# Heisenberg's uncertainty principle

1. Dec 6, 2013

### bmrick

1. The problem statement, all variables and given/known data
So here's the question

An electron is confined within a region of width 1 X 10^-10 meters.
Estimate the uncertainty in the x component of the electrons momentum

2. Relevant equations
ΔPΔx$\geq$(1/2)(h/2pi)

3. The attempt at a solution
The problem appears pretty straight forward. But my teacher has the solution as using the equation
ΔPΔx$\geq$h/2pi
So I'm lost as to why he got rid of the one half. I looked at the problem thinking, oh this is simple, and then after seeing that he did that I literally have no idea where to start. Can someone explain?

2. Dec 6, 2013

### bmrick

i should add, there is a diagram that shows the length restriction on the electron to be in the y direction

3. Dec 6, 2013

### BOYLANATOR

Have a look at this

Are you certain about the y direction? If we know nothing about the the x position then we can know the x component of momentum with certainty.

4. Dec 7, 2013

### vanhees71

The point is that the Heisenberg uncertainty relation is valid for any quantum state of the particle. It reads
$$\Delta x \Delta p \geq \frac{\hbar}{2}.$$
One can also show that the only (pure) states, were the equality sign holds are Gaussian wave functions.

Here the question is a bit different. It says "the particle is constraint within a region of $10^{-10} \; \mathrm{m}$". A bit idealized this means it's trapped in a potential pot with infinitely high walls on an interval of this length.

Unfortunately the question is very vague, and the answer is not unique. What might be behind the question is to estimate the above uncertainty product for the bound states of the so trapped particles in the potential pot (of course with infinitely high walls, there are only bound-state solutions of the energy-eigenvalue problem).

So I guess, what's supposed to be done is to evaluate the energy eigensolutions and then (falsely, but that's another pretty subtle issue!) to assume that the momentum operator is given by the usual expression $\hat{p}=-\mathrm{i}\hbar \mathrm{d}_x$ and evaluate the standard deviation for both position and momentum for the particle in the energy eigenstates and check the uncertainty relation.