1 Physics Concepts Classical Mechanics Study of how things move Newton’s laws Conservation laws Solutions in different reference frames (including rotating and accelerated reference frames) Lagrangian formulation (and Hamiltonian form.) Central force problems – orbital mechanics Rigid body-motion Oscillations lightly Chaos :04

2 Mathematical Methods Vector Calculus Differential equations of vector quantities Partial differential equations More tricks w/ cross product and dot product Stokes Theorem “Div, grad, curl and all that” Matrices Coordinate change / rotations Diagonalization / eigenvalues / principal axes Lagrangian formulation Calculus of variations “Functionals” and operators Lagrange multipliers for constraints General Mathematical competence :06

3 Correlating Classical and Quantum Mechanics Correspondence Principle In the limit of large quantum numbers, quantum mechanics becomes classical mechanics. First formulated by Niels Bohr, one of the leading quantum theoreticians We will illustrate with Particle in a box Simple harmonic oscillator Equivalence principle is useful Prevents us from getting lost in “quantum chaos”. Allows us to continue to use our classical intuition as make small systems larger. Rule of thumb. System size>10 nm, use classical mechanics. :02

4 1-D free particle :02 Classical Lagrangian and Hamiltonian for free 1-D particle Schroedinger’s equation for free particle

5 Hydrogen Atom :02 Classical Lagrangian and Hamiltonian Schroedinger’s equation for hydrogen

6 Hydrogen Atom :02 Schroedinger’s equation for hydrogen

7 Particle in a box :02

8 Particle in a box :02  no match between quantum and classical probability  Averaged quantum probability approaches classical constant probability.

9 Simple harmonic oscillator (SHO) :02

10 Expectation values :02 Bra-ket notation and Matrix formulation of QM All wave functions may be written as linear combination of eigenfunctions. Thus effect of operator can be replaced by a matrix showing effect of operator on each eigenfunction. All QM operators (p, L, H) have real eigenvalues – They are “Hermitian” operators

11 Expectation values :02 Bra-ket notation and Matrix formulation of QM All wave functions may be written as linear combination of eigenfunctions. Thus effect of operator can be replaced by a matrix showing effect of operator on each eigenfunction. All QM operators (p, L, H) have real eigenvalues – They are “Hermitian” operators

12 Spin Matrix :02

13 Wind up :02 Classical mechanics is valid for In other words … almost all of human experience and endeavor. Use it well!