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1、第七章 谐振子第1页，共30页，编辑于2022年，星期一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 writtenStationa
2、ry Schrodinger equation第2页，共30页，编辑于2022年，星期一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，共30页，编辑于2022年，星期一Insert(2)to(1),getThis is Hermite(厄米)differential equati
3、onAt the vicinity of =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)第4页，共30页，编辑于2022年，星期一Energy eigenvalue of harmonic oscillator1.Energy level is discr
4、ete.2.The energy gap is identical.3.The energy level of ground state(zero point energy)is not zero.第5页，共30页，编辑于2022年，星期一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，共30页，编辑于2022年，星期一Some most simple Hermite pol
5、ynomialsH0=1,H1=2,H2=422,H3=83 12,The basic properties of Hermite polynomials(The definition)Two important and useful relations第7页，共30页，编辑于2022年，星期一n=0:n=1:n=2:The first three eigenfunctions of harmonic oscillator第8页，共30页，编辑于2022年，星期一The symmetry propertyWhen n is even,positive parity(n 为偶数，偶宇称)When
6、 n is odd,negative parityIn general第9页，共30页，编辑于2022年，星期一Ground 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
7、is maximum,hence the interval which it delays at x=0 is shortest.第10页，共30页，编辑于2022年，星期一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第11页，共30页，编辑
8、于2022年，星期一Zero point energy is a direct consequence of the uncertainty relationSince the integrand（被积函数）is an odd function,第12页，共30页，编辑于2022年，星期一We can write uncertainty relation againThe mean energyThe minimum energy is zero point energy,which is compatible with uncertainty principle.第13页，共30页，编辑于2
9、022年，星期一The normalization eigenfunction of harmonic oscillator According to these relations,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）第14页，共30页，编辑于2022年，星期一Hence(1)(2)第15页，共30页，编辑于2022年，星期一By addition or subtraction of(1)and(2),we getW
10、e define the operatorsHence is called the lowering operator(降幂算符),+the raising operator(升幂算符).第16页，共30页，编辑于2022年，星期一The number operator(数算符)第17页，共30页，编辑于2022年，星期一By successively operator+on,we can calculate all the eigenfunctions,staring from the ground state.For n=0The eigenfunction of ground state
11、The normalized eigenfunction 第18页，共30页，编辑于2022年，星期一One-dimension Hamiltonian harmonic oscillatorWe introduceHence 3.Representation of the Oscillator Hamiltonian in Terms of and+第19页，共30页，编辑于2022年，星期一According to the definitions of and+,getWe obtain a simple Hamiltonian representationEigenvalue 第20页，
12、共30页，编辑于2022年，星期一 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，共30页，编
13、辑于2022年，星期一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，共30页，编辑于2022年，星期一Example 1Using the recursion of Hermite polynomials Prove the following expressions,An
14、d according to these,prove 第23页，共30页，编辑于2022年，星期一Solution:n(x)is the eigenfunction of harmonic oscillator,and can be written 第24页，共30页，编辑于2022年，星期一Hamitonnian of the coupling harmonic oscillator can be written Example 2wherex1,p1 and x2,p2 belong to different freedom degree,and set Problem:the energ
15、y level of this coupling harmonic oscillator.第25页，共30页，编辑于2022年，星期一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 transform the above question into the question of
16、 two independent one-dimension harmonic oscillator,then its energy level and eigenfunction are whereis energy eigenfunction of one-dimension oscillator 第26页，共30页，编辑于2022年，星期一For the coupling harmonic oscillator,we can simplify it two independent harmonic oscillator using coordinate transformation,so
17、 we setWe can easily prove the following expressions第27页，共30页，编辑于2022年，星期一Therefore Hamitanian becomes Where Hence 第28页，共30页，编辑于2022年，星期一1.Using the recursion of Hermite polynomials Prove the following expressions,And according to these,prove Exercisewhere第29页，共30页，编辑于2022年，星期一2.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，共30页，编辑于2022年，星期一