Quantum Mechanics for Applied Physics

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Quantum Mechanics for Applied Physics

• Lecture VII Interaction of matter and light
• Coherent states
• Pure States
• Interaction of TLS atom and EM field
• Dipole Interaction, selection rules
• Spontaneous emission

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Coherent States

• Goal Approximate classical state
• Definition
• Annihilation eigenvector
• Exercise Show that has no eigenvectors

unitary
Roy Glauber
complex!
note is a coherent state
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Uncertainties in coherent states

• Expectation values in energy states
• energy states dont have a good semiclassical
  limit

Roy Glauber
Completely delocalized (also in momentum space)
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Uncertainties in coherent states

• Expectation values in coherent states
• Uncertainty product

Roy Glauber
Independent of
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minimum uncertainty states!
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Dynamics and wavefunctions

• Time evolution of coherent states
• Exercise Establish wavefunction

up to phase
Coherent states persist
Roy Glauber
2005 Nobel Physics Prize!
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The density matrix

• Definition Density matrix of an isolated system
• Example
• Measurement
• Expectation value
• Dynamics

information about phase lost
opposite sign to Heisenberg picture
Can formulate QM exclusively with density matrices
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Mixed states

• Density matrix of a subsystem
• Properties
• Positivity
• Normalization
• Decoupling
• projection

Environment
System
Probability to find subsystem in state
pure state
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Mixed states Examples

• System in equilibrium with heat bath
• Partition function

Temperature T heat bath
System
Singlet state
Coupled spin 1/2
System
Envnment
Ludwig Boltzmann
Completely disordered!
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Driving by a classical source

• Hamiltonian of harmonic oscillator driven by
  classical force

Interaction picture
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Driving by a classical source

• Int. Hamiltonian
• Solution Let
• Then

Exercise Derive solution
Classical source generates coherent states
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Dipole interaction

• Long wavelength approximation
• EM wavelength system dimensions
• Gauge transformation

optical wavelength Bohr radius
Coulomb potential
constant
dipole operator
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Absorption and emission

• 2-level system in resonant
• monochromatic EM field
• Radiation-induced transition amplitude
• Absorptn/emission rate
• validity Target state is in continuous spectrum
• or rate much slower than natural width

frequency ?
polarization
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Perturbation theory in QED
Coulomb gauge Hamiltonian

• eigenstates
• single photon interaction 2-photon intrctn

atomic state
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Feynman diagrams

• Processes mediated by

absorption
emission
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Feynman diagrams

• Processes mediated by

2-photon absorption etc.
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Spontaneous emission

• 1st order amplitude
• Sponetaneous emission rate
• differential
• total

Long wavelength approx.
differential density of states
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Absorption stimulated emission
coherent light of frequency

• Normalized n-photon wavepacket
• Absorption
• Emission

amplitude of electric field
Exercise Establish exact correspondence for
the case where the field is in a coherent state
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