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Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative .
These notes are intended to be used for the course “Advanced Quantum Mechanics”, which is part of the master programme in Applied Physics at Delft University of Technology. They are been the result of teaching activities in the field of quantum mechanics over a range of years and at different levels.
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That theory is known as quantum mechanics, and it is now the basic framework for understanding atomic, nuclear, and subnuclear physics, as well as condensed-matter (or ”solid-state”) physics.
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Derive the Heisenberg equations of motion for the creation and annihilation operators in the simple harmonic oscillator and show that their solution is. a(t) = a(0)e i!t ; ay(t) = ay(0)ei!t : (370) From these, obtain equations of motion for the position and momentum operators.
Quantum mechanics can be baffling, yes, surprising, definitely, and certainly counter-intuitive. That’s because quantum mechanics lives outside of our everyday lives and any attempt to explain quantum phenomena using classical physics fails. Quantum mechanics just is, and it’s awesome!
Key Features of Quantum Mechanics: Linearity of the Equations of Motion, Complex Numbers are Essential, Loss of Determinism, Quantum Superpositions, Entanglement (PDF) 2 Experiments with Photons: Mach-Zehder Interferometer, Elitzur-Vaidman Bombs (PDF)
Lecture 1: An overview of quantum mechanics. L1.1 Quantum mechanics as a framework. Defining linearity (17:46) L1.2 Linearity and nonlinear theories. Schrödinger’s equation (10:01) L1.3 Necessity of complex numbers (07:38) L1.4 Photons and the loss of determinism (17:20) L1.5 The nature of superposition. Mach-Zehnder interferometer (14:30)
