Calculate number of quantum states of a particle

1. Sep 8, 2014

leroyjenkens

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

A particle is moving in one dimension, estimate the number of quantum states available to that particle if it is an electron confined in a region 10-9m long with speed less than 107 m/s (less than meaning velocity is between 107 and -107 m/s)

2. Relevant equations
[x][px]/h

x being the total accessible area in phase space
px being the particle's momentum
h being Plank's constant

3. The attempt at a solution

This seemed pretty straight forward. I plugged in the values the question gives x = 10-9
I broke down px into mvx and plugged in the value of those given in the problem. I plugged in 107 for vx, even though it's "less than" that value. I just made up for that by making my answer "number of quantum states ≤ 13.7", which is my answer.
Now the answer in the solution manual (no solution provided) gives an answer of 27. I'm thinking something is wrong with the velocity part.
And are those brackets supposed to mean "expectation value"?
Thanks.

2. Sep 8, 2014

vela

Staff Emeritus
Did you account for the fact the electron can move in both directions?

3. Sep 8, 2014

leroyjenkens

The question says the particle is moving in one dimension. I assumed that meant it could only move, for example, to the right. I don't know why I thought that. But how would I account for the ability of the particle to move left or right? Would I simply multiply the formula by 2 to account for both directions? Similarly, would I multiply by 4 if the particle was moving in 2 dimensions?

4. Sep 8, 2014

vela

Staff Emeritus
It just means it can move along a line, as opposed to, say, in a plane. Also note that the given limits of the velocity are both positive and negative.

5. Sep 8, 2014

leroyjenkens

Does that mean I take each case individually; the positive and negative values of the velocity, and then add them together? That sounds right.