35 线性谐振子课件.pptVIP

Chapter 7 The Harmonic Oscillator  (谐振子);Many complicated potential can be approximated in the vicinity of their equilibrium points by a harmonic oscillator. The Taylor expansion of V(x) at equilibrium point x = a is ;Referencing the book edited by曾谨言，we solve the Schrodinger equation. ;Insert (2) to (1), get;Energy eigenvalue of harmonic oscillator;The solution of equation (3) is Hermite polynomials (厄米多项式).;Some most simple Hermite polynomials;n=0:;The symmetry property;Ground state ;In classical mechanics, a particle with ground state energy E0 motions in the range ;Zero point energy is a direct consequence of the uncertainty relation;We can write uncertainty relation again;The normalization eigenfunction of harmonic oscillator ;Hence ;By addition or subtraction of (1) and (2), we get;The number operator (数算符) ;By successively operator a+ on ?, we can calculate all the eigenfunctions, staring from the ground state.;One-dimension Hamiltonian harmonic oscillator;According to the definitions of a and a+, get; 基态ψ0所具有的零点能量为?ω/2, 而且我们知道谐振子的能量是等间隔的, ψn所具有的能量大于n?ω, 我们将该能量以能量量子?ω分成n份(谐振子场中的量子), 称为声子(phonons), 那么将ψn称为n声子态(n-phonon state), 在Dirac’s 表象中表示为;由于;Example 1;Solution: ?n(x) is the eigenfunction of harmonic oscillator, and can be written ;Hamitonnian of the coupling harmonic oscillator can be written ;Solution: if the coupling term ?x1x2 is not exists, the coupling harmonic oscillator becomes two-dimension oscillator, and then its Hamitanian is given by ;For the coupling harmonic oscillator, we can simplify it two independent harmonic oscillator using coordinate transformation, so we set;Therefore Hamitanian becomes ;1. Using the recursion of Hermite polynomials ;2. A particle is in the ground state of one-dimension harmonic oscillating potential