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„There was a time when newspapers said that only twelve men understood the theory of relativity. I do not believe that there ever was such a time... On the other hand, A very elementary approach to Quantum mechanics R.P. Feynman The Character of Physical Law (1967) I think it is safe to say that no one understands quantum mechanics“ let´s approach some aspects of qm anyway

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Experimental facts: Light has wave (interference) and particle properties Plot from Existence of photons Energy of the quantum Planck’s const. Frequency Radiation modes in a hot cavity provide a test of quantum theory

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Energy of a free particle where Consider photons andwith or Dispersion relation for light

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Electrons (particles) have wave properties Today: LowEenergyElectronDiffraction standard method in surface science Figures from applicable for “particles” de Broglie LEED Fe 0.5 Zn 0.5 F 2 (110) 232 eV top view (110)-surface

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Implications of the experimental facts Electrons described by waves: Wave function ( complex for charged particles like electrons ) Probability to find electron at (x,t) Which equation describes the temporal evolution of Schroedinger equation Erwin Schroedinger Can’t be derived, but can be made plausible Let’s start from the wave nature of, e.g., an electron: and take advantage of

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In complete analogy we find the representation of E

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Schroedinger equation for 1 free particle Hamilton function of classical mechanics ; H=E total energy of the particle 1-dimensional In 3 dimensions where

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Schroedinger equation for a particle in a potential Classical Hamilton function: Hamilton operator Time dependent Schroedinger equation If independent of time like only stationary Schroedinger equation has to be solved Proof: Ansatz: (Trial function) Stationary Schroedinger equation

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Solving the Schroedinger equation (Eigenvalue problem) Solution requires: -Normalization of the wave function according Physical meaning: probability to find the particle somewhere in the universe is 1 -Boundary conditions of the solution: have to be continuous when merging piecewise solutions Note: boundary conditions give rise to the quantization Particle in a box: x Eigenfunctions Eigenenergies Quantum number Details see homework

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Heisenberg‘s uncertainty principle It all comes down to the wave nature of particles Wave function given by a single wavelength Momentum p precisely known, but where is the particle position -P precisely given -x completely unknown

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Wave package Particle somewhere in the region Fourier-analysis Particle position known with uncertainty Particle momentum known with uncertainty Fourier-theorem In analogy

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