P2424 Test #2 March 2003

1. When a charge q moves through a potential difference V, it gains energy eV. Suppose an electron is accelerated through a 100 kV potential difference. Given that the rest energy of an electron is 0.511 MeV, find the final speed of the electron. If we ignored relativistic effects, what would the final speed of the electron be?

2. The reaction shown below can only if the protons, p, are moving and in the reaction convert kinetic energy to rest energy. Find out how much kinetic energy is needed if the protons hit head on with equal speeds and the pion, p⁺, and deuteron, d don’t move. The rest masses of the proton, deuteron, and pion are respectively 1.007825 u, 2.014102 u, and 0.149867 u.

p + p → p⁺ + d .

3. A particle is moving with speed u = 0.5 c with respect to an observer O. Its momentum is measured to be 310.0 GeV/c.

(a) What is its rest energy and kinetic energy particle?

(b) An Observer O' is moving towards O and the oncoming particle at v = 0.25 c. What does O' measure for the particle’s momentum and energy?

(c) An Observer O'' is moving away from O and the retreating particle at v = 0.25 c. What does O'' measure for the particle’s momentum and energy?

4. A finite potential square well has width 0.85 nm and a height of 14.17 eV. An electron is in the well.

(a) Find the energy of the lowest bound state.

(b) Find the energy of the highest bound state.

(c) Which level, n, is the highest bound state?

5. Sketch the n = 9 wavefunction for the potential well below. Explain the features of your sketch.

6. Sketch the potential well which produces the wavefunction below. Be sure to explain all the features. Which energy level n is this wavefunction for?

Useful Constants

1 u = 931.49401 MeV/c² c = 2.99792458 ´ 10⁸ m/s h = 4.135667 ´ 10^-15 eV×s.

m_e = 0.5119989 MeV/c²