How Do You Normalize a Wave Function and Verify Its Momentum Space?

[Tales Roberto]

Homework Statement

Consider the wave packet $$\psi\left(x\right)=\Psi\left(x,t=0\right)$$ given by $$\psi=Ce^{\frac{ip_{0}x}{h}-\frac{\left|x\right|}{2\Delta x}$$ where C is a normalization constant:

(a) Normalize $$\psi\left(x\right)$$ to unity

(b) Obtain the corresponding momentum space wave function $$\phi\left(p_{x}\right)$$ and verify that it is normalized to unity according to: $$\int^{\infty}_{-\infty}\left|\phi\left(p_{x}\right)\right|^{2} dp_{x}=1$$

(c) Suggest a reasonable definition of the width $$\Delta p_{x}$$ and show that $$\Delta x \Delta p_{x} \geq h$$

The Attempt at a Solution

(a) is easy to solve and we find that $$C=\frac{1}{\sqrt{2\Delta x}}$$ assuming that C is real. This way $$\psi=\frac{1}{\sqrt{2\Delta x}}e^{\frac{ip_{0}x}{h}-\frac{\left|x\right|}{2\Delta x}$$

I attempt to use Fourier Transform to calculate (b):

$$\phi\left(p_{x}\right)=\left(2\Pi h \right)^{-\frac{1}{2}} \int e^{\frac{-ip_{x}x}{h}} \psi dx$$

$$\phi\left(p_{x}\right)=\left(4\Pi h \right\Delta x)^{-\frac{1}{2}} \int e^{\frac{-i\left(p_{x}-p_{0}\right)x}{h}} e^{\frac{-\left|x\right|}{2\Delta x}} dx$$


$$\phi\left(p_{x}\right)=\left(4\Pi h \right\Delta x)^{-\frac{1}{2}} \left[\int_{0}^{\infty} e^{-\left(\frac{ip}{h}} + \frac{1}{2\Delta x}\right)x} dx + \int_{-\infty}^{0} e^{-\left(\frac{ip}{h} - \frac{1}{2\Delta x}\right)x} dx\right]$$

where $$p=p_{x} - p_{0}$$. To simplify let's write:

$$\beta_{1}=\left(\frac{ip}{h}} + \frac{1}{2\Delta x}\right)$$ $$\beta_{2}=\left(\frac{ip}{h} - \frac{1}{2\Delta x}\right)$$

Then:

$$\phi\left(p_{x}\right)=\left(4\Pi h \right\Delta x)^{-\frac{1}{2}} \left[\int_{0}^{\infty} e^{-\beta_{1}x} dx + \int_{-\infty}^{0} e^{-\beta_{2}x} dx\right]$$

This integral does not converge since arguments are complex. My "feeling" is that my solution is completely wrong, please help!

 

[lanedance]

are you sure those integrals don't converge?

 

[Tales Roberto]

I calculated the integral, however i anchieved a "weird" result. I think it was wrong, i expected an exp term. Here my result:

$$\phi\left(p_{x}\right)=-\left(4\Pi h \Delta x\right)^{-\frac{1}{2}}\left[\frac{1}{\frac{ip}{h}+\frac{1}{2\Delta x}}+\frac{1}{\frac{ip}{h}- \frac{1}{2\Delta x}} \right]$$

I have nothing to do with this, may wrong!

 

[lanedance]

i haven't tried it in detail, but it looks reasonable to me... the exponentials disappear when you take the infinite limit

try simplifying, by putting it all over the same denominator, it at least makes sense in that it goes to zero for large |p|

now calculate the magnitude square (worth plotting) to get the momentum probability density function and check if its normalised...

 

[Redbelly98]

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Please post new questions by starting a new thread.

The OP's new question can be found here:
https://www.physicsforums.com/showthread.php?t=355884