Assignment:
1. Consider a running cat and an electron The electron is moving at a velocity of I 5510s t g ft 10^ m/s. Choose a reasonable mass and velocity for the cat, and a tea: onablo estimate as to how well you know the velocity of the cat.
a. What is the uncertainty in the position of the cat and the electron under Halsonberga uncortatnty principle?
b. How does this compare to typically relevant spatial scales for the objects in question (say, a cat door for the cat and an atom for the electron)?
2. Consider a hydrogen atom in the ground (n=i) state. In this state, the Bohr radius is the most probable location for the electron in the hydrogen atom.
a. What is the total energy of the electron at this distance according to Bohr's formula? (remember it has a minus sign).
b. Classically the total energy E= KE - ke2/t What is the kinetic energy of the electron? (remember that your units must match. so E must be in Joules.)
c. What is the (classical) velocity of the electron in this state?
d. Let the Bohr radius be a reasonable estimate of the uncertainty of the electron's position. What is the uncertainty in its velocity? How does this compare to its classical velocity?
e. What is the electron's actual speed according to the Copenhagen Interpretation of quantum mechanics?
3. The source of energy from our sun is fusion of hydrogen nuclei into helium. In the first step of this process, two protons fuse together into a deuterium nucleus (one proton and one neutron). In order to gel close enough to fuse, the protons must overcome the repulsive electric force between them so that they can be bound together as a nucleus.
a. A typical deuterium nucleus has a radius of 10-15 meters. Electric potential energy is given by the formula PE= kq1q2/r. What is the electric potential energy between two protons at r this distance?
b. In order to get this close, a proton must have an initial kinetic energy of this amount. What is the velocity of a proton with this kinetic energy? (The mass of a proton is 1.67x10'27 kg.)
c. For temperatures at the center of the sun, protons have an average velocity of 5x105 mis. This is clearly not high enough for the protons to get close enough to fuse. What is the de Broglie wavelength of a proton moving with velocity 5x1Os m/s? Is this larger or smaller than the size of a deuterium nucleus?
d. How is fusion able to lake place in the sun even though the protons aren't moving with enough energy to overcome the repulsive force of their positive charges?