1、Chapter 4 Mathematics Foundations of Quantum Mechanics I (量子力学中的数学基础）,Linear operator,4.1 Properties of Operaotors,Operator in quantum mechanics denotes an operation of wave function, such as,What is operator?,Linear operator is self-adjoint (自共轭) or Hermitian (厄密的),In quantum mechanics, all operate
2、s are Hermitian operators,Hermitian operator (厄密算符),x, px, V(x) Hermitian operator,In any quantum states, the mean value of Hermitian operator is real,(1) Sum,(2) product,4.1-2 Combining Two Operators,(3) Unit operator I,(4) operator commutator (算符对易性),So,In general, the product of two operators do
3、not commute (对易).,For example,We can similarly obtain,But,In summary,operator commutator satisfy,4.1-3 Bra and Ket Notation（左矢和右矢符号）,A scalar product of the square-integrable function can be expressed,The orthonormality relation of two wave functions,The mean value of L,(1) position and momentum ope
4、rators （P73）,Position operator,Its components,Momentum operator,Its components,4.1- 4 Operator in quantum mechanics,Commutator of x and p,(2) Angular-momentum operators (角动量算符) P75,It components,i, j, k = (1, 2, 3), 123 任何两个角标对换，改变正负号，若两个角标相同则为零。,The first term becomes,The square of the angular mome
5、ntum operator,The second term,The sum of two terms is 0, so,Similarly,In Cartesian coordinate,In polar coordinate,(3) Kinetic operator,If V(r ) is only the function of distance, V=V(r ), and L is dependent on and , so L,V( r)=0. We can easily obtain L, p2=0,so,L, p2=0,L,V( r)=0,(4) Total energy oper
6、ator (Hamiltonian operator),4.2 Eigenvalue and Eigenfunction (本征值和本征函数),If L is a constant value, its deviation L=0, so we can find the corresponding wave function L,在波函数L中, 我们将平均值换成测量值.,该方程称为本征值方程, L称为本征值, L称为本征函数.,一个算符L, 本征值为L ，相应的本征函数L有无数多个。 L可以是分离的，也可以是连续的。当它为连续时，它可以取Ln L Ln+1之间的任何值。,厄密算符本征函数的性质
7、(只考虑分立谱),可以证明属于不同本征值的本征函数之间是正交的.,本征值L为实数, 我们取第一个方程的复共轭,subtract,Integrate over the whole volume,由于,是厄密算符, 那么左边两个积分在整个空间的积分,是相等的. 即,We required LnLm, hence,Two wave function are orthogonal,分立谱的本征函数是平方可积的.,结合上面的结果,我们得到,对一个本征值Ln, 若同时有几个本征函数与之对应, 我们将这样的态称之为简并态(degenerate states). 为精确起见, 如果不同的本征函数n1、n2、n
8、3属于同一本征值Ln, 我们称之为n重简并(n-fold degeneracy).,简并态的物理意义为观测量L的某个确定值的几率可以在不同态中实现.,前面我们已经证明了属于不同本征值的波函数是正交的, 那么属于同一个本征值Ln的简并波函数nk, 有,一般来说，nk不正交. 但找到正交函数的几率总是存在的。,2. Translate the text contents on the page P6869.,1. Prove the following formulas: (1),(2),L, p2=0,Exercise,The proof of Hermiticity of momentum o
9、perator,This proves that obeys the Hermiticity relation.,The Commutator of Position-and Momentum Operators,so,Position-and Momentum Operators do not commute, so they cant be exactly measured at the same time.,Momentum Eigenfunction,C is constant, the spectrum of momentum is continuous.,量子力学中最常见的几个物理
10、量：位置、动量、角动量和能量，其中位置和动量的取值(本征值)是连续变化的，角动量和能量的本征值是分立的，而能量的本征值则兼而有之。,以动量本征态为例,一维粒子的本征值为p的本征函数,p可以取-+中连续变化的一切实数，不难看出，C取决于p值。为了确定C，我们考虑积分,Normalization of the eigenfunction of momentum operator (P100),According to the definition of (x),那么三维空间的动量算符写为,Example 1,A particle with the mass m motions in the pot
11、ential field V(r),Problem: estimate its ground state energy using the uncertainty principle.,Solution: total energy operator is,The eigenvalue of H under any energy eigenfunction is given by,For the estimation the radius in ground state is approximately regarded as rR,Due to the mean value of moment
12、um of bound state being zero, we can obtain,using the uncertainty principle,The value R must be satisfy the condition that E is minimum extremum, i.e.,So we get,Finally we obtain the approximate energy value of ground state,Example 2,Problem: calculate the values of the following expression,Where n
13、is positive integer, is parameter variable.,Solution: In the following calculation, we leave out the superscript () of operator.,Let n transform to (n-1),So we get,Repeat above calculating,According to Thalers expansion,Set (n-1) = m, we can obtain,Exercise,Using uncertainty principle, please estima
14、te the ground state energy of helium atom (氦原子). Prove the following equations,3. Set the mechanics quantity satisfy the most simple algebra (代数) equation:,Where C1, C2, , Cn are constant coefficients. Please prove that the number of engenvalue of is n, and they are the roots of equation f(x) = 0,Imply: set ,prove , then integrate.,
