Lecture 6 - Harmonic oscillator Algebraic approach...

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Harmonic oscillator Algebraic approach

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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 0 0 2 2 0 0 2 2 0 0 0 2 0 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

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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 time-dependent factors, fit it to an initial condition, and we are done. Third step: Solve it.

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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 0 1/2 /2 1/2 2 /2 1/2 3 /2 1 1 0 ( ) 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

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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 = = + = + h h 1 1 2 2 p x n ∆ ∆ = + h h Uncertainty relation Ground state has minimum possible uncertainty Kinetic energy 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 n p p K K n n E m m ma
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