# Wave Functions and Uncertainty

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.

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#### Solution Summary

This physics questions addresses smallest range of speeds from sodium atoms and the distribution of speeds.

Estimate the ground state energy of a particle in the potential V(x) = lambda *(x)^4 using variational methods and the uncertainty principle

Estimate the ground state energy of a particle of mass m moving in the potential

V(x) = lambda *(x)^4

by two different methods.

a. Using the Heisenberg Uncertainty Principle;

b. Using the trial function

psi(x)=N*e^{[- abs(x)]/(2a)}

where a is determined by minimizing (E)

*Note abs = absolute value

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