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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 19: Quantization of the simple harmonic oscillator Phy851 Fall 2009
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = 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 + =
Background image of page 2
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 + = osc osc E h = 2 2 osc osc m E λ h = 2 2 osc osc m h h = osc osc m h = osc osc μ h = osc osc m h = The SHO introduces a single new parameter, which must govern all of the physics
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 + = h h P P P osc osc h μ = = 2 2 2 1 2 1 X P H + = osc osc osc m m m h h h h = = 2 2 2 osc osc osc osc osc m m m h h = = 2 2 2 2 2 2 1 2 1 X P H osc osc osc h h h + =
Background image of page 4
Dimensionless Commutation Relations Let’s compute the commutator for the new variables: X , P [ ] = X P P X λ X X X X = = P P P P h h = = X , P [ ] = X P h P h X ( ) PX XP = h 1 [ ] P X , 1 h = X , P [ ] = i We have stopped writing the subscript ‘osc’ Since the new
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.

Page1 / 17

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online