Waves and quantum behavior

 

 Module 4: Understanding processes

4.1 Waves and quantum behavior   
3.2 Describe and explain:

a)    Describe and explain:

I) Production of standing waves by waves travelling in opposite directions

II) Interference of waves from two slits

III) Refraction of light at a plane boundary in terms of the changes in the speed of light and explanation in terms of the wave model of light

IV) Diffraction of waves passing through a narrow aperture

V) Diffraction by a grating

VI) Evidence that photons exchange energy in quanta E = hf (for example, one of light emitting diodes, photoelectric effect and line spectra)

VII) Quantum behavior: quanta have a certain probability of arrival; the probability is obtained by combining amplitude and phase for all possible paths

VIII) Evidence from electron diffraction that electrons show quantum behavior.

b) Make appropriate use of:

I) The terms: phase, phasor, amplitude, probability, interference, diffraction, superposition, coherence, path difference, intensity, electron-volt, refractive index, work function, threshold frequency

c) Make calculations and estimates involving:

I) Wavelength of standing waves

II) [math]n = \frac{\sin i}{\sin r} = \frac{C_{\text{1st medium}}}{C_{\text{2nd medium}}}[/math]

III) Path differences for double slits and diffraction grating, for constructive interference [math]n\lambda = d \sin\theta[/math] (both limited to the case of a distant screen)

IV) The energy carried by photons across the spectrum ,[math]E = hf[/math]

V) The wavelength of a particle of momentum p,[math]\lambda = \frac{h}{p}[/math]

d) Demonstrate and apply knowledge and understanding of the following practical activities (HSW4):

I) Using an oscilloscope to determine frequencies

II) Determining refractive index for a transparent block

III) Superposition experiments using vibrating strings, sound waves, light and microwaves

IV) Determining the wavelength of light with a double-slit and diffraction grating

V) Determining the speed of sound in air by formation of stationary waves in a resonance tube

VI) Determining the Planck constant using different colored LEDs.

  • a) Describe and Explain:

  • I. Production of Standing Waves by Waves Traveling in Opposite Directions

  • ⇒ Concept:
  • Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere.
  • ⇒ Explanation:
  • The superposition of the two waves produces points of constructive interference (antinodes) and destructive interference (nodes).
  • Figure 1 Constructive and destructive interference
  • At nodes, the amplitude is zero due to complete destructive interference.
  • At antinodes, the amplitude is maximum due to constructive interference.
  • Standing waves are characterized by stationary patterns where energy oscillates but does not propagate.
  • ⇒ Example:
  • Vibrating strings on a musical instrument.
  • Air columns in a closed or open tube (e.g., organ pipes).

  • II. Interference of Waves from Two Slits

  • ⇒ Concept:
  • Two coherent sources of waves produce an interference pattern of bright and dark fringes on a screen.
  • ⇒ Explanation:
  • Coherent Sources: Waves from the two slits have a constant phase difference and identical frequency.
  • Constructive Interference: Occurs where the path difference is an integral multiple of the wavelength ([math]n\lambda[/math]), forming bright fringes.
  • Destructive Interference: Occurs where the path difference is an odd multiple of half the wavelength ([math]\left( n + \frac{1}{2} \right) \lambda[/math]), forming dark fringes.
  • Young’s Double-Slit Experiment:
  • – Path difference:
  • [math]\Delta x = d \sin\theta[/math]
  • – Fringe spacing:
  • [math]\Delta y =\frac{\lambda L}{d}[/math]
  • Where d is slit separation, L is the screen distance, and [math]\lambda[/math] is the wavelength.
  • Figure 2 Young’s Double slit experiment

  • III. Refraction of Light at a Plane Boundary

  • ⇒ Concept:
  • Refraction occurs when light passes between media of different refractive indices, causing a change in its speed and direction.
  • ⇒ Explanation:
  • Wave Model of Light: As light crosses the boundary, the wavelength changes but the frequency remains constant.
  • Speed decreases in denser media:
  • [math]v=\frac{c}{n}[/math]
  • Where c is the speed of light in a vacuum and n is the refractive index.
  • Snell’s Law:
  • [math]n_1 \sin\theta_1 = n_2 \sin\theta_2[/math]
  • Where [math]\theta_1  \ \text{and} \ \theta_2[/math] are angles of incidence and refraction, respectively.
  • ⇒ Example:
  • A straw appearing bent in a glass of water due to the change in light’s speed at the air-water boundary.

