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Lecture 9: Macroscopic Quantum Model Outline 1. Development of Quantum Mechanics 2. Schrödinger's Equation • • • 3. Physical meaning of Wavefunction • 4. Free particle With forces With Electromagnetic force Probability density and Probability Current density Macroscopic Quantum Model October 2, 2003 Massachusetts Institute of Technology 6.763 2003 Lecture 9 Macroscopic Quantum Model Superconductivity is a Quantum Phenomenon on a macroscopic length scale. Massachusetts Institute of Technology 6.763 2003 Lecture 9 Development of Quantum Mechanics Free Particle (no forces or potentials) Wave-like properties frequency wavenumber Particle-like properties energy momentum Planck’s constatnt Combine wave and particle properties, to find the dispersion relation: Massachusetts Institute of Technology 6.763 2003 Lecture 9 Schrödinger's Equation (free particle) Assume that this results from a uniform plane wave solution Then a good guess of the differential equation that gives the dispersion relation is This guess is justified by experimental confirmation; this is not a derivation. Massachusetts Institute of Technology 6.763 2003 Lecture 9 Schrödinger's Equation (with forces) We present a plausibility argument, not a derivation, relating the classical formulation to the quantum formulation. The energy for a particle in a force is, classically, Energy is conserved since the potential is independent of time. 0 Massachusetts Institute of Technology 6.763 2003 Lecture 9 Canonical Momentum & Schrödinger's Equation V Here, the canonical momentum equals the kinematic momentum; and the generalized potential, the scalar potential. Schrödinger's Equation Massachusetts Institute of Technology 6.763 2003 Lecture 9 Rules for Classical to Quantum Massachusetts Institute of Technology 6.763 2003 Lecture 9 The wave function Ψ : Real or Complex? Compare the plane wave solutions for QM and E&M Ψ must be a complex function; it is not a mathematical convenience. What does it mean to have a complex wave function? H is a real function; it is only a mathematical convenience to consider it a complex function. The real part must be taken. Massachusetts Institute of Technology 6.763 2003 Lecture 9 The physical meaning of the wave function Ψ The absolute phase of a plane wave should not influence the overall physics of a system. So Max Born hypothesized in ~1927 that the square of the magnitude of the wave function Ψ was equal to the probability of a quantum mechanical particle to be at the location r at time t. With the normalization condition (particle must be somewhere) Massachusetts Institute of Technology 6.763 2003 Lecture 9 Evolution of Probability Multiply the S-Eqn by Ψ* and (S-Eqn)* by Ψ - Massachusetts Institute of Technology 6.763 2003 Lecture 9 Probability Current Therefore we find that the probability and the probability current satisfy a continuity relation Massachusetts Institute of Technology 6.763 2003 Lecture 9 Schrödinger's Equation with E&M Fields For a charged particle, we want the classical equations such that Start with the Lorentz Force Law and use the vector and scalar potentials and to find Massachusetts Institute of Technology 6.763 2003 Lecture 9 Canonical Momentum and Energy V where the canonical momentum is Kinematic momentum Field momentum The energy follows from the above to be Massachusetts Institute of Technology 6.763 2003 Lecture 9 Classical to Quantum The energy can be written as The transition to quantum mechanics is done as before and The S-Eqn becomes With probability current Massachusetts Institute of Technology 6.763 2003 Lecture 9 Macroscopic Quantum Model 1. The wave function describes the whole ensemble of superelectrons such that density and Total number 2. The flow of probability becomes the flow of particles, with the physical current density given by Massachusetts Institute of Technology 6.763 2003 Lecture 9 MQM cont. 3. This macroscopic quantum wavefunction follows Writing , we find The Supercurrent Equation Massachusetts Institute of Technology 6.763 2003 Lecture 9 The Supercurrent Equation Js is a unique physical quantity, but A and θ are not. If a new vector and scalar potential are found in another gauge such that Then and B, E, and J are gauge invariant, A, φ, and θ are not. Massachusetts Institute of Technology 6.763 2003 Lecture 9