Part I: Diffraction and the Uncertainty Principle

Sample calculation to obtain uncertainty in momentum for red laser with micrometer setting of 0.4 mm

= (.0004 m)(sin(.00678º)) = 4.73333 m 4.73333= p = 1.39986 kg m/s px = sin = sin(.00678º) = 1.6565 kg m/s

ΔxΔpx = 0.0004 = m2 kg/s

Table 1: Momentum Calculations

L (m)

3.38

Red

Δx (m) spot width (m) half-width (m) θ wavelength (m) p (kg m/s)

0.0004

0.008

0.004

0.0678

4.73333E-07

1.39986E-27

0.0002

0.02

0.01

0.1695

5.91666E-07

1.11989E-27

0.0001

0.03

0.015

0.2543

4.43836E-07

1.49289E-27

0.00005

0.06

0.03

0.5085

4.43744E-07

1.4932E-27

Green

Δx (m) spot width (m) half-width (m) θ wavelength (m) p (kg m/s)

0.0004

0.01

0.005

0.0848

5.92015E-07

1.11923E-27

0.0002

0.02

0.01

0.1695

5.91666E-07

1.11989E-27

0.0001

0.045

0.0225

0.3814

6.65664E-07

9.95398E-28

0.00005

0.05

0.025

0.4238

3.69832E-07

1.79162E-27

According to the Heisenberg Uncertainty Principle, it is impossible to know the momentum and position of a particle with exact precision. This is demonstrated above, where as the uncertainty in position decreases, the spot width, which corresponds to the uncertainty in momentum, increases. The relationship ΔxΔp ≥ h, where h is Planck’s constant, holds; each product of uncertainty in position and uncertainty in momentum is equal to h. Figures 1-8 in the appendix show the resulting central bright spots from the lasers shone through the different micrometer width apertures.

Part II – Diffraction and Interference from Optical Transform Slides

Sample calculation to obtain the spacing between the lines of pattern (a) where = 638.2 nm

638.2 nm = dsin(0.369º)

d = 1.729 · 10-3 mm = 1.729 · 10-6 m

Table 2: Data & Calculations for Spacing Between Lines

Slide Pattern

Distance between spots (m)

Spacing between lines (m)

Discovery a

1.6 · 10-3

0.369

1.729 · 10-6

Discovery b

1.75 · 10-3

0.404

1.578 · 10-6

Discovery c

2.45 · 10-3

0.566

1.127 · 10-6

Discovery d

2.55· 10-3

0.589

1.083 · 10-6

Unit d

2.0 · 10-3

0.462

1.381· 10-6

In the appendix, Figures 9, 10, 11, and 12 correspond to the diffraction patterns resulting from light shone through discovery slide patterns a, b, c, and d, respectively. As the slide for a was rotated, the spots rotated with it. The general pattern of a through d was lines traveling across the slide, the differences being 1) the orientation (either horizontal or vertical), and 2) the spacing between them (largest in a and decreasing to the smallest spacing between lines in d). Although the space between the lines on the slide pattern decreased from a to d, the diffraction pattern showed an increase in distance between the central bright spot and the nearest spot. This was further proven by quantitatively calculating the spacing between the lines in the pattern, the results of which are in the fourth column in Table 2; in the diffraction equation, if the spacing d decreases, the angle at which the light is diffracted must increase and thus, the locations at which constructive interference occurs are farther away from each other. It was also noted that if the pattern on the slide involved horizontal slits, the diffraction pattern produced vertical spots, because constructive interference occurred in those regions.

If diffraction patterns a and c are compared, the bright spots both have a vertical orientation but the distance between bright spots is greater in c than a because c had a smaller spacing between lines. If diffraction patters b and d are compared, the bright spots both have a horizontal orientation but the distance between them in d was greater than those in b because d had a smaller spacing between lines. These characteristics are summarized in Table 3 below.

Table 3: Part II Quantitative Results – Discovery Slide a-d

Scattering Pattern Characteristic

Resulting Diffraction Pattern Characteristic

Horizontal orientation