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1、第七章 谐振子第1页，本讲稿共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 Schrodi
2、nger equation第2页，本讲稿共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)第3页，本讲稿共30页Insert(2)to(1),getThis is Hermite(厄米)differential equationAt the vicinity of =0,u()is
3、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)第4页，本讲稿共30页Energy eigenvalue of harmonic oscillator1.Energy level is discrete.2.The energy gap is identical.3.The
4、energy level of ground state(zero point energy)is not zero.第5页，本讲稿共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第6页，本讲稿共30页Some most simple Hermite polynomialsH0=1,H1=2,H2=422,H3=83 12,The basic properties of He
5、rmite polynomials(The definition)Two important and useful relations第7页，本讲稿共30页n=0:n=1:n=2:The first three eigenfunctions of harmonic oscillator第8页，本讲稿共30页The symmetry propertyWhen n is even,positive parity(n 为偶数，偶宇称)When n is odd,negative parityIn general第9页，本讲稿共30页Ground state The energy and wave f
6、unction 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 shortest.第10页，本讲稿共30页In classical
7、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第11页，本讲稿共30页Zero point energy is a direct consequence of the uncertainty relationSince the integrand（被积函数）is an
8、odd function,第12页，本讲稿共30页We can write uncertainty relation againThe mean energyThe minimum energy is zero point energy,which is compatible with uncertainty principle.第13页，本讲稿共30页The normalization eigenfunction of harmonic oscillator According to these relations,we getThe description of the Harmonic
9、Oscillator by Creation and Annihilation operators（产生算符和湮灭算符产生算符和湮灭算符）第14页，本讲稿共30页Hence(1)(2)第15页，本讲稿共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).第16页，本讲稿共30页The number operator(数算符)第17页，本讲稿共30页By succe
10、ssively operator+on,we can calculate all the eigenfunctions,staring from the ground state.For n=0The eigenfunction of ground stateThe normalized eigenfunction 第18页，本讲稿共30页One-dimension Hamiltonian harmonic oscillatorWe introduceHence 3.Representation of the Oscillator Hamiltonian in Terms of and+第19
11、页，本讲稿共30页According to the definitions of and+,getWe obtain a simple Hamiltonian representationEigenvalue 第20页，本讲稿共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
12、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+第21页，本讲稿共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 two pohonons谐振子的能级和声子的湮灭、产生示意图En/7/25/23/21/2x第22页，本讲稿共30页Exampl
13、e 1Using the recursion of Hermite polynomials Prove the following expressions,And according to these,prove 第23页，本讲稿共30页Solution:n(x)is the eigenfunction of harmonic oscillator,and can be written 第24页，本讲稿共30页Hamitonnian of the coupling harmonic oscillator can be written Example 2wherex1,p1 and x2,p2
14、belong to different freedom degree,and set Problem:the energy level of this coupling harmonic oscillator.第25页，本讲稿共30页Solution:if the coupling term x1x2 is not exists,the coupling harmonic oscillator becomes two-dimension oscillator,and then its Hamitanian is given by Using separating variable,we can
15、 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 第26页，本讲稿共30页For the coupling harmonic oscillator,we can simplify it two independent harmonic oscil
16、lator using coordinate transformation,so we setWe can easily prove the following expressions第27页，本讲稿共30页Therefore Hamitanian becomes Where Hence 第28页，本讲稿共30页1.Using the recursion of Hermite polynomials Prove the following expressions,And according to these,prove Exercisewhere第29页，本讲稿共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.第30页，本讲稿共30页