Assignment 5

Problem 1:

A two-state system has a Hamiltonian H = E0 

    1    2i  
  -3i    6  

  in the {|1>, |2>} basis.

Find the eigenvalues and normalized eigenvectors of this Hamiltonian.

Problem 2

A cylindrical solenoid L = 50 cm long with a radius of r = 3 mm has N = 500 tightly-wound turns of low-resistance wire uniformly distributed along its length.  The solenoid is connected, using low-resistance wires, in series with a RS = 20 ohm resistor, a V = 9-volt battery, and a switch, which is initially open.  Around the middle of the solenoid is a two-turn rectangular loop l = 3 cm by w = 2 cm made of resistive wire having a total resistance of RL = 150 ohms.  The plane of the loop is perpendicular to the axis of the solenoid and the centers of the loop and solenoid coincide.  The switch is closed at t = 0.
Showing all your work, develop an expression, in terms of the above symbolic quantities, for the current as a function of time in the rectangular loop, and evaluate that current at t = 1 microsecond after the switch is closed.

Problem 3:

A particle of mass m and energy E > 0 whose wave function is a one-dimensional plane wave moving in the +x direction is incident on a one-dimensional square potential U(x) = -|U0|, 0 ≤ x ≤ a,  U(x) = 0 everywhere else.
There are certain discrete energies for which the probability density of the transmitted wave is equal to the probability density of the incident wave.  Find an expression for the values of these resonant energies.

Problem 4:

imageRefer to the figure.  One end of a conducting rod rotates with angular velocity ω in a circle of radius a making contact with a horizontal, conducting ring of the same radius.  The other end of the rod is fixed.  Stationary conducting wires connect the fixed end of the rod (A) and a fixed point on the ring (C) to either end of a resistance R.  A uniform vertical magnetic field B passes through the ring.

(a)  Find the current I flowing through the resistor and the rate at which heat is generated in the resistor.
(b)  What is the sign of the current, if positive I corresponds to flow in the direction of the arrow in the figure?
(c)  What torque must be applied to the rod to maintain its rotation at the constant angular rate ω? 
What is the rate at which mechanical work must be done?

Problem 5:

A particle of mass m moves in a one-dimensional potential given by
V(x) = -W for |x| < a,  V(x) = 0 for |x| ≥ a.
Demonstrate that this potential has at least one even bound state.