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1、第七章 谐振子现在学习的是第1页，共30页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 Hamitonnian function of an oscillator with mass m and oscillating frequency 0 can be writtenStationary Schr
2、odinger equation现在学习的是第2页，共30页Referencing the book edited by曾谨言，we solve the Schrodinger equation.Introduce the no-dimension parameters(无量纲参数)We get(boundary condition),(1)(2)We get an asymptotic solution(试探解)现在学习的是第3页，共30页Insert(2)to(1),getThis is Hermite(厄米)differential equationAt the vicinity of
3、=0,u()is expanded the Taylor series.Only will satisfies the boundary condition()(4)Therefore the condition(4)is satisfied,we can get the solution which is allowed in physics field.According to(3)现在学习的是第4页，共30页Energy eigenvalue of harmonic oscillator1.Energy level is discrete.2.The energy gap is iden
4、tical.3.The energy level of ground state(zero point energy)is not zero.现在学习的是第5页，共30页The solution of equation(3)is Hermite polynomials(厄米多项式).The eigenfuction and energy of harmonic oscillator are Normalized constant现在学习的是第6页，共30页Some most simple Hermite polynomialsH0=1,H1=2,H2=422,H3=83 12,The basi
5、c properties of Hermite polynomials(The definition)Two important and useful relations现在学习的是第7页，共30页n=0:n=1:n=2:The first three eigenfunctions of harmonic oscillator现在学习的是第8页，共30页The symmetry propertyWhen n is even,positive parity(n 为偶数，偶宇称)When n is odd,negative parityIn general现在学习的是第9页，共30页Ground
6、state The energy and wave function of ground state(n=0)The probability finding a particle at x=0 is maximum,which is contrary to classical particle.For a classical harmonic oscillator,when x=0,its potential is minimum and kinetic energy is maximum,hence the interval which it delays at x=0 is shortes
7、t.现在学习的是第10页，共30页In classical mechanics,a particle with ground state energy E0 motions in the range According to quantum mechanics,the probability finding a particle outside the classical allowed range isn15xW(x)wclwqu现在学习的是第11页，共30页Zero point energy is a direct consequence of the uncertainty relati
8、onSince the integrand（被积函数）is an odd function,现在学习的是第12页，共30页We can write uncertainty relation againThe mean energyThe minimum energy is zero point energy,which is compatible with uncertainty principle.现在学习的是第13页，共30页The normalization eigenfunction of harmonic oscillator According to these relations
9、,we getThe description of the Harmonic Oscillator by Creation and Annihilation operators（产生算符和湮灭算符产生算符和湮灭算符）现在学习的是第14页，共30页Hence(1)(2)现在学习的是第15页，共30页By addition or subtraction of(1)and(2),we getWe define the operatorsHence is called the lowering operator(降幂算符),+the raising operator(升幂算符).现在学习的是第16页，
10、共30页The number operator(数算符)现在学习的是第17页，共30页By successively operator+on,we can calculate all the eigenfunctions,staring from the ground state.For n=0The eigenfunction of ground stateThe normalized eigenfunction 现在学习的是第18页，共30页One-dimension Hamiltonian harmonic oscillatorWe introduceHence 3.Representa
11、tion of the Oscillator Hamiltonian in Terms of and+现在学习的是第19页，共30页According to the definitions of and+,getWe obtain a simple Hamiltonian representationEigenvalue 现在学习的是第20页，共30页 基态0所具有的零点能量为/2,而且我们知道谐振子的能量是等间隔的,n所具有的能量大于n,我们将该能量以能量量子分成n份(谐振子场中的量子),称为声子(phonons),那么将n称为n声子态(n-phonon state),在Diracs 表象中
12、表示为表示声子数，零声子态(zero-phonon state)。称为真空。应用上面的表述,算符 和+作用于波函数可表示成解释:如果 作用于波函数,则湮灭(annihilate)了一个声子,因而称为湮灭算符;+作用于函数,则产生一个声子,+产生算符.4.Interpretation of and+现在学习的是第21页，共30页由于称为声子数算符(phonon number operator),n为相应态的子数.声子表象的引入被称为二次量子化,而谐振子波场中的量子正是声子.如果与光子相类比的话,就更清楚了.|3 annihilation of a phonon+2|1 creation of t
13、wo pohonons谐振子的能级和声子的湮灭、产生示意图En/7/25/23/21/2x现在学习的是第22页，共30页Example 1Using the recursion of Hermite polynomials Prove the following expressions,And according to these,prove 现在学习的是第23页，共30页Solution:n(x)is the eigenfunction of harmonic oscillator,and can be written 现在学习的是第24页，共30页Hamitonnian of the co
14、upling harmonic oscillator can be written Example 2wherex1,p1 and x2,p2 belong to different freedom degree,and set Problem:the energy level of this coupling harmonic oscillator.现在学习的是第25页，共30页Solution:if the coupling term x1x2 is not exists,the coupling harmonic oscillator becomes two-dimension osci
15、llator,and then its Hamitanian is given by Using separating variable,we can transform the above question into the question of two independent one-dimension harmonic oscillator,then its energy level and eigenfunction are whereis energy eigenfunction of one-dimension oscillator 现在学习的是第26页，共30页For the
16、coupling harmonic oscillator,we can simplify it two independent harmonic oscillator using coordinate transformation,so we setWe can easily prove the following expressions现在学习的是第27页，共30页Therefore Hamitanian becomes Where Hence 现在学习的是第28页，共30页1.Using the recursion of Hermite polynomials Prove the foll
17、owing expressions,And according to these,prove Exercisewhere现在学习的是第29页，共30页2.A particle is in the ground state of one-dimension harmonic oscillating potential Now,k1 is abruptly changed to k2,i.e,and immediately measure the energy of a particle,k1 and k2 are positive real number.Solve the probability finding that a particle is the ground state of new potential V2.现在学习的是第30页，共30页