1、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,Hamitonnian function of an oscillator with mass m and oscillating frequency 0 can be
2、 written,Stationary Schrodinger equation,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),Insert (2) to (1), get,This is Hermite (厄米) differential equation,At the
3、 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),Energy eigenvalue of harmonic oscillator,Energy level is discrete. The energ
4、y gap is identical. The energy level of ground state (zero point energy) is not zero.,The solution of equation (3) is Hermite polynomials (厄米多项式).,The eigenfuction and energy of harmonic oscillator are,Normalized constant,Some most simple Hermite polynomials,H0=1, H1=2, H2=422, H3=83 12, ,The basic
5、properties of Hermite polynomials,(The definition),Two important and useful relations,n=0:,n=1:,n=2:,The first three eigenfunctions of harmonic oscillator,The symmetry property,When n is even, positive parity (n 为偶数，偶宇称),When n is odd, negative parity,In general,Ground state,The energy and wave func
6、tion 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.,In classical me
7、chanics, 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 is,Zero point energy is a direct consequence of the uncertainty relation,Since the integrand （被积函数） is an odd function,We can wr
8、ite uncertainty relation again,The mean energy,The minimum energy is zero point energy, which is compatible with uncertainty principle.,The normalization eigenfunction of harmonic oscillator,According to these relations, we get,The description of the Harmonic Oscillator by Creation and Annihilation
9、operators （产生算符和湮灭算符）,Hence,(1),(2),By addition or subtraction of (1) and (2), we get,We define the operators,Hence, is called the lowering operator (降幂算符), + the raising operator (升幂算符).,The number operator (数算符),By successively operator + on , we can calculate all the eigenfunctions, staring from
10、the ground state.,For n = 0,The eigenfunction of ground state,The normalized eigenfunction,One-dimension Hamiltonian harmonic oscillator,We introduce,Hence,3. Representation of the Oscillator Hamiltonian in Terms of and +,According to the definitions of and +, get,We obtain a simple Hamiltonian repr
11、esentation,Eigenvalue,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. Interpret
12、ation of and +,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.,Example 1,Using the recursion of Hermite polynomials,Prove the following expressions,And according to these, prove,Solution: n(x) is the eigenfunction of harmonic oscillator, and can be w
13、ritten,Hamitonnian of the coupling harmonic oscillator can be written,Example 2,where,x1, p1 and x2, p2 belong to different freedom degree, and set,Problem: the energy level of this coupling harmonic oscillator.,Solution: if the coupling term x1x2 is not exists, the coupling harmonic oscillator beco
14、mes two-dimension oscillator, 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,where,is energy eigenfunction of one-dimension oscillat
15、or,For the coupling harmonic oscillator, we can simplify it two independent harmonic oscillator using coordinate transformation, so we set,We can easily prove the following expressions,Therefore Hamitanian becomes,Where,Hence,1. Using the recursion of Hermite polynomials,Prove the following expressi
16、ons,And according to these, prove,Exercise,where,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.,
