THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 3:
 
QUANTUM MECHANICS
 
19 Jan. 2000 - finish by 26 Jan. 2000

1.

Photon Drive: A laser generates 1 kW of power continuously in a beam of light of wavelength $\lambda = 600$ nm. The beam uniformly illuminates a circular spot 1 cm in radius.

(a)

How many photons does the beam contain per unit volume?

(b)

What is the momentum of a single photon in the beam?

(c)

If the beam is completely absorbed by a blackbody, what thrust (force) does it exert on the blackbody?

plus Tipler Ch. 25, problems 26, 35, 38 & 51;

 
 

. . . and Tipler Ch. 26, problems 8, 10, 12 & 13.

JUST FOR FUN: Guns vs. Lasers - Somewhere in deep space a battle is being fought between two adversaries 1,000 km apart. One side has an ultraviolet laser cannon that ``shoots'' a beam of 10.204 eV photons through a circular aperture of diameter a = 2 mm. The other side is using an ``ideal rifle'' in which individual lead atoms are used as ``bullets.'' These atoms are prepared in a plane wave state where their positions are completely uncertain but their velocity (v = 5000 m/s) and direction are precisely determined; these bullets are then ``collimated'' through a 2 mm diameter hole in a steel plate. [Treat the bullets as ideal point particles.] Each weapon has just enough beam power to burn a 10 cm hole in the adversary's hull in 1 s at close range.

1.

Using the Rayleigh formula

$$\sin \theta_1 \; = \; 1.22 \; {\lambda \over a}$$

for the angle $\theta_1$ of the first minimum of the diffraction pattern from light of wavelength $\lambda$ passing through a circular aperture of diameter a, calculate the diameter of the laser beam at its ``target'' (twice the distance from the central maximum to the first minimum).

2.

Convert Rayleigh's criterion into an uncertainty relation between the uncertainty $\Delta p_y$ in the transverse momentum of the photons and the uncertainty $\Delta y$ in their transverse position at the ``muzzle'' of the cannon. Compare the result with the standard form $\Delta y \; \Delta p_y \; \ge \; \hbar/2$ and briefly discuss any differences.

3.

How good a marksman can one be with the ``ideal rifle?'' I.e., what will the ``pattern'' of holes in the target look like? Sketch of the distribution of ``hit'' positions, including scale.

4.

If you were unfortunate enough to be caught in such a battle, which weapon would you choose? Explain your reasoning.