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Principles of Quantum Mechanics

Principles of Quantum Mechanics. Physics 123. Concepts. De Broigle waves Heisenberg’s uncertainty principle Schrödinger’s equation Particle in a box Boundary conditions Unitarity condition. Wave – Particle duality.

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Principles of Quantum Mechanics

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  1. Principles of Quantum Mechanics Physics 123 Lecture XV

  2. Concepts • De Broigle waves • Heisenberg’s uncertainty principle • Schrödinger’s equation • Particle in a box • Boundary conditions • Unitarity condition Lecture XV

  3. Wave – Particle duality • If light exhibits both wave and particle properties then particles (e.g. electrons) must also exhibit wave properties – e.g. interference. • Matter (de Broglie) waves l=h/p p=mv Lecture XV

  4. Interference of electrons • Send electron beam (a lot of electrons) on crystal structure • Interference pattern is determined by l=h/p • Double slits distance d~1nm • Interference pattern • Maxima (more e): d sinq = m l m=0,1,2,3,…. • Minima (no e): d sinq = (m+½ ) l Lecture XV

  5. Matter waves • Particle position in space cannot be predicted with infinite precision • Heisenberg uncertainty principle • (Wave function Y of matter wave)2dV=probability to find particle in volume dV. • Laws of quantum mechanics predict Y for a given system • Given Y one can estimate probability for certain outcomes of experiment Lecture XV

  6. Schrödinger’s equation • Equivalent of energy conservation equation in classical mechanics. • Predicts the shape of the wave function. • System is defined by potential energy, boundary conditions Lecture XV

  7. Particle in a box • Infinite potential well • Particle mass m in a box length L • U(x)=0, if 0<x<L, • U(x)=∞, if x<0 –or- x>L • Boundary conditions on y: • Y(0)=0=Y(L) Lecture XV

  8. Particle in a box • Second derivative proportional to the function with “-” sign • Possible solutions: sin(kx) and cos(kx) k-wave vector Lecture XV

  9. Particle in a box • Let’s satisfy boundary conditions Wave vector is quantized! Lecture XV

  10. Particle in a box • Quantum number n Energy is quantized! We are not done yet, We don’t know A Lecture XV

  11. Particle in a box • We know for sure that the particles is somewhere in the box • Probability to find the particle in 0<x<L is 1: • Unitarity condition: Lecture XV

  12. Particle in a box Lecture XV

  13. Count knots 2 knots in the box (N-1) knots in the box 0 knots in the box 1 knot in the box Lecture XV

  14. Wave function Probability Probability to find particle between x and x+dx Probability to find particle between x1 and x2 Probability density Lecture XV

  15. Symmetry considerations The observable: probability density must respect the symmetry of the system System is symmetric around x=L/2 This means that the probability density function does not change if I replace x with (L-x): Odd harmonics have even number of knots and are symmetric Even harmonics have odd number of knots and are antisymmetric Wave function is either symmetric or antisymmetric Lecture XV

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