# Quantum Physics Study Notes ## Learning Objectives By the end of this chapter, you should be able to: - Understand the nature of photons and their energy. - Explain the photoelectric effect and its implications. - Understand why the photoelectric effect cannot be explained by the wave theory of light. - Describe Compton scattering and its significance. - Explain the concept of matter waves and calculate the de Broglie wavelength. - Describe experiments that demonstrate the wave nature of electrons. --- ## 22.1 Photons and the Photoelectric Effect ### **Photons: The Particle Nature of Light** Light is traditionally understood as an electromagnetic wave, with: - **Wavelength** ($\lambda$) - **Frequency** ($f$) - **Speed of light** ($c$), related by $c = f\lambda$ However, phenomena like the **photoelectric effect** challenge this wave-only model. Einstein proposed that light consists of discrete packets of energy called **photons**. Each photon has energy: $$E = hf$$ Where: - $h$ = Planck’s constant ($6.63 \times 10^{-34} \, \text{Js}$) - $f$ = frequency of light #### Key Properties of Photons: - **Energy**: $E = hf$ - **Momentum**: $p = \frac{E}{c} = \frac{h}{\lambda}$ - **Mass**: Photons have zero rest mass. **Example**: A photon of blue light ($\lambda = 450 \, \text{nm}$) has more energy than a photon of red light ($\lambda = 700 \, \text{nm}$) because $E = hf$ and blue light has a higher frequency. --- ### **The Photoelectric Effect** The photoelectric effect occurs when light incident on a metallic surface causes electrons to be emitted. Key observations include: 1. **Stopping Voltage ($V_s$)**: The voltage required to stop the emitted electrons. 2. **Critical Frequency ($f_c$)**: Below this frequency, no electrons are emitted, regardless of light intensity. #### Experimental Setup: - **Photo-surface**: Emits electrons when light strikes it. - **Collecting Plate**: Measures the current produced by emitted electrons. - **Variable Power Supply**: Adjusts the voltage to control electron flow. #### Key Relationships: - Maximum kinetic energy of emitted electrons: $$E_{\text{max}} = eV_s$$ Where: - $e$ = charge of an electron ($1.6 \times 10^{-19} \, \text{C}$) - $V_s$ = stopping voltage #### Einstein’s Explanation: Einstein proposed that: - A single photon transfers its energy ($hf$) to a single electron. - The electron uses part of this energy to escape the metal (work function $\phi$), and the rest becomes its kinetic energy: $$E_K = hf - \phi$$ **Tip**: The work function ($\phi$) is the minimum energy required to free an electron from the metal. #### Key Observations: 1. **Intensity does not affect electron energy**: Higher intensity increases the number of emitted electrons but not their energy. 2. **Frequency determines energy**: Higher frequency light produces electrons with greater kinetic energy. 3. **Threshold frequency ($f_c$)**: Below $f_c$, no electrons are emitted, regardless of intensity. 4. **No time delay**: Electrons are emitted instantaneously when light strikes the surface. **Common Mistake**: Confusing the role of intensity and frequency. Intensity affects the number of electrons emitted, while frequency affects their energy. --- ### **Worked Examples** #### **Example 1**: Calculating Photon Emission A 60 W lamp emits photons of wavelength $5.0 \times 10^{-7} \, \text{m}$. If 10% of the lamp’s energy goes into photons of this wavelength, calculate the number of photons emitted per second. **Solution**: Energy emitted per second = $6.0 \, \text{J}$ Energy of one photon = $E = \frac{hc}{\lambda}$ Number of photons = $\frac{\text{Total Energy}}{\text{Energy per Photon}}$ $$N = \frac{6.0}{\frac{6.63 \times 10^{-34} \times 3.0 \times 10^8}{5.0 \times 10^{-7}}} = 1.5 \times 10^{19} \, \text{photons/second}$$ --- ## 22.2 Compton Scattering ### **What is Compton Scattering?** Compton scattering occurs when photons collide with electrons, transferring energy and momentum. The scattered photon has a longer wavelength than the incident photon. #### Key Formula: The change in wavelength ($\Delta \lambda$) is given by: $$\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)$$ Where: - $m_e$ = mass of the electron ($9.11 \times 10^{-31} \, \text{kg}$) - $\theta$ = scattering angle #### Observations: - Larger scattering angles result in greater wavelength shifts. - The **Compton wavelength** ($\frac{h}{m_e c}$) is approximately $2.4 \times 10^{-12} \, \text{m}$. **Example**: A photon of energy $54 \, \text{keV}$ scatters off an electron at an angle of $82^\circ$. Shift in wavelength: $$\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta) = 2.4 \times 10^{-12} \times (1 - \cos 82^\circ) = 2.1 \times 10^{-12} \, \text{m}$$ **Note**: Compton scattering provides strong evidence for the particle nature of light. --- ## 22.3 Matter Waves ### **The de Broglie Hypothesis** Louis de Broglie proposed that particles, like electrons, exhibit wave-like properties. The wavelength associated with a particle is: $$\lambda = \frac{h}{p}$$ Where: - $p$ = momentum of the particle #### Key Concept: Wave-Particle Duality Particles can exhibit wave phenomena like **diffraction** and **interference**. **Example**: An electron accelerated through a potential difference of $54 \, \text{V}$ has a de Broglie wavelength: $$\lambda = \frac{h}{\sqrt{2 m_e q V}} = \frac{6.63 \times 10^{-34}}{\sqrt{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19} \times 54}} = 1.7 \times 10^{-10} \, \text{m}$$ --- ### **Experimental Evidence** #### **Davisson-Germer Experiment** Electrons were scattered off a nickel crystal, producing diffraction patterns consistent with wave behavior. The observed wavelength matched the de Broglie prediction. #### **Bohr Model Connection** The allowed orbits in the Bohr model correspond to integer multiples of the electron’s de Broglie wavelength: $$2\pi r = n\lambda$$ **Tip**: The de Broglie wavelength is significant for small particles like electrons but negligible for macroscopic objects. --- ## Key Takeaways - **Photons**: Light behaves as particles with discrete energy and momentum. - **Photoelectric Effect**: Demonstrates the particle nature of light and the role of frequency in electron emission. - **Compton Scattering**: Provides further evidence for photons as particles. - **Matter Waves**: Particles exhibit wave-like properties, as shown by diffraction and interference experiments. --- ## Practice Problems 1. Calculate the energy of a photon with wavelength $450 \, \text{nm}$. 2. A beam of light of intensity $25 \, \text{W/m}^2$ and wavelength $620 \, \text{nm}$ strikes a surface. How many photons are incident per second per unit area? 3. Determine the de Broglie wavelength of a proton with kinetic energy $200 \, \text{MeV}$. **Reflection**: Can you explain the photoelectric effect and its implications for the nature of light? How does the de Broglie hypothesis connect particles and waves?