  • IV. Diffraction of Waves Passing Through a Narrow Aperture

  • ⇒ Concept:
  • Diffraction occurs when waves spread out after passing through a narrow aperture or obstacle.
  • ⇒ Explanation:
  • Diffraction is most pronounced when the aperture width (a) is comparable to the wavelength (λ).
  • The wave spreads into regions beyond the geometric shadow of the aperture.
  • For light, the central maximum is the brightest and widest region, with intensity decreasing in subsequent fringes.
  • ⇒ Example:
  • Sound waves spreading around corners.
  • Light spreading through a narrow slit in a dark room.

  • V. Diffraction by a Grating

  • ⇒ Concept:
  • When light passes through a diffraction grating with many closely spaced slits, an interference pattern forms with bright and dark fringes.
  • Figure 3 Diffraction grading
  • ⇒ Explanation:
  • The diffraction pattern is sharper than for a double slit due to the higher number of slits.
  • The condition for maxima is:
  • [math]d \sin\theta = n\lambda[/math]
  • Where d is the grating spacing, n is the order of the diffraction, and λ is the wavelength.
  • ⇒ Example:
  • Diffraction gratings used in spectrometers for separating light into its component wavelengths.

  • VI. Evidence that Photons Exchange Energy in Quanta (E = hf)

  • ⇒ Concept:
  • Photons transfer energy in discrete packets proportional to their frequency (f).
  • ⇒ Evidence:
  • Photoelectric Effect:
  • – Light above a certain threshold frequency ejects electrons from a metal surface.
  • – Energy of photons:
  • [math]E = hf[/math]
  • Where h is Planck’s constant.
  • Figure 4 Photoelectric effect
  • Light-Emitting Diodes (LEDs):
  • – Electrons emit photons with energy hf as they recombine with holes.
  • Line Spectra:
  • – Atoms emit or absorb light at discrete frequencies, corresponding to energy transitions.

  • VII. Quantum Behavior: Probability of Arrival

  • ⇒ Concept:
  • Quantum particles (e.g., photons, electrons) exhibit probabilistic behavior. The probability is determined by the wave function.
  • ⇒ Explanation:
  • Amplitude and Phase:
  • – The probability density is proportional to the square of the wave function’s amplitude.
  • Interference:
  • – Combining amplitudes for all possible paths explains phenomena like double-slit interference.
  • ⇒ Example:
  • Double-slit experiment with single electrons shows an interference pattern over time, despite individual electrons behaving like particles.
  • VIII. Evidence from Electron Diffraction
  • ⇒ Concept:
  • Electron diffraction demonstrates the wave nature of electrons.
  • ⇒ Evidence:
  • When a beam of electrons passes through a thin crystal lattice, it produces a diffraction pattern similar to that of X-rays.
  • The pattern confirms the de Broglie hypothesis, which states:
  • [math]\lambda = \frac{h}{p}[/math]
  • where h is Planck’s constant and p is the electron’s momentum.
  • ⇒ Example:
  • The Davisson-Germer experiment: Electrons diffracted off a nickel crystal confirm their wave-like behavior.

  • b) Make Appropriate Use of:

  • Below are explanations and applications of the terms in wave mechanics, quantum phenomena, and light behavior:

