Physics Concepts

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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

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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

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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.

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