Review: Quantum mechanics of the harmonic oscillator
Transcript of Review: Quantum mechanics of the harmonic oscillator
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Review: Quantum mechanics of the
harmonic oscillator
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Molecular vibrations
Molecular vibrations: may involve complex motions of all atoms
E.g. vibrations of HFCO
Luckily the equations of motion can be made isomorphic with the equations of motions of a simple harmonic oscillator
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Harmonic oscillator • Normal modes (we will discuss this in detail later)
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Harmonic oscillator • Normal modes (we will discuss this in detail later)
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Harmonic oscillator • Normal modes (we will discuss this in detail later)
• Classical description
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Quantum mechanical description
Hamiltonian
Schrödinger equation
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Quantum mechanical description
Hamiltonian (explicit operators)
Schrödinger equation
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Eigenvalues
Solutions
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Eigenfunctions (explicit form):
Solutions
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Ladder operator formalism
• We will no longer deal with the explicit solutions
• Instead we use
• And we define
Annihilation (lowering) operator
Creation (raising) operator
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Ladder operator formalism
• Work out the following on your own
€
ˆ a ±, ˆ a [ ] =1
Annihilation (lowering) operator
Creation (raising) operator
(use )
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Ladder operator formalism • The following properties are useful
(raising operation)
(lowering operation)
You can show this by brute force by using the expressions for
E.g. see: Brandsen and Joachain, Introduction to Quantum Mechanics
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Why are ladder operators useful?
• We are typically interested in expressions such as
or
Plug in
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Suggested reading • More elegant solution of the quantum harmonic oscillator (Dirac’s method)
All properties of the quantum harmonic oscillator can be derived from:
€
ˆ a ±, ˆ a [ ] =1
E.g. see: Sakurai, Modern Quantum Mechanics
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Transitions induced by light
Oscillating electric field induces transitions
• Only single-quantum transitions are allowed (Δn = ±1)
Electric dipole Hamiltonian
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Transitions induced by light
Oscillating electric field induces transitions
• Only single-quantum transitions are allowed (Δn = ±1)
Electric-dipole Hamiltonian
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Transitions induced by light
Oscillating electric field induces transitions
• Only single-quantum transitions are allowed (Δn = ±1)
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Transitions induced by light
Oscillating electric field induces transitions
• Only single-quantum transitions are allowed (Δn = ±1)
operator time dependence
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For those interested
350 page derivation of the Light-matter Hamiltonian
Cohen-Tannoudji, Dupont-Roc & Grynberg
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Effect of perturbation Solve time-dependent Schrödinger equation
First order perturbation theory: Fermi’s golden rule
Bohr condition:
Ek
El
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Effect of perturbation Solve time-dependent Schrödinger equation
First order perturbation theory: Fermi’s golden rule
Ek
El
Transition probability per second
(on resonance)
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Effect of perturbation
Ek
El
Absorption probability = stimulated emission probability
Net energy absorption
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Selection rules
Classically
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Selection rules
Quantum
Operator character switches to
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Selection rules
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Selection rules
Transition dipole moment
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Selection rules
Transition dipole should be parallel to the electric field
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Selection rules
Dipole moment should change with vibration
E.g. symmetric stretch of CO2 is not infrared active
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Selection rules
Only single-quantum transitions are allowed
remember
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Selection rules Only single-quantum transitions are allowed
If this were true water would not be blue!
H2O D2O
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Selection rules Only single-quantum transitions are allowed
H2O D2O
4-quantum vibrational transition
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Multiquantum transitions
1) Anharmonic potential (mechanical anharmonicity)
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Multiquantum transitions
2) Nonlinear dependence of dipole moment (electrical anharmonicity)
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Summary ‘Quantum Harmonic Oscillator’
• Harmonic oscillator Hamiltonian
• Properties of the solutions
• Ladder operator formalism
• Transitions induced by light
• Selection rules
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Lambert-Beer law
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Molecular quantity
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Lambert-Beer law
Connects the microscopic to the macroscopic world
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Lambert-Beer law
Connects the microscopic to the macroscopic world
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Lambert-Beer law
Connects the microscopic to the macroscopic world
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Average over unit sphere
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Transition probability per second
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Transition probability per second
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Power absorbed per second by a single molecule
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Lambert-Beer law
Connects the microscopic to the macroscopic world
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Lambert-Beer law
Connects the microscopic to the macroscopic world
power light intensity = power/area
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Lambert-Beer law
Connects the microscopic to the macroscopic world
Area (=cross section)
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Lambert-Beer law
Connects the microscopic to the macroscopic world
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Lambert-Beer law
Relate energy dissipation in the slab to the in- and outgoing intensities
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Lambert-Beer law
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Lambert-Beer law
N = concentration per unit area
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Lambert-Beer law
Absorbance is linear with concentration
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Lambert-Beer law
Why do we call σ the cross section?
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Lambert-Beer law
Why do we call σ the cross section?
Number of molecules per surface area (m2)
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Lambert-Beer law
Why do we call σ the cross section?
Fraction of light blocked in slab of thickness dx
Number of molecules per surface area (m2)
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Lambert-Beer law
Why do we call σ the cross section?
If we represent a molecule by an opaque disk, it should have a surface area
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Lambert-Beer law Example
• Dye solution (1 µM, 1 cm pathlength) • Gives absorbance of 0.1
Transmission
~10% of light blocked
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Lambert-Beer law Example
• Dye solution (1 µM, 1 cm pathlength) • Gives absorbance of 0.1
~10% of light blocked
• Number of molecules per cm2 = 10-6 x 6.02·1023 x 10-3 = 6·1014
• These molecules apparently cover ~0.1 cm2
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Summary ‘Lambert Beer’
• Lambert-Beer law: connection between molecular quantity (µ= transition dipole) and macroscopic observable (σ=cross section)
• Absorbance is linear with concentration (per unit area)
• Absorption cross section can be interpreted as the physical cross section of the molecule to light