# A momentum problem

1. Jan 19, 2005

### Hyperreality

(a) In the infinite one-dimensional well, what is $$p_{av}$$?

(b) What is $$(p^2)_{av}$$?

(c) What is $$\Delta p = \sqrt{(p^2)_av - (p_av)^2}$$?

(d) Compute $$\Delta p \Delta x$$, and compare with the Heisenberg uncertainty relationship.

Here's my working:

(a) $$p_{av}=0$$.

(b)$$(\frac{p^2}{2m})_{av} = E_{n} = \frac{\hbar^2\pi^2n^2}{2mL^2}$$.
There fore $$(p^2)_{av}=(\frac{\hbar\pi^2n^2}{L})^2$$

(c)Therefore,
$$\Delta p = \frac{\hbar\pi n}{L}$$.

(d)$$\Delta p\Delta x = \frac{\hbar}{2}\sqrt{2n^2\pi^2 -1}$$

Part (d) it seems the most suspicious, that is, the uncertainty increases with n^2. Have I done anything wrong?

Last edited: Jan 19, 2005
2. Jan 19, 2005

### dextercioby

Actually it increases with "n"...You have square root form a "n^{2}"...It looks okay...Though you didn't show the calculations leading to $\Delta x$...

Daniel.