# Wave functions problems

1. Nov 4, 2009

### Tales Roberto

1. The problem statement, all variables and given/known data

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 $$\Delta x \Delta p_{x} \geq h$$

3. 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 lets 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!

2. Nov 5, 2009

### lanedance

are you sure those integrals don't converge?

3. Nov 5, 2009

### 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!

4. Nov 5, 2009

### lanedance

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

5. Nov 18, 2009

### Redbelly98

Staff Emeritus