This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Harmonic oscillator Algebraic approach Classical harmonic oscillator ( 29 2 2 ; ; cos ; d x k k F kx x x A t dt m m ϖ δ ϖ =  =  = + = A block attached to an elastic spring is a typical example of a harmonic oscillator. One can find position as a function of time using Hook’s law in combination with 2 nd Newton’s law Energy of harmonic oscillator consists of kinetic and potential energies given as 2 2 2 2 2 1 1 1 ; ; 2 2 2 2 2 mv mv U kx K E kx kA = = = + = Generalization of harmonic oscillator 2 2 2 2 2 1 ( ) ( ) ( ) ( ) 2 x x x x x x x x dU d U k dx dx dU d U U x U x x x x x dx dx = = = = = = ≈ + + + 1 4 442 4 4 43 1 442 4 43 K In most cases harmonic oscillators appear as a result of an approximation of a potential energy with a minimum at some points by its series expansion about this point Classical harmonic oscillator (continue) K=0 turning point K=0 turning point K<0 classically forbidden region K<0 classically forbidden region K>0 classically allowed region Classical harmonic oscillator exhibits so called finite motion: particle’s position is limited by some finite region of space, while remaining of it remains forbidden. Quantum harmonic oscillator First step: Replace classical momentum and coordinate with the respective operators in order to find quantum Hamiltonian 2 2 2 2 2 2 2 2 ˆ 1 1 ˆ ˆ 2 2 2 2 p d H m x m x m m dx ϖ ϖ = + =  + h Second step: Set up stationary Schrödinger equation ˆ ( ) ( ) H x E x ψ ψ = 2 2 2 2 2 1 2 2 d m x E m dx ψ ϖ ψ ψ + = h The ultimate task is to find wave function of a HO at any time given its initial state. The first step in achieving is goal is finding stationary states and respective energy levels. When those are known we can present a general solution as a linear combination of the stationary states with respective timedependent factors, fit it to an initial condition, and we are done. Third step: Solve it. Energy levels Energy levels 1 , 0,1,2 2 n E n n ϖ = + = K h Energy levels of quantum harmonic oscillator are discrete and equidistant Stationary states 2 2 2 2 2 2 2 2 1/2 /2 1/2 /2 1/2 2 /2 1/2 3 /2 1 1 ( ) 2 3 1 1 2 2 2 5 1 2 2 4 2 8 7 1 3 12 8 2 48 x a x a x a x a n E x e a x n E e a a x n E e a a x x n E e a a a ϖ ψ π ϖ π ϖ π ϖ π = = = = = = = = = h h h h 1. Wave functions are either symmetric (even) or antisymmetric (odd) 2. Number of zeroes is again level number minus one 3. Wave functions are a Gaussian function multiplied by a polynomial a m ϖ = h Characteristic length or “size” of the wave function Expectation values coordinate 2 2 1 1 0; ; 2 2 x x n a x a n = = + ∆ = + momentum 2 2 2 1 1 0; ; 2 2 p p n p n a a = = + ∆ = +...
View
Full
Document
This note was uploaded on 09/27/2010 for the course PHYSICS 365 taught by Professor Deych during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 deych

Click to edit the document details