Physicists use laser beams to create an atom trap in which...

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 $1 \mathrm{mm}$. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately $1 \mathrm{nK},$ which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a one-dimensional model of a sodium atom in a 1.0 -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 speed $v_{\max }$ of the atoms in the trap is half the value you found in part a. Use this $v_{\operatorname{mat}}$ to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.

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 $1 \mathrm{mm}$. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately
$1 \mathrm{nK},$ which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a one-dimensional model of a sodium atom in a 1.0 -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 speed $v_{\max }$ of the atoms in the trap is half the value you found in part a. Use this $v_{\operatorname{mat}}$ to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.

Key Concepts

Uncertainty Principle

This is a fundamental principle in quantum mechanics stating that the product of the uncertainties in position and momentum cannot be smaller than a set minimum value (approximately ?/2). In confined systems, such as an atom in a small box, the limited position leads to an inherent uncertainty in momentum, setting a lower bound on the range of possible speeds.

Particle in a Box Model

This is a simple quantum mechanical model where a particle is confined within a potential well with rigid or infinitely high boundaries. It provides a framework to understand how confinement in a limited spatial region quantizes the momentum and energy levels, helping to estimate the minimum momentum (and hence speed) of a particle.

Quantum Confinement

This concept describes the effects observed when particles such as atoms or electrons are confined to very small dimensions, such that their wave-like properties become significant. The confinement leads to discrete energy levels and a nonzero minimum kinetic energy due to the uncertainty principle.

Kinetic Energy and Temperature Relationship

In both classical and quantum contexts, kinetic energy is related to the temperature of a system. For atoms, estimating the kinetic energy (here influenced by the uncertainty-induced momentum spread) allows one to approximate the temperature, typically using relationships like 1/2mv² ? kBT for each degree of freedom.

Transcript

00:01 So here we are given temperature, which is one nanocalcalfin in the size of the box that the particle is in.

00:09 That's one millimeter.

00:15 So it's this.

00:19 And this atom is subject to the eisenberg uncertainty principle, as everything is, which is delta x times delta p.

00:35 It's always bigger or equal than the blanks constant over two.

00:40 Okay, and delta b is just the mass times delta v.

00:48 Okay.

00:50 And now we wanna know the smallest range of speeds for sodium metham in this particular box.

01:01 So we can just calculate that the range of speeds, or the uncertainty in the speed, should i say, is always bigger than the plant's constant over two times.

01:19 Times the mass of sodium times this deltax, which really is d 'l.

01:32 Sodium's standard atomic weight is around 23 atomic mass units.

01:39 So we'll just take the, as an approximation, we'll take the mass of a proton and multiply it by, which is almost the mass of the neutron, will multiply it by 23.

01:52 23 to get the mass this one.

01:58 And so this is just the planx constant, which is 6 .63 times 10 to the minus 34.

02:07 We get 2 times 23 times the mass of a proton, which is around 1 .67 times 10 to the minus 27 kilograms.

02:20 And then the 1 millimeter, which is 10 to the minus 3 meters.

02:28 That's the uncertainty in the position.

02:31 And this gives around 8 .6 times 10 to the minus 6 meters per second.

02:48 So if we assume that the velocity, the medium, the average velocity is centered, is zero.

02:57 So the velocities will be centered around zero...

Please give Ace some feedback