  • I. Terms and Their Applications

  • 1. Phase
  • ⇒ Definition:
  • The position of a point in the wave cycle, expressed in degrees or radians.
  • A complete wave cycle corresponds to [math]360^\circ \quad \text{or} \quad 2\pi[/math] radians.
  • ⇒ Application:
  • Used to determine constructive ([math]\Delta \phi = 0, 2\pi[/math]) and destructive interference([math]\Delta \phi = \pi, 3\pi[/math]) in wave superposition.
  • 2. Phasor
  • ⇒ Definition:
  • A rotating vector that represents the amplitude and phase of a sinusoidal wave.
  • ⇒ Application:
  • Phasors simplify calculations involving multiple waves by combining amplitudes and phase differences using vector addition.
  • 3. Amplitude
  • ⇒ Definition:
  • The maximum displacement of a wave from its equilibrium position.
  • ⇒ Application:
  • In interference, the amplitude determines the intensity ([math]I \propto A^2[/math]) of the resultant wave.
  • 4. Probability
  • ⇒ Definition:
  • In quantum mechanics, the square of the wave function’s amplitude gives the probability density of finding a particle in a given region.
  • ⇒ Application:
  • Used to predict outcomes in quantum experiments like the double-slit experiment with electrons.
  • 5. Interference
  • ⇒ Definition:
  • The phenomenon of waves overlapping and combining, leading to regions of constructive and destructive interference.
  • ⇒ Application:
  • Explains the fringe patterns in Young’s double-slit experiment and thin-film interference.
  • 6. Diffraction
  • ⇒ Definition:
  • The bending and spreading of waves when they encounter an obstacle or aperture.
  • ⇒ Application:
  • Demonstrates wave behavior in phenomena like diffraction gratings and single-slit diffraction patterns.
  • 7. Superposition
  • ⇒ Definition:
  • The principle that when two or more waves overlap, the resultant wave is the algebraic sum of the individual wave displacements.
  • ⇒ Application:
  • Describes the formation of standing waves, interference patterns, and beats.
  • 8. Coherence
  • ⇒ Definition:
  • A property of wave sources that have a constant phase difference and the same frequency.
  • ⇒ Application:
  • Coherence is essential for stable interference patterns, such as in laser light or Young’s double-slit experiment.
  • 9. Path Difference
  • ⇒ Definition:
  • The difference in the distances traveled by two waves from their sources to a given point.
  • ⇒ Application:
  • Determines the type of interference:
  • Constructive:
  • [math]\Delta x =  n \lambda[/math]
  • Destructive:
  • [math]\Delta x = \left( n + \frac{1}{2} \right) \lambda[/math]
  • 10. Intensity
  • ⇒ Definition:
  • The power per unit area carried by a wave, proportional to the square of the amplitude ([math]I \propto A^2[/math])
  • ⇒ Application:
  • Used to describe brightness in optical interference patterns and loudness in sound waves.
  • 11. Electron-Volt (eV)
  • ⇒ Definition:
  • A unit of energy equal to the energy gained by an electron when it is accelerated through a potential difference of 1 volt.
  • [math]1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}[/math]
  • ⇒ Application:
  • Commonly used in quantum mechanics to express photon energy (E = hf) or the work function in the photoelectric effect.
  • 12. Refractive Index
  • ⇒ Definition:
  • The ratio of the speed of light in a vacuum to its speed in a medium:
  • [math]n = \frac{c}{v}[/math]
  • ⇒ Application:
  • Describes light bending at interfaces (Snell’s Law) and critical angles for total internal reflection.
  • 13. Work Function
  • ⇒ Definition:
  • The minimum energy required to eject an electron from a material’s surface.
  • ⇒ Application:
  • In the photoelectric effect, the energy of incident photons must exceed the work function ([math]hf \geq \phi[/math]) to release electrons.
  • 14. Threshold Frequency
  • ⇒ Definition:
  • The minimum frequency of light required to eject electrons from a material in the photoelectric effect.
  • ⇒ Application:
  • Threshold frequency (​[math]f_0[/math]) relates to the work function (ϕ) through:
  • [math]f_0 = \frac{\phi}{h}[/math]

  • II. Sketching and Interpreting:

