DP IB Physics: SL
E. Nuclear and Quantum Physics
E.2 Quantum Physics
DP IB Physics: SLE. Nuclear and Quantum PhysicsE.2 Quantum PhysicsUnderstandings | |
|---|---|
| a) | the photoelectric effect as evidence of the particle nature of light |
| b) | that photon of a certain frequency, known as the threshold frequency, are required to release photoelectrons from the metal |
| c) | Einstein’s explanation using the work function and the maximum kinetic energy of the photoelectrons as given by [math]E_{\text{max}} = h f – \Phi[/math] where Φ is the work function of the metal |
| d) | diffraction of particles as evidence of the wave nature of matter |
| e) | that matter exhibits wave–particle duality |
| f) | the de Broglie wavelength for particles as given by [math]\lambda = \frac{h}{p}[/math] |
| g) | Compton scattering of light by electrons as additional evidence of the particle nature of light |
| h) | that photons scatter off electrons with increased wavelength |
| i) | the shift in photon wavelength after scattering off an electron as given by
[math]\lambda_f – \lambda_i = \Delta \lambda = \frac{h}{m_e c} (1 – \cos \theta)[/math] |
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a) The Photoelectric Effect: Evidence for the Particle Nature of Light
- The photoelectric effect:
- The photoelectric effect is the emission of electrons (called photoelectrons) from the surface of a metal when it is exposed to light (electromagnetic radiation) of a certain frequency or higher.
- ⇒ Observations:
- Electrons are emitted instantly when light hits the metal — there is no time delay.
- There is a minimum frequency (threshold frequency) below which no electrons are emitted, regardless of the light intensity.
- Increasing the intensity of light increases the number of electrons emitted, not their energy.
- Higher frequency light results in more energetic electrons being ejected.
- Figure 1 Photoelectric effect
- Surprising:
- Classical wave theory of light predicted:
- – Energy should depend on intensity (brightness), not frequency.
- – Given enough time, any frequency should eventually emit electrons.
- But this was not observed. These results could not be explained using the wave theory.
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b) Einstein’s Explanation: Light Acts Like Particles (Photons)
- Einstein proposed that light is made of packets of energy called photons.
- Each photon has energy given by:
- [math]E = hf[/math]
- Where:
- – E is the energy of the photon,
- – h is Planck’s constant (63×10−34 Js),
- – f is the frequency of the light.
- Figure 2 Einstein’s experiment
- ⇒ The Threshold Frequency:
- To eject an electron, a photon must have at least the minimum energy required to break it free from the metal. This energy is called the work function, Φ
- The threshold frequency [math]f_o[/math] is the frequency where:
- [math]hf_0 = Φ[/math]
- If [math]f < f_0[/math], no photoelectrons are emitted.
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c) Maximum Kinetic Energy of the Photoelectrons
- If a photon’s energy is greater than the work function, the excess energy becomes kinetic energy of the ejected electron.
- Einstein’s photoelectric equation:
- [math]E_{\text{max}} = h f – \Phi[/math]
- Where:
- – [math]E_{\text{max}}[/math] is the maximum kinetic energy of the photoelectron,
- – hf is the energy of the photon,
- – Φ is the work function of the metal.
- ⇒ Important Points:
- The kinetic energy of emitted electrons depends on the frequency of the incident light, not its intensity.
- No electrons are emitted if [math]f < f_0[/math], regardless of intensity.
- More intense light means more photons, which can eject more electrons, but doesn’t increase their energy.
- Figure 3 Photoelectric effect
- Graphical Representation:
- A graph of [math]E_{max}[/math] vs frequency f shows:
- – A straight line with slope = h,
- – x-intercept at [math]f_o = f[/math],
- – y-intercept at [math]- Φ[/math].
- Real-World Significance:
- This experiment was crucial in establishing quantum theory.
- Demonstrated that light has both wave and particle properties — foundation of wave-particle duality.
- Earned Albert Einstein the Nobel Prize in Physics (1921).
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d) Wave–Particle Duality
- Wave–Particle Duality:
- In classical physics:
- – Light was thought to be a wave (based on diffraction, interference).
- – Particles (like electrons) were considered to have only mass and momentum, with no wave properties.
- However, modern quantum physics reveals:
- All matter and radiation exhibit both wave-like and particle-like properties, depending on how they are observed. This is called wave–particle duality.
