Question

I'm working on homework for quantum mechanics and I am currently stuck on a question about uncertainty. I have two equations:

(delta)x*(delta)p> (h-bar)/2
and
p=h/(lambda)

I don't know how to approach this problem. Any help would be appreciated. The problem is attached in the post below.

Answer 1

The thing to realize here is that
$$\Delta x \Delta p > \frac{\hbar}{2}$$
actually means the difference of the expectation value of the square of an operator and the square of the expectation value of an operator:
$$(\Delta A)^2 = <A^2> - <A>^2$$
So the way you would get the uncertainties written fully is:
\begin{align*}
(\Delta x)^2 & = \! \int\!\! dx\, ||\psi_x(x)||^2 x^2 -
\left(\int \!\!dx \, \|\psi_x(x)||^2 x \right)^2 \\
(\Delta p)^2 & = \! \int \!\!dp \, ||\psi_p(p)||^2 p^2 -
\left(\int\!\!dp \, ||\psi_p(p)||^2 p \right)^2
\end{align*}
Where the two wave functions psix and psip for the electron are defined over the basic psi but in different representations:
\begin{align*} \psi_p(x) &=<x|\psi> \\
\psi_p(p) & = <p|\psi>
=\frac{1}{\sqrt{2\pi \hbar} }\int \! \! dx\, e^{-ipx/\hbar} \psi(x)
\end{align*}
I hope this helps because I have no idea what relative uncertainty deltaP/P means and I don't actually know how to do this task.

Answer 2

Oh, I think it's safe to assume that the psi is that one of a free electron in 2D? But I only wrote x as in 1D. In any case for generalizations:
$$\psi = A e^{-i\vec k \cdot \vec x/\hbar} \\
\vec k_i = \frac{2\pi n_i}{L} $$
n is some positive or negative integer.

Answer 3

This is actually an algebra problem. They give you:

\[\Delta x \Delta p \ge \frac{\hbar}{2}\]\[p=\frac{h}{\lambda}\]
with:

\[\frac{\Delta p}{p} = .01\]\[\Delta x = 10^{-9} m\] and they're asking to solve for \(\lambda\).

The trick is to divide the equation by the inequality like this:

\[\Delta x \frac{\Delta p}{p} \ge \frac{\hbar \lambda}{2h}\]
Now you can solve for what \(\lambda\) can be and do whatever else the question asks, just make sure you keep
\(\frac{\Delta p}{p} = .01\) together