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Quantum Mechanics Exam II: Delta Function Potential, Finite Square Potential, Hermitian Operators

Review for the Quantum Mechanics Exam II covering topics such as the delta function potential, finite square potential, Hermitian operators, eigenvalues, eigenvectors, and the uncertainty principle.

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Quantum Mechanics Exam II: Delta Function Potential, Finite Square Potential, Hermitian Operators

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  1. Physics 451 Quantum mechanics I Fall 2012 Review 2 Karine Chesnel

  2. Quantum mechanics EXAM II When: Tu Oct 23 – Fri Oct 26 Where: testing center • Time limited: 3 hours • Closed book • Closed notes • Useful formulae provided

  3. Quantum mechanics EXAM II 1. The delta function potential 2. The finite square potential (Transmission, Reflection) 3. Hermitian operator, bras and kets 4. Eigenvalues and eigenvectors 5. Uncertainty principle

  4. Physical considerations Symmetry considerations Quantum mechanics Square wells and delta potentials V(x) Scattering States E > 0 x Bound states E < 0

  5. is continuous is continuous is continuous is continuous except where V is infinite y a æ ö d 2 m ( ) D = - y ç ÷ 0 h 2 dx è ø Quantum mechanics Square wells and delta potentials Continuity at boundaries Delta functions Square well, steps, cliffs…

  6. For Quantum mechanics The delta function potential

  7. Quantum mechanics The delta function well Bound state

  8. Scattering state A F B x 0 Transmission coefficient Reflection coefficient “Tunneling” Quantum mechanics Ch 2.5 The delta function well/ barrier

  9. Symmetry considerations The potential is even function about x=0 The solutions are either even or odd! Quantum mechanics The finite square well Bound state V(x) x -V0

  10. Quantum mechanics The finite square well Bound states where

  11. C,D A F B (1) (2) (3) Quantum mechanics The finite square well Scattering state (2) V(x) (1) (3) +a -a x -V0

  12. The well becomes transparent (T=1) when Quantum mechanics The finite square well V(x) x A F B -V0 Coefficient of transmission

  13. Wave function Vector Operators Linear transformation (matrix) Observables are Hermitian operators Quantum mechanics Formalism

  14. For a given transformation T, there are “special” vectors for which: is an eigenvector of T l is an eigenvalue of T Quantum mechanics Eigenvectors & eigenvalues

  15. Find the N roots Find the eigenvectors Spectrum Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a Nth polynomial in l: characteristic equation

  16. Infinite- dimensional space Inner product Orthonormality Hilbert space: functions f(x) such as Schwarz inequality Quantum mechanics Hilbert space N-dimensional space

  17. Position - momentum Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables

  18. Derived from the Heisenberg’s equation of motion Special meaning of Dt Quantum mechanics The uncertainty principle Energy - time

  19. Quantum mechanics The Dirac notation Different notations to express the wave function: • Projection on the energy eigenstates • Projection on the position eigenstates • Projection on the momentum eigenstates

  20. = inner product = matrix (operator) Operators, projectors projector on vector en Quantum mechanics The Dirac notation Bras, kets

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