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principles of quantum mechanics

9/24/2011. Lecture XVII. 2. Concepts . De Broigle wavesHeisenberg's uncertainty principleSchr

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principles of quantum mechanics

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    1. 9/24/2011 Lecture XVII 1 Physics 114 Principles of Quantum Mechanics

    2. 9/24/2011 Lecture XVII 2

    3. 9/24/2011 Lecture XVII 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

    4. 9/24/2011 Lecture XVII 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+˝ ) l

    5. 9/24/2011 Lecture XVII 5 Matter waves Particle position in space cannot be predicted with infinite precision Heisenberg uncertainty principle (Wave function Y of matter wave)2 dV=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

    6. 9/24/2011 Lecture XVII 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

    7. 9/24/2011 Lecture XVII 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)=8, if x<0 –or- x>L Boundary conditions on y: Y(0)=0=Y(L)

    8. 9/24/2011 Lecture XVII 8 Particle in a box Second derivative proportional to the function with “-” sign Possible solutions: sin(kx) and cos(kx)

    9. 9/24/2011 Lecture XVII 9 Particle in a box Let’s satisfy boundary conditions

    10. 9/24/2011 Lecture XVII 10 Particle in a box Quantum number n

    11. 9/24/2011 Lecture XVII 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:

    12. 9/24/2011 Lecture XVII 12 Particle in a box

    13. 9/24/2011 Lecture XVII 13 Count knots

    14. 9/24/2011 Lecture XVII 14 Wave Function (Wave function Y of matter wave)2 dV=probability to find particle in volume dV . In 1-dimentional case probability P to find particle between x1 and x2 is Unitarity condition (probability to find particle somewhere is one): Schrödinger equation predicts wave function for a system System is defined by potential energy, boundary conditions

    15. 9/24/2011 Lecture XVII 15 Wave function ?Probability

    16. 9/24/2011 Lecture XVII 16 Symmetry considerations

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