  • 1. Young’s Double-Slit Interference Graph
  • ⇒ Graph:
  • Intensity (I) vs. Position (x) on the screen shows bright (constructive interference) and dark (destructive interference) fringes.
  • ⇒ Interpretation:
  • The separation of fringes depends on the wavelength, slit separation, and distance to the screen.
  • Figure 5 Young’s double slit interference graph
  • 2. Diffraction Patterns
  • ⇒ Single Slit:
  • Central maximum is the brightest and widest.
  • ⇒ Diffraction Grating:
  • Sharper and more distinct bright fringes.
  • ⇒ Interpretation:
  • Diffraction patterns provide information about the wavelength and grating spacing.
  • 3. Photoelectric Effect Graphs:
  • ⇒ Graph 1:
  • Kinetic energy ([math]E_K[/math]) of ejected electrons vs. Frequency (f):
  • [math]E_k = hf – \phi[/math]
  • Linear graph with slope h and intercept -ϕ.
  • Graph 2:
  • Photoelectron current vs. Intensity of light:
  • Current increases with intensity for frequencies above [math]f_0[/math].
  • Figure 6 Photoelectric effect
  • 4. Electron Diffraction Patterns
  • ⇒ Graph:
  • Ring-like diffraction pattern from a polycrystalline material.
  • ⇒ Interpretation:
  • The pattern confirms the wave nature of electrons.
  • 5. Refractive Index and Refraction
  • ⇒ Diagram:
  • Light ray bending at the boundary between two media.
  • ⇒ Interpretation:
  • The angle of refraction depends on the refractive index of the media.

  • c) Detailed Explanation of Calculations and Estimates

  • I) Wavelength of Standing Waves

  • Standing waves are formed when two waves traveling in opposite directions interfere.
  • For a string fixed at both ends:
  • – The wavelength (λ) is determined by the length (L) of the string and the number of harmonics (n):
  • [math]\lambda = \frac{2L}{n}[/math]
  • – Where n=1,2,3,… (harmonic number).
  • Figure 7 Standing wave
  • For open or closed pipes:
  • – Open at both ends:
  • [math]\lambda = \frac{2L}{n}[/math]
  • – Closed at one end:
  • [math]\lambda = \frac{4L}{n}[/math]
  • Where n=1,3,5,… (odd harmonics).
  • ⇒ Example Calculation:
  • String length L=1.0 m, fundamental mode (n=1):
  • [math]\lambda = \frac{4L}{n} \\
    \lambda = \frac{2(1)}{1} \\
    \lambda = 2.0 \text{ m}[/math]

  • II) Snell’s Law

  • Snell’s Law relates the angles of incidence (i) and refraction (r) when light passes between two media:,
  • [math]n = \frac{\sin i}{\sin r} \\
    n = \frac{C_1}{C_2}[/math]
  • Where:
  • – n is the refractive index,
  • ​- [math]C_1 , C_2[/math] are the speeds of light in the respective media.
  • Figure 8 Snell’s law
  • ⇒ Example Calculation:
  • Light enters glass (n=5) from air (n≈1.0).
  • Angle of incidence [math]i = 30^0[/math]
  • [math]n = \frac{\sin i}{\sin r}[/math]
  • Rearrange for [math]\sin r[/math]
  • [math]\sin r = \frac{\sin 30^\circ}{1.5} \\
    \sin r = \frac{0.5}{1.5} \\
    \sin r = 0.333[/math]
  • Using[math]\arcsin r = \arcsin (0.333) \approx 19.5^\circ[/math]

  • II) Path Differences for Double Slits and Diffraction Gratings

  • Path difference determines constructive or destructive interference.
  • For constructive interference in double slits or diffraction gratings:
  • [math]n\lambda = d \sin\theta[/math]
  • where:
  • – n = order of diffraction,
  • – λ = wavelength of light,
  • – d = slit separation,
  • – θ = angle of diffraction.
  • ⇒ Example Calculation:
  • Wavelength [math]\lambda = 600 \text{ nm} = 600 \times 10^{-9} \text{ m} \text{Slit separation } (d) = 0.01 \text{ mm} = 1 \times 10^{-5} \text{ m} \text{First order } (n) = 1[/math]
  • [math]\sin\theta = \frac{n\lambda}{d} \\
    \sin\theta = \frac{(1)(600 \times 10^{-9})}{1 \times 10^{-5}} \\
    \sin\theta = 0.06 \\
    \theta = \arcsin(0.06) \\
    \theta \approx 3.4^\circ[/math]

