Description
1. Submit your answer to problem (4) from the previous homework set. (This is the long
homework problem dealing with Fµν.)
2. Consider the function
f(x) = 4x
3 − 32x
2 + 66x − 18
(a) Solve f(x) = 0 for x, obtaining both symbolic and numeric answers.
(b) Solve f
0
(x) = 0 for x, obtaining both symbolic and numeric answers.
(c) Plot the function and verify that your roots and extrema are correct.
3. Plot the Taylor series expansion of the sine function over the interval 0 < x < 2π and
determine how many terms you need in order to get reasonable accuracy. (Hint: You
can use the Table command to generate a list of the terms and the Total command
to sum up the list.)
4. Consider the 10 × 10 matrix Aij = sin (ij) and vector 10 × 1 column vector bi = i.
Solve the numerical problem
A~x = ~b
for the unknown vector ~x using the command LinearSolve. ( Hint: If this takes a
while to calculate on your computer, then you have gotten caught in one of the traps
I warned you about in lecture.)
5. Euler method breakdown: Consider a simple harmonic oscillator governed by
x¨ = −x
with x(0) = 1, and ˙x(0) = 0. (We have set the spring constant and the mass to one).
Define v ≡ x˙.
(a) Consider the curve (x(t), v(t)). What should it look like for the exact solution?
(b) Solve this with a standard Euler approach:
vi+1 = vi − xi∆t
xi+1 = xi + vi∆t
(Convince yourself that this is a reasonable discretization scheme and is consistent
with the equation of motion.)
Plot the orbit (xi
, vi) for 0 ≤ t ≤ 8π for some
reasonable choice of step size ∆t. What type of orbit do you get?
(c) Alter the algorithm so that you instead use:
vi+1 = vi − xi∆t
xi+1 = xi + vi+1∆t
Thus the new position depends upon the new velocity. Plot the orbits again. Does
this give a better result? Note that this is partially an implicit scheme, in that
we use some information at time i + 1 to calculate the solution at time i + 1.
(d) The energy Ei at time i is given by (x
2
i + v
2
i
)/2. Calculate analytically the energy
at time i + 1 in the algorithm of part (b), in terms of xi and vi
. What do you get
for (Ei+1 − Ei)/∆t ∼ E˙ ? Is this consistent with your graph? Do the same for the
algorithm of part (c). Why is it better?
6. Assume you wish to solve for the eigenstates of an electron in an infinite 1D quantum
well with an electric field, E applied across the well. Your Hamiltonian is given by:
−
h¯
2
2m
∂
2ψ
∂x2
− eEx ψ = E ψ
where 0 ≤ x ≤ L. You are interested in finding out how the energy of the first two
eigenstates changes as a function of the strength of the electric field.
(a) What is the unit in which you will measure distance?
(b) What is the unit with which you will measure the energy? What is its physical
meaning?
(c) What is the dimensionless parameter that is the “control knob” that corresponds
to increasing the field? What is its meaning?
(d) At what value of this parameter would you expect to see deviations from the
infinite square well solutions?
(e) Solve the problem numerically. Plot the energy of the lowest ten eigenstates for
a “small” value of the applied field. How do they vary with n, the number of the
eigenstate?
(f) Solve the problem numerically. Plot the energy of the lowest ten eigenstates for
a “large” value of the applied field. How do they vary with n, the number of the
eigenstate?
(g) Plot the energy of the lowest two eigenstates as a function of the applied field
sweeping from small to large values of the applied field. Is your estimate above
for the critical values of the field correct?
Figure 1: Cartoon by Z. Weinersmith, “Saturday Morning Breakfast Cereal”.