Exploring the Limit of Quantum Cooling in an Atom Trap

[natalie.*]

Homework Statement

Physicists use laser beams to create an "atom trap" in which atoms are confined within a spherical region of space with a diameter of about 1 mm. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately 1 nK, which is extremely close to absolute zero, but it would be interesting to know if this temeprature is close to any limit set by quantum physics. We can explore this issue with a 1-D model of a sodium atom in a 1-mm-long box.

a) Estimate the smallest range of speeds you might find for a sodium atom in this box.

b) Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in part a. Because there's a distribution of speeds, suppose we estimate that the root-mean-square spped v_rms of the atoms in the trap is half the value you found in part a. Use this v_rms to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.

HINT: use the equation:

v_rms = Sqrt[(3(k_B)T)/m], where k_B = 1.38 x 10^-23 J/K is Boltzmann's constant.

Homework Equations

For a particle in a box, the allowed energies are:
$$E_{n}=\frac{1}{2}mv_{n}^{2}=n^2\frac{h^2}{8mL^2}$$
which means that the allowed velocities are:
$$v_{n}=n\frac{h}{2L}$$

Heisenberg Uncertainty Principle
$$\Delta x \Delta p_{x}\geq\frac{h}{2}$$

The Attempt at a Solution

Using the uncertainty principle I found that $$\Delta v=\frac{h}{2 \Delta x m}$$ where I think that $$\Delta x=L$$, the length of the box. I don't know what this range is around though? The answer for a) has to be entered as the lower speed and then the higher speed.

 

[natalie.*]

For anyone else googling this problem or anything:

I ended up figuring it out. The delta v you can find with the Heisenberg uncertainty principle is around 0 m/s, so:
-0.5*(delta v) <= v <= 0.5*(delta v)

Then your v_rms=0.25*(delta v) and you sub that into the equation given to find T (in Kelvin). Remember that the mass of a sodium atom is the mass in amu=g/mol divided by Avogadro's number divided by 1000 (SI units is kg not g).