Wave function,probability,normalization,etc.

[prehisto]

Homework Statement

-∞<p_i<∞
N-normalization multiple

f(p_x)=N*exp(-$\alpha$p_x)
1)normalize wave function to 1
2) find probability distrubution p_x
2) find avarage value and dispersion for p_x

Homework Equations

So for normalization i have to find N value when $\int$ (from -∞ to ∞)[N*exp(-$\alpha$p_x)]^2=1
Am I right ?
either way when I am tried to solve this integral, i obtained -> -N^2=1
Could it be right ?

The Attempt at a Solution

 

[Goddar]

Your integral diverges, are you sure there's no square or absolute value involved?

 

[prehisto]

Your integral diverges, are you sure there's no square or absolute value involved?

Right now i have only this example ,checked - there is nothing more to it.
I will check the source and let you know.

 

[vela]

Homework Statement

-∞<p_i<∞
N-normalization multiple

f(p_x)=N*exp(-$\alpha$p_x)
1)normalize wave function to 1
2) find probability distrubution p_x
2) find avarage value and dispersion for p_x

Homework Equations

The Attempt at a Solution

So for normalization i have to find N value when $\int$ (from -∞ to ∞)[N*exp(-$\alpha$p_x)]^2=1
Am I right?

Yes, that would be the normalization condition.

Either way when I tried to solve this integral, I obtained -> -N^2=1
Could it be right ?

No. First, as Goddar noted, the integral you wrote down diverges. Second, even if it didn't, your N would cause the integral to be negative, not +1.

 

[paranormal]

Yes, you are correct in your approach to finding the normalization constant, N. The integral you have written is correct. However, the result you have obtained, -N^2 = 1, is not correct. The normalization constant, N, should be a positive value, and the integral should result in a positive value as well.

To find the normalization constant, you can use the following steps:

1) Substitute the given function, f(px), into the integral and expand it as follows:

∫ (from -∞ to ∞) [N*exp(-\alphapx)]^2 dx

= N^2 ∫ (from -∞ to ∞) exp(-2\alphapx) dx

= N^2 [∫ (from -∞ to 0) exp(-2\alphapx) dx + ∫ (from 0 to ∞) exp(-2\alphapx) dx]

2) Use the substitution u = -2\alphapx for the first integral and u = 2\alphapx for the second integral.

3) After substitution, the first integral becomes:

∫ (from ∞ to 0) -1/2\alpha exp(u) du = -1/2\alpha [exp(u)] (from ∞ to 0) = 1/2\alpha

And the second integral becomes:

∫ (from 0 to ∞) 1/2\alpha exp(-u) du = 1/2\alpha [exp(-u)] (from 0 to ∞) = 1/2\alpha

4) Substituting these values back into the original integral, we get:

N^2 [1/2\alpha + 1/2\alpha] = N^2/α

5) To normalize the wave function, this integral should equal 1. Therefore, we can set N^2/α = 1 and solve for N, giving us:

N = √(α)

Now that we have the normalization constant, we can find the probability distribution, px, by squaring the wave function:

px = [N*exp(-\alphapx)]^2 = N^2*exp(-2\alphapx) = α*exp(-2\alphapx)

To find the average value and dispersion for px, we can use the