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Chapter 5: Quantum Mechanics Limitations of the Bohr atom necessitate a more general approach

Chapter 5: Quantum Mechanics Limitations of the Bohr atom necessitate a more general approach de Broglie waves –> a “new” wave equation “probability” waves classical mechanics as an approximation Wave Function Y probability amplitude. Mathematical properties of the wave function.

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Chapter 5: Quantum Mechanics Limitations of the Bohr atom necessitate a more general approach

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  1. Chapter 5: Quantum Mechanics • Limitations of the Bohr atom necessitate a more general approach • de Broglie waves –> a “new” wave equation • “probability” waves • classical mechanics as an approximation • Wave Function Y • probability amplitude

  2. Mathematical properties of the wave function

  3. More mathematical properties of the wave function

  4. The classical wave equation as an example of a wave equation:

  5. Time dependent Schrödinger Equation • linear (in Y) partial differential equation

  6. Expectation values (average values)

  7. If the potential energy U is time independent, • Schrödinger equation can be simplified by “factoring” • separation of variables • Total energy can have a constant (and well defined) value • Consider plane wave: An eigenvector, eigenvalue problem!

  8. The time independent Schrödinger equation • Allowed values for (some) physical quantities such as energy are related to the eigenvalues/eigenvectors of differential operators • eigenvalues will depend on the details of the wave equation (especially in U) and on the boundary conditions

  9. U Particle in a box: (infinite) potential well L V0 x

  10. Wavefunction normalization

  11. Example 5.3 Find the probability that a particle trapped in a box L wide can be found between .45L and .55L for the ground state and for the first excited state. Example 5.4 Find <x> for a particle trapped in a box of length L

  12. U Particle in a box: finite potential well L V0 E x I II III

  13. U Boundary Conditions L V0 E x I II III

  14. c = 100 c = 1600 c = 4

  15. Tunneling U L V0 E x I II III

  16. U Boundary Conditions L V0 E x I II III

  17. Example 5.5: Electrons with 1.0 eV and 2.0 eV are incident on a barrier 10.0 eV high and 0.50 nm wide. • (a) Find their respective transmission probabilities. • (b) How are these affected if the barrier is doubled in width?

  18. Harmonic Oscillator: classical treatment

  19. Quantum Oscillator

  20. Quantum Harmonic Oscillator Solutions

  21. Operators

  22. Example 5.6: An eigenfunction of the operator d 2 /dx 2 is y = e2x. Find the corresponding eigenvalue.

  23. Chapter 5 exercises: 4, 5, 6, 11, 23

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