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Foundations of Quantum Mechanics: Photoelectric Effect, Tunneling, and Wavefunctions

This unit delves into key concepts of quantum mechanics, including the photoelectric effect, Compton effect, and matter waves. It covers the uncertainty principle and the Schrödinger equation in one dimension, focusing on square well potentials and quantum tunneling. Students will learn to solve the Schrödinger equation, apply boundary conditions, and normalize wavefunctions to determine allowed energies and probability densities. The course also compares finite and infinite wells and explores tunneling phenomena, reflection, and transmission of electron waves.

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Foundations of Quantum Mechanics: Photoelectric Effect, Tunneling, and Wavefunctions

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  1. Quantum mechanics unit 1 • Foundations of QM • Photoelectric effect, Compton effect, Matter waves • The uncertainty principle • The Schrödinger eqn. in 1D • Square well potentials and 1D tunnelling • The harmonic oscillator www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

  2. Last time • Solving the Schrödinger equation (for given ) • Examine the physics • Write down the solution to the S.E. (which will generally contain some arbitrary constants) • Apply the boundary conditions and normalise the wavefunctionto find the unknown constants • Find the allowed energies and the probability density

  3. Finite square well www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

  4. Graphical solution: Even parity states

  5. Graphical solution: Odd parity states

  6. Compare infinite to finite well Well half width, Å, Finite well depth, eV Infinite well eV eV eV … Finite well eV eV eV eV

  7. Quantum tunnelling

  8. Quantum tunnelling

  9. Tunnelling of classical waves Tippler – 35.4 Reflection and transmission of electron waves: Barrier penetration

  10. Tunnelling of classical waves

  11. Tunnelling through a barrier

  12. Tunnelling through a barrier • Write down solutions to S.E. • Apply boundary conditions at , • Eliminate coefficients - see notes at www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures • Find

  13. Å eV

  14. Tunnelling through a barrier • Transmission probability is • Large barrier, then • General case if

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