《物理学教学》26 introduction to quantum mecha

1、Introduction to quantum mechanicsChapter 26.26.1 Operator and physical quantity26.2 Schrdinger equation26.3 One dimensional potential well26.4 Hydrogen atom 26.5 Quantum numbers and atomical orbitals.Operator represents certain kind of mathematical operation which applies