Title: Physics Concepts
1Physics 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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2Mathematical 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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3Correlating 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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41-D free particle
Classical Lagrangian and Hamiltonian for free 1-D
particle
Schroedingers equation for free particle
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5Hydrogen Atom
Classical Lagrangian and Hamiltonian
Schroedingers equation for hydrogen
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6Hydrogen Atom
Schroedingers equation for hydrogen
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7Particle in a box
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8Particle in a box
N1, no match between quantum and classical
probability
N51, Averaged quantum probability approaches
classical constant probability.
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9Simple harmonic oscillator (SHO)
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10Expectation 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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11Expectation 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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12Spin Matrix
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13Wind up
Classical mechanics is valid for In other
words almost all of human experience and
endeavor. Use it well!
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