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Lect19_SHOintro

# Lect19_SHOintro - Lecture 19 Quantization of the simple...

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Lecture 19: Quantization of the simple harmonic oscillator Phy851 Fall 2009

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Systems near equilibrium The harmonic oscillator Hamiltonian is: Or alternatively, using Why is the SHO so important? Answer: any system near a stable equilibrium is equivalent to an SHO 2 2 2 2 1 2 X m m P H ω + = 2 2 2 1 2 kX m P H + = m k = ω A Random Potential Stable equilibrium points Definition of stable equilibrium point: 0 ) ( 0 = x V Expand around x 0 : K + + + = 2 0 0 0 0 0 ) )( ( 2 1 ) )( ( ) ( ) ( x x x V x x x V x V x V 0 x x y = 2 0 2 1 ) ( y k V y V + =
Analysis of energy and length scales The parameters available in the SHO Hamiltonian are: The frequency defines a quantum energy [J] scale via: The frequency also defines a quantum length scale via: This length scale then defines a quantum momentum scale: osc m ω , , h 2 2 2 2 1 2 X m m P H osc ω + = osc osc E ω h = 2 2 osc osc m E λ h = 2 2 osc osc m λ ω h h = osc osc m ω λ h = osc osc λ μ h = osc osc m ω μ h = The SHO introduces a single new parameter, which must govern all of the physics

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Dimensionless Variables To solve the QM SHO it is very useful to introduce the natural units: – Let osc X X λ = osc H H ω h = 2 2 2 2 2 2 2 1 2 1 X m P m H osc osc osc osc λ ω λ ω + = h h P P P osc osc h λ μ = = 2 2 2 1 2 1 X P H + = osc osc osc m m m ω ω λ h h h h = = 2 2 2 osc osc osc osc osc m m m ω ω ω λ ω h h = = 2 2 2 2 2 2 1 2 1 X P H osc osc osc ω ω ω h h h + =
Dimensionless Commutation Relations Let’s compute the commutator for the new variables: X , P [ ] = X P P X λ λ X X X X = = P P P P h h λ λ = = X , P [ ] = X λ λ P h λ P h

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Lect19_SHOintro - Lecture 19 Quantization of the simple...

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