AS

Ch 6 Lecture 2

Quantum Mechanics Lecture Notes

Photoelectric Effect

  1. Max Planck and Quantized Energy

    • Introduced to solve the blackbody radiation problem.

  2. Albert Einstein's Contribution

    • Worked on the photoelectric effect, crucial for photovoltaic cells.

  3. Planck's Equation for Energy

    • Energy (E) is quantized: E = H u, where:

      • H = Planck's constant (6.63\cdot10^{-34}{ J s} )

      • u = frequency of light

    • Light incident on a metal ejects electrons by converting light to electrical energy (similar to solar panels).

  4. Kinetic Energy of Ejected Electrons

    • Ejected electrons have kinetic energy dependent on the energy of the light and metal's work function (\phi ).

    • Einstein's equation: E=K+\phi , where:

      • K = kinetic energy of ejected electrons

      • E = incident light energy

      • \phi = work function (how tightly electrons are held in the atom)

      • Example: E=K+\phi , or K=E-\phi .

    • Notation: φ = work function represented by a zero with a line through it, the Greek letter phi.

Graphical Representation

  1. Graph of Energy vs. Frequency

    • Straight line graph; E = H
      u is the equivalent to y = mx + b.

    • Slope of graph is Planck's constant (H).

    • Frequency (ν) correlates to the amount of energy needed to eject electrons from different materials.

Problem Solving: Kinetic Energy Calculations

  1. Question Example: For cerium (work function = 2.14 eV):
    a. Light wavelength = 750 nm,
    b. Light wavelength = 250 nm.

  2. Calculating Kinetic Energy for 750 nm Light

    • Use equation: K=E-\phi .

    • Total energy calculated as: E = rac{H imes C}{ ext{λ}} where:

      • H = Planck's constant, C = speed of light, ext{λ} = wavelength converted to meters.

    • For {λ}=750{ nm} :

      • Convert: 1{ m}=1\cdot10^9ext{ nm}

      • Energy = {(6.63\cdot10^{-34}{ J s})(3\cdot10^8{ m/s})}{750\cdot10^{-9}{ m}}=2.65\cdot10^{-19}{ J}

      • Work function \phi=2.14{ eV} = 2.14\cdot1.62\cdot10^{-19}{ J}=3.47\cdot10^{-19}{ J}

    • Therefore, K=2.65\cdot10^{-19}{ J}-3.47\cdot10^{-19}{ J}=-0.82\cdot10^{-19}{ J} (no ejection).

  3. Calculating Kinetic Energy for 250 nm Light

    • Follow previous steps with a new wavelength of 250 nm:

    • Energy computed as: E = rac{H imes C}{ ext{λ}} = 7.956 imes 10^{-19} ext{ J}.

    • K = 7.956 imes 10^{-19} ext{ J} - 3.47 imes 10^{-19} ext{ J} = 4.486 imes 10^{-19} ext{ J} (successful ejection).

  4. Calculating Speed of Ejected Electrons

    • Using equation: K = rac{mv^2}{2}, solve for v:
      v = ext{sqrt} rac{2K}{m}.

    • Ejected speed calculation gives v
      ightarrow 9.97 imes 10^5 ext{ m/s}.

    • Important note: joules represented in ext{kg} ext{m}^2/ ext{s}^2.

Quantum Mechanics Concepts

Particle in a Box Model

  1. Fundamentals

    • The particle can be thought of as an electron confined in an atom.

    • Exhibits wave and particle duality; behaves as matter waves when confined.

  2. Wave Function and Wavelength

    • Wavelength equation: {λ}={2L}{n} , where:

      • L = length of the box, n = quantum number (1, 2, 3, …).

    • Explanation of states based on varying n values.

  3. Quantizing of Energy

    • Energy expression in a box: E = rac{n^2 H^2}{8mL^2}.

    • Notable that as n increases, energy quantization increases, aligned with the nature of quantum mechanics.

    • Relation to the ground state and excited states within quantized systems.

Louis de Broglie's Hypothesis

  1. Matter Waves and Momentum

    • Wavelength derived from: ext{λ} = rac{H}{p} (momentum p = mv).

    • Links between particles' movement and their wave properties described in quantum mechanics.

Energy Calculation Based on Box Lengths

  1. Example Calculations for Energy Differences

    • Analyze energy in confined versus unconfined states using distinct box lengths (1 Å versus 30 cm), leading to significant inferences about wave-like versus particle-like behavior.

Heisenberg's Uncertainty Principle

  1. Definition and Implication

    • Equation: ext{Δx Δp} ≥ rac{ ext{ħ}}{2}.

    • Indicates the fundamental limits to measuring position ( ext{Δx}) and momentum ( ext{Δp}) simultaneously.

    • Stresses the limitations of classical mechanics in quantum systems.

  2. Classical Mechanics Limitation

    • Classical view of electrons spiraling into nuclei leading to atom collapse refuted by quantum mechanical principles and stable energy levels.

  3. The Importance of Uncertainty

    • Ties back to the necessity of quantum mechanics for accurate atomic models and behaviors.

Conclusion

  1. Reinforcement of Concepts

    • Strong emphasis on wave-particle duality, quantization of energy levels, and the critical implications of Heisenberg's principle.

    • Review of successful learning experiences from problem-solving through practical applications of the discussed principles.

  2. Study Recommendations

    • Students are encouraged to repeatedly review the material to solidify understanding, especially focusing on key equations, principles, and their applications in problem-solving.