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Lecture 2: Fundamental Principles of Quantum Mechanics Website with slides and handouts: http://home.uchicago.edu/ matthewroberts/compton/index.html Black body radiation Light comes in individual particles, explaining deviation from Rayleigh-Jeans law at short wavelengths - waves are particles! Double-slit experiment Experiments confirm that electron behave like waves and interfere with themselves - particles are waves! Schrödinger Wave Function ψ(t, x) is a complex-valued function of space and time. (Recall complex numbers have real and imaginary part, or a magnitude and phase, z = x + iy = reiθ .) The magnitude of the wave function tells you the probability distribution of where you might find the particle, ψ ∗ (x)ψ(x) = |ψ(x)|2 R= P (x). Since the sum of all probabilities must be one, the amplitude of the wave is fixed, |ψ(x)|2 = 1. Schrödinger’s Equation 2 ~ Schrödinger’s equation, i~∂t ψ = Hψ = − 2m ∂x2 ψ + V (x)ψ, tells you how the wave function 2 ~ evolves in time. − 2m ∂x2 is the kinetic energy and V is the potential energy, and therefore H is the total energy in the system. The phase of the wave function oscillates at a frequency fixed by the energy, ω = E/~. Expectation Values You can R calculate the statistical Raverage for different observations, for instance position hxi = ψ ∗ xψdx or energy hEi = ψ ∗ Hψdx. Energy Spectrum By separation of variables, ψ(t, x) = ψ(x)e−iEt/~ , we can calculate the energy levels of a system. For instance for a Hydrogen atom, we can find that there are energy levels labelled by an integer n with energy En = −1 Ry × n12 , 1 Ry = 13.6 eV. This allows us to predict the energy of photons emitted when electrons transition, Ephoton = 1 Ry × 1 n21 − 1 n22 . Uncertainy Principle You can’t simultaneously know the exact position and and momentum of a quantum object: ∆x · ∆p ≥ ~. You also can’t measure the energy of a state precisely in a short amount of time, ∆E · ∆t ≥ ~. Electron spin and entanglement Electrons have spin 12 and therefore can be spinning up or down (|+i or |−i). We say a state made of more than one particle is entangled if the state is not a trivial product state: | + +i is not entangled, as each particle is spin up regardless of the other. The state | + −i + | − +i is entangled, as if we measure the first electron to be spin up the second one must then be spin down, and vice versa. While this is “spooky action at a distance”, you can not send information faster than the speed of light with it.