Calculate number of quantum states of a particle

[leroyjenkens]

Homework Statement

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)

Homework Equations

[x][p_x]/h

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

The Attempt at a Solution

This seemed pretty straight forward. I plugged in the values the question gives x = 10^-9
I broke down p_x into mv_x and plugged in the value of those given in the problem. I plugged in 10⁷ for v_x, 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.

 

[vela]

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

 

[leroyjenkens]

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

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?

 

[vela]

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.

 

[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.