- Figure 4 Wave-Particle duality
- ⇒ Examples of Duality:
| Entity | Particle Behavior | Wave Behavior |
|---|---|---|
| Light | Photoelectric effect, Compton effect | Interference, diffraction |
| Electrons | Collisions, charge, mass | Electron diffraction (e.g., by crystals) |
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e) Diffraction of Particles: Wave Nature of Matter
- ⇒ Experiment: Electron Diffraction
- When a beam of electrons passes through a thin crystal, such as graphite, a diffraction pattern is observed.
- This diffraction pattern (rings or fringes) is identical to what we expect from waves passing through a grating or crystal lattice.
- Figure 5 Diffraction of particle
- Conclusion:
- Electrons interfere and diffract, confirming they behave like waves in certain conditions.
- This experiment is direct evidence of the wave nature of matter.
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f) de Broglie Hypothesis
- In 1924, Louis de Broglie proposed that particles (like electrons, protons, even atoms) also have an associated wavelength, just like photons.
- He derived this wavelength using Planck’s constant:
- [math]\lambda = \frac{h}{p}[/math]
- Where:
- – λ is the de Broglie wavelength,
- – h is Planck’s constant (63×10−34Js),
- – p is the momentum of the particle ([math][/math]).
- Figure 6 de Broglie hypothesis
- ⇒ Implications:
- Faster (more massive or higher velocity) particles have shorter wavelengths.
- Lighter or slower particles (like electrons) have longer wavelengths, making their wave behavior easier to observe.
- ⇒ Example Calculation:
- If an electron moves with speed [math]v = 1 × 10^6 m/s[/math], and has mass m=9.11 × 10−31kg, then:
- [math]p = mv \\
p = (9.11 \times 10^{-31}) \times (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] - This is about the size of an atom → why electron diffraction works with crystals.
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g) Compton Scattering:
- Compton scattering is the phenomenon where X-rays or gamma rays (high-energy photons) collide with free or loosely bound electrons, causing the photons to scatter and change direction — and importantly, to lose energy.
- First observed by:
- – Arthur Compton in 1923.
- – His experiment showed that after interacting with electrons, light behaves like a particle carrying momentum and energy.
- Figure 7 Compton effect
- ⇒ Description of the Phenomenon
- Before collision:
- A high-energy photon (e.g. an X-ray) travels toward an electron.
- The electron is either at rest or moving slowly (often treated as initially at rest).
- ⇒ During collision:
- The photon transfers energy and momentum to the electron — like a billiard ball collision.
- After collision:
- The electron recoils, gaining kinetic energy.
- The photon scatters at an angle θ and comes out with:
- – Less energy
- – Longer wavelength
- – Reduced frequency
- This shows that photons have momentum and can “bounce off” particles like tiny physical objects — a key particle-like behavior.
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h) Compton Wavelength Shift Formula
- The change in wavelength of the photon after scattering is given by:
- [math]\Delta \lambda = \lambda_f – \lambda_i \\
\Delta \lambda = \frac{h}{m_e c} \left(1 – \cos \theta \right)[/math] - Where:
| Symbol | Meaning |
|---|---|
| [math]λ_f[/math] | Final wavelength of the photon after scattering |
| [math]λ_i[/math] | Initial wavelength of the photon before scattering |
| h | Planck’s constant ≈ 6.63 × 10−34Js |
| [math]m_e[/math] | Mass of the electron ≈ 9.11 × 10−31kg |
| c | Speed of light ≈ 3.00 × 108m/s |
| θ | Scattering angle (angle between incoming and outgoing photon direction) |
- ⇒ Results:
- If [math]θ = 0^0[/math]: No wavelength shift (photon continues straight, no energy lost).
- If [math]θ = 180^0[/math]: Maximum shift — photon bounces directly backward.
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i) Compton Wavelength Constant
- The factor [math]\lambda_C = \frac{h}{m_e c}[/math] is called the Compton wavelength of the electron, denoted:
- [math]\lambda_C = \frac{h}{m_e c} \\
\lambda_C \approx 2.43 \times 10^{-12} \ \text{m}[/math] - This is a fundamental constant and represents the maximum possible wavelength shift in Compton scattering.
- ⇒ Evidence for Particle Nature of Light
- Compton scattering could not be explained by wave theory of light.
- – Classical wave theory predicted that energy would be spread across the whole wavefront and no specific momentum transfer would occur.
- – But experimental results showed individual photons transferring specific amounts of energy and momentum, as particles
- Figure 8 Compton effect
- Conclusion:
- – Compton scattering confirmed that photons have momentum, and hence light behaves like a particle in interactions with matter.