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Visual Quantum Mechanics

Dive into the fascinating world of Quantum Mechanics with an exploration of the double slit experiment, particle motion, eigenstates, eigenvalues, and dynamics. Understand complex numbers visualization, free particles, boundary conditions, harmonic oscillators, scattering, and tunneling phenomena. Unravel the mysteries of eigenfunctions, quantum particles in potential wells, and computational methods for analyzing quantum systems. Engage with interactive programs and applets to deepen your understanding of quantum concepts.

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Visual Quantum Mechanics

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  1. Visual Quantum Mechanics Hans-Jürgen Korsch Fachbereich Physik TU Kaiserslautern

  2. Outline • The Quantum Double Slit experiment • Motion of Quantum Wave Packets • Eigenstates & Eigenvalues • Eigenstates & Dynamics

  3. Quantum Double Slit Experiment I One of the major paradoxes that we use to introduce Quantum Mechanics is the double slit experiment for classical particles, classical waves and quantum objects. bullets

  4. Quantum Double Slit Experiment II Particles: `bullets´ Waves:

  5. Quantum Double Slit Experiment III Quantum particles (electrons): after 10 electrons after 25 electrons Only single electrons are emitted !!! after 100 electrons after 1000 electrons

  6. Outline • The Quantum Double Slit experiment • Motion of Quantum Wave Packets • Eigenstates & Eigenvalues • Eigenstates & Dynamics

  7. Motion of Quantum Wave Packets • Visualization of complex numbers • Free particles • Boundary conditions • Harmonic oscillator • Scattering – tunneling and STM Programs from the book „Visual Quantum Mechanics“ by Bernd Thaller http://www.kfunigraz.ac.at/imawww/vqm/

  8. Outline • The Quantum Double Slit experiment • Motion of Quantum Wave Packets • Eigenstates & Eigenvalues • Eigenstates & Dynamics

  9. The are determined by the wavefunction at time Eigenstates & Eigenvalues I • Hamiltonian • Eigenfunctions • General solution `phasor´

  10. Eigenstates & Eigenvalues II • Hamiltonian • eigenvalues • eigenfunctions Hermite functions

  11. Eigenstates & Eigenvalues III • matrix representation of operators • matrix elements

  12. Eigenstates & Eigenvalues IV • quantum particle in a potential V(x): • Hamiltonian • matrix representation of position and momentum : • matrix representation of H

  13. Eigenstates & Eigenvalues V Computational recipe: • represent x and p by (nxn)-matrices (n large) • evaluate matrix H(x,p) • compute eigenvalues of matrix H simple numerical method for computation of eigenvalues and eigenfunctions

  14. Matlab Program– eigen.m - % position and momentum operator % represented in HO-eigenfunctions n=1:99; m=sqrt(n); aminus = diag(m,1); aplus = diag(m,-1); x= 1/sqrt(2)*(aminus + aplus); p=-i/sqrt(2)*(aminus - aplus); % Hamilton operator H1=0.5*p^2+0.5*x^2; % computation of eigenvalues [V,D] = eig(H1); [E,index] = sort(diag(D)); eigenvalues=E(1:10)

  15. Outline • The Quantum Double Slit experiment • Motion of Quantum Wave Packets • Eigenstates & Eigenvalues • Eigenstates & Dynamics

  16. Eigenstates & Dynamics eigenfunctions & wavefuction dynamics for several exactly solvable potentials • 1D: box potential • double well • harmonic oscillator • 2D: box potential • harmonic oscillator • 3D: hydrogen atom Applets written by Paul Falstad http://www.falstad.com/mathphysics.html

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