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Physics Concepts

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Title: Physics Concepts


1
Physics Concepts
  • Classical Mechanics
  • Study of how things move
  • Newtons 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 sizegt10 nm, use classical
    mechanics.

02
4
1-D free particle
Classical Lagrangian and Hamiltonian for free 1-D
particle
Schroedingers equation for free particle
02
5
Hydrogen Atom
Classical Lagrangian and Hamiltonian
Schroedingers equation for hydrogen
02
6
Hydrogen Atom
Schroedingers equation for hydrogen
02
7
Particle in a box
02
8
Particle in a box
N1, no match between quantum and classical
probability
N51, Averaged quantum probability approaches
classical constant probability.
02
9
Simple harmonic oscillator (SHO)
02
10
Expectation values
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
02
11
Expectation values
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
02
12
Spin Matrix
02
13
Wind up
Classical mechanics is valid for In other
words almost all of human experience and
endeavor. Use it well!
02
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