  • III) Energy Carried by Photons Across the Spectrum

  • The energy (E) of a photon is related to its frequency (f) by:
  • [math]E = hf[/math]
  • Where:
  • – [math]h = 6.63 \times 10^{-34} \text{ Js}[/math](Planck’s constant),
  • – [math]f = \frac{c}{\lambda}[/math](frequency is speed of light divided by wavelength).
  • Figure 9 Energy levels and photon emission
  • ⇒ Example Calculation:
  • Wavelength([math]\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m}[/math])
  • [math]f = \frac{c}{\lambda} \\ f = \frac{3 \times 10^8}{500 \times 10^{-9}} \\ f = 6 \times 10^{14} \text{ Hz} \\E = hf \\ E = (6.63 \times 10^{-34}) (6 \times 10^{14}) \\ E = 3.98 \times 10^{-19} \text{ J}[/math]

  • IV) Wavelength of a Particle of Momentum

  • The de Broglie wavelength (λ) of a particle is:
  • [math]\lambda = \frac{h}{p}[/math]
  • Where:
  • – [math]h = 6.63 \times 10^{-34} \text{ Js}[/math]
  • – [math]p = m[/math](momentum is mass times velocity).
  • ⇒ Example Calculation:
  • Electron([math](m=9.11\times 10^{-31} \text{ kg}) \text{ traveling at } v=1\times 10^6[/math])
  • [math]p = mv \\
    p = (9.11 \times 10^{-31}) (1 \times 10^6) \\
    p = 9.11 \times 10^{-25} \text{ kg m/s} \\
    \lambda = \frac{h}{p} \\
    \lambda = \frac{6.63 \times 10^{-34}}{9.11 \times 10^{-25}} \\
    \lambda = 7.28 \times 10^{-10} \text{ m}[/math]

  • V) Sampling Theorems

  • Nyquist Criterion: The sampling rate must be greater than twice the maximum frequency of the signal:
  • [math]f_s > f_{\text{max}}[/math]

  • d)   Detailed Explanation of Practical Activities

  • I. Using an Oscilloscope to Determine Frequencies

  • Objective:
  • To measure the frequency of a periodic signal.
  • Figure 10 Using an oscilloscope to determine frequencies
  • Equipment:
  • – Oscilloscope
  • – Signal generator (or periodic signal source)
  • – Connecting cables
  • Procedure:
  • Connect the signal generator to the oscilloscope.
  • Set the oscilloscope to display the waveform (adjust time base and amplitude settings).
  •  Measure the time period (T) of one complete cycle by counting divisions on the horizontal scale:
  • [math]T = \text{Number of divisions} \times \text{Time per division}[/math]
  • Calculate the frequency (f) using:
  • [math]f = \frac{1}{T}[/math]
  • ⇒ Example:
  • If the time period of the waveform is 2 ms , the frequency is:
  • [math]f = \frac{1}{T} \\  f = \frac{1}{2 \times 10^{-3}} \\
    f = 500 \text{ Hz}[/math]

  • II. Determining Refractive Index for a Transparent Block

  • Objective:
  • To calculate the refractive index of a transparent material using Snell’s law.
  • Figure 11 Determine refractive index for a transparent block
  • Equipment:
  • – Glass or acrylic block
  • – Ray box
  • – Protractor
  • – Ruler
  • Procedure:
  • – Place the block on a sheet of paper and outline it.
  • – Shine a ray of light at an angle to the block’s surface.
  • – Mark the incident and refracted rays and measure the angles of incidence (i) and refraction (r).
  • – Calculate the refractive index (n) using Snell’s law:
  • [math]n = \frac{\sin i}{\sin r}[/math]
  • Example:
  • For an incident angle ([math]i = 40^\circ[/math]) and a refracted angle([math]r = 25^\circ[/math]):
  • [math]n = \frac{\sin i}{\sin r} \\
    n = \frac{\sin 40^\circ}{\sin 25^\circ} \\
    n = 1.5[/math]

  • III. Superposition Experiments Using Vibrating Strings, Sound Waves, Light, and Microwaves

