exp(-i*pi*x))/2 to simplify the integral. Another option is to use cos x=Re[exp(ix)] and split the integral into real and imaginary parts.
  • #1
wakko101
68
0
I need to normalize the following wave function:

psi= Cexp(-abs(x))exp(-iwt)cos(pix)

I know that when squaring it, the time dependent part drops out, which is good, but what I seem to be left with is

Psi^2=C^2exp(-2abs(x))cos^2(pix)

Which seems like a fairly complicated integral to compute. I'm thinking that there is something that I'm missing about this particular wave function that will make it easier to integrate?

Any help?

Cheers,
wakko =)
 
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  • #2
Split it into x>0 and x<0 parts. For the positive part drop the abs. The negative part is the same since the integrand is even.
 
  • #3
That's not really the problem I'm having...I understand that I can double the integral over 0 to infinity, I'm just wondering if there is a simpler way to to figure out the integral itself.

Thanks anyway.
 
  • #4
cos(pi*x)=(exp(i*pi*x)+exp(-i*pi*x))/2. If you do it that way you can turn the whole thing into one big exponential. Otherwise you can integrate by parts. It IS a somewhat complicated integral to compute. But not the worst.
 
  • #5
but if I do the conversion, I end up with an integrand that has i still in it, don't I? that doesn't seem right to me...
 
  • #6
It will seem right when all of the i's cancel in the end.
 
  • #7
It's a bit easier to use cos x=Re[exp(ix)]
 

1. What is the purpose of normalizing a wave function with cosine?

The purpose of normalizing a wave function with cosine is to ensure that the total probability of finding the particle in any location is equal to 1. This is a fundamental requirement in quantum mechanics and allows for accurate predictions of the behavior of the particle.

2. How is normalization of a wave function with cosine achieved?

Normalization of a wave function with cosine is achieved by dividing the original wave function by the square root of the integral of the absolute value squared of the wave function. This ensures that the total probability of finding the particle within the given space is equal to 1.

3. Can any wave function with cosine be normalized?

Yes, any wave function with cosine can be normalized as long as the integral of the absolute value squared of the wave function is finite. This is a requirement for any wave function to be considered physically valid.

4. What happens if a wave function with cosine is not normalized?

If a wave function with cosine is not normalized, the total probability of finding the particle within the given space will be less than 1. This can lead to inaccurate predictions and violate the fundamental principles of quantum mechanics.

5. Is normalizing a wave function with cosine always necessary?

In most cases, normalizing a wave function with cosine is necessary to ensure accurate predictions and uphold the principles of quantum mechanics. However, there are some cases where normalization may not be necessary, such as when the wave function represents an infinite plane wave.

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