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King Fahd University of Petroleum and Minerals
Physics Department
PHYS 373: Computational Physics
Spring 2005
Midterm Exam
Tuesday. Feb.26.2005
Due date: Saturday Apr.30,2005
Problem 1: Simple Quantum Mechanics
Consider the one-dimensional Schrodinger equation
EMBED Equation.3
where the potential is anharmonic and given by
EMBED Equation.3
where EMBED Equation.3 are constants.
Find the lowest three eigenvalues (two even and one odd) of the anharmonic potential, and plot the potential and the eigenfunctions.
Hint: See the textbook discussion on this problem.
Problem 2: Random Walks
Consider a gas molecule in a two-dimensional EMBED Equation.3 box performing random walk motion each step of length one unit. Suppose that there is a hole randomly placed on one side of the box of length 1 unit.
On the average, how many steps are needed for the gas molecule to escape.
How is the length of the step affecting this average?
How would this value change with the size of the hole (try 2 and 4 units)?
If instead of one hole you have two holes randomly placed on the perimeter, how many steps are needed for it to escape? Are there discrepancies in the results?
What if the container is a circle with an area similar to that of the rectangle?
Can you relate this to the effusion problem in statistical mechanics?
Problem 3: Spring System
A spring and mass system is governed by differential equation:
EMBED Equation.3 ; x(0)=2 and x(4)=6
where k = 16 N/m and m = 4 Kg. Solve the differential equation by Finite Difference Method for t=1 sec. Compare your results with the analytic solution.
Problem 4: One-Dimensional Motion
The speed of a bicycle racer is governed by the differential equation:
EMBED Equation.3
where P is the power, v is the speed and m is the mass of the racer; k is a constant is the friction coefficient between the tires and the road.
Evaluate the speed of the racer by Euler Method during 5 s for the values of P /p}~# $ 7 8 9 : l m :
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