  • Objective:
  • To observe constructive and destructive interference in different waveforms.
  • Figure 12 Superposition and stationary waves
  • Equipment:
  • For strings: Vibrating string apparatus
  • For sound waves: Two speakers connected to a signal generator
  • For light: Laser and double-slit
  • For microwaves: Microwave transmitter and receiver with a metal plate.
  • Procedure:
  • Strings:
  • – Set the string vibrating and observe nodes (points of no displacement) and antinodes (maximum displacement).
  • Sound:
  • – Use two speakers to create an interference pattern. Walk around and observe areas of loud (constructive interference) and soft sound (destructive interference).
  • Light:
  • – Shine a laser through a double-slit and observe interference fringes.
  • Microwaves:
  • – Direct microwaves toward a metal plate to observe standing waves as regions of maximum and minimum intensity.
  • Example:
  • Measure fringe separation (x) in light interference:
  • [math]\lambda = \frac{xd}{L}[/math]
  • Where:
  • – d=slit separation,
  • – L=distance to screen.

  • IV. Determining the Wavelength of Light with a Double-Slit and Diffraction Grating

  • Objective:
  • To calculate the wavelength of light.
  • Equipment:
  • – Laser
  • – Double-slit or diffraction grating
  • – Screen
  • – Ruler
  • Procedure:
  • Set up the laser to pass through the double-slit or diffraction grating.
  • Measure the distance (L) from the slits to the screen.
  • Measure the fringe spacing (x) between bright spots.
  • For a double-slit:
  • [math]\lambda = \frac{xd}{L}[/math]
  • For a diffraction grating:
  • [math]n\lambda = d \sin\theta[/math]
  • Where d is the slit separation and θ is the diffraction angle.
  • Example:
  • If x=0.5 cm L=1.0 m d=0.1 mm x = 0.5
  • [math]\lambda = \frac{(0.5 \times 10^{-2}) (0.1 \times 10^{-3})}{1} \\
    \lambda = 500 \text{ nm}[/math]

  • V. Determining the Speed of Sound in Air by Formation of Stationary Waves in a Resonance Tube

  • Objective:
  • To measure the speed of sound in air using resonance.
  • Equipment:
  • – Resonance tube
  • – Tuning fork
  • – Water reservoir
  • Procedure:
  • – Strike the tuning fork and hold it above the tube.
  • – Adjust the water level until resonance is heard.
  • – Measure the distance (L) between successive resonance points.
  • – Calculate the speed of sound (v):
  • [math]v = fλ[/math]
  • Where:
  • – f=frequency of tuning fork.
  • – λ=2L (for the first harmonic).
  • Example:
  • [math]f = 256 \text{ Hz}, \quad L = 0.67 \text{ m} \\
    \lambda = 2 \times 0.67 = 1.34 \text{ m} \\
    v = 256 \times 1.34 = 343 \text{ m/s}[/math]

  • VI. Determining the Planck Constant Using Different Colored LEDs

  • Objective:
  • To calculate the Planck constant (h).
  • Equipment:
  • – LEDs of different colors
  • – Voltmeter
  • – Resistor
  • – Power supply
  • Procedure:
  • – Connect the LED in series with a resistor and a power supply.
  • – Gradually increase the voltage until the LED just begins to emit light (threshold voltage, V).
  • – Measure the wavelength (λ) of the emitted light.
  • – Calculate h using:
  • [math]eV = hf[/math]
  • Where:
  • [math]f = \frac{c}{\lambda}[/math]
  • ⇒ Example:
  • For an LED with [math]V = 2.0 \text{ V}, \quad \lambda = 600 \text{ nm} = 600 \times 10^{-9} \text{ m}[/math]
  • [math]f = \frac{3 \times 10^8}{600 \times 10^{-9}} \\
    f = 5 \times 10^{14} \text{ Hz} \\
    h = \frac{eV}{f} \\
    h = \frac{(1.65 \times 10^{-19}) (2)}{5 \times 10^{14}} \\
    h = 6.6 \times 10^{-34} \text{ Js}[/math]
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