Section 1

Question 1

Relativistic photon energy

Answer 1

E = pc

Question 2

Energy of a classical particle

Answer 2

E = p^2/(2m)

Question 3

Energy of a relativistic particle

Answer 3

E = sqrt(m^2c^4 + p^2c^2)

Question 4

Work function in photoelectric effect

Answer 4

E_(k-max) = ħω - ϕ

Question 5

Describe Compton scattering

Answer 5

Compton scattering refers to the scattering of high energy x-ray photons from free electrons. This helped establish that photons have quantised momentum as well as quantised energy. Monochromatic x-rays scatter from a free electron in a metallic sample. Experiments show that the scattered x-rays have longer wavelengths. Classical electromagnetism predicts no wavelength shift.

Question 6

Derive the equation:

Δλ = λ_C(1 - cosθ); λ_C = h/(m_e*c)

Answer 6

See page 7 of section 1 (2016-2017 notes).

Question 7

Show that E = ħω combined with relativistic photon energy gives p = ħk

Answer 7

ħω = pc
p = ħ * ω/c
p = ħ * 2π * f/c
p = ħ * 2π/λ
p = ħk

Question 8

Qualitatively describe the double slit experiment.

Answer 8

Coherent waves (of light or matter) pass through two slits.
Interference fringes are detected on a screen.
Interference is observed even if only one particle goes through the slits at a time!
The arrival location of an individual particle cannot be predicted.
The probability of a particle arriving at a point is proportional to the intensity of the associated interference pattern.

Question 9

Derive fringe separation:

Δy ≈ Dλ/d

Answer 9

See page 21 of section 1 (2017-2018 notes).

Question 10

Expression for uncertainty principle showing that it applies to simultaneous measurement of position and momentum.

Answer 10

Δx * Δp_x ≥ ħ/2

Question 11

Qualitatively describe how the uncertainty principle arises from wave particle duality by considering a wave packet.

Answer 11

Consider we know the momentum precisely.
Definite momentum implies definite wavelength, and we can represent our particle by a plane wave corresponding to this wavelength. However, a plane wave of definite wavelength is delocalised over all space!

We can construct a localised wave packet by summing together many different waves, each having a slightly different frequency.

A strongly localised wave packet requires more frequencies.

A perfectly localised particle requires an infinite spread of momenta.