Phys 312 - Assignment 3 - Due 5 Feb 98

1.

Given the complex number z = 1-i/2, find z², z³, z⁴, z⁵ and draw them as "vectors" in the complex plane.
Find 1/z, 1/z², 1/z³, 1/z⁴, 1/z⁵ and draw them as "vectors" in the complex plane.
Given the complex number z = exp(i/5), find its first twelve powers and draw them as "vectors" in the complex plane.
Do the same for z = exp(-i/5).

Find the real and imaginary parts of the complex ratio (7+3i)/(5-4i). Find its magnitude (absolute value) and phase angle. Draw numerator,denominator, and ratio as "vectors" in the complex plane.

2.

Two coupled oscillators (pendulums in the class demo) obey the equations

$$m\,\frac{d^{2}x_{1}}{dt^{2}} =-m\omega _{0}^{2}x_{1}+K(x_{2}-x_{1})$$

$$m\,\frac{d^{2}x_{2}}{dt^{2}} =-m\omega _{0}^{2}x_{2}+K(x_{1}-x_{2})$$

Verify that these equations have the solution (normal mode)

$$x_{1}=\cos (\omega _{s}t)\quad \quad \quad x_{2}=\cos (\omega _{s}t)$$

for the appropriate value of $\omega _{s}$, and find $\omega _{s}$. Describe in words the motion corresponding to this solution.

Similarly, verify that the equations have the solution

$$x_{1}=\cos (\omega _{a}t)\quad \quad \quad x_{2}=-\cos (\omega _{a}t)$$

for the appropriate value of $\omega _{a}$, and find $\omega _{a}$. Describe in words the motion corresponding to this solution.

The most general solution is described by a linear combination of the eigenstates we have found:

$$x_{1} =a\cos (\omega _{s}t)+b\cos (\omega _{a}t)$$

$$x_{2} =a\cos (\omega _{s}t)-b\cos (\omega _{a}t)$$

In particular, consider the solution with a = b. Show that this describes the situation where, initially, pendulum 1 is held at distance 2a from the vertical, and released with zero speed; what are the initial position and speed of pendulum 2? Plot and describe in words the resulting motion of each pendulum when $K/m\omega _{0}^{2}=0.1$ (weak coupling) and $K/m\omega _{0}^{2}=1$ (strong coupling).

For full credit, you must give accurate plots, by MAPLE or otherwise, and they must extend over several oscillations of each pendulum.

3.

The coupled pendulums of problem 2 behave in a way similar to two coupled quantum states. The wavefunction of two coupled quantum states is

$$\psi =c_{1}(t)\phi _{1}+c_{2}(t)\phi _{2}$$

and the time evolution is given by

$$i\hbar \frac{d}{dt}c_{1} =E_{1}c_{1}+Vc_{2} \\$$

$$i\hbar \frac{d}{dt}c_{2} =E_{2}c_{2}+Vc_{1}$$

Consider only the "degenerate" case when E₁=E₂ (identical atoms, for instance), analogously to the identical pendulums of problem 2. Use E₀ to denote the common value of E₁ and E₂.

Verify that these equations have the solution (eigenstate)

$$c_{1}=\exp (-i\omega _{s}t)\quad \quad \quad c_{2}=\exp (-i\omega _{s}t)$$

for the appropriate value of $\omega _{s}$, and find $\omega _{s}$. As a function of time, what is the probability that the quantum system is in the state $\phi _{1}$? In the state $\phi _{2}$?

Similarly, verify that the equations have the solution

$$c_{1}=\exp (-i\omega _{a}t)\quad \quad \quad c_{2}=-\exp (-i\omega _{a}t)$$

for the appropriate value of $\omega _{a}$, and find $\omega _{a}$. As a function of time, what is the probability that the quantum system is in the state $\phi _{1}$? In the state $\phi _{2}$?

The most general solution is described by a linear combination of the eigenstates we have found:

$$c_{1} &=&a\exp (-i\omega _{s}t)+b\exp (-i\omega _{a}t) \\$$

$$c_{2} &=&a\exp (-i\omega _{s}t)-b\exp (-i\omega _{a}t)$$

In particular, consider the solution with a=b. Show that this describes the situation where, initially, the system is in the state $\phi _{1}.$ Plot (accurately) and describe in words $\left\vert c_{1}\right\vert^{2}$ and $\left\vert c_{2}\right\vert ^{2}$, as they vary with time when V/E₀=0.1 (weak coupling) and V/E₀=1 (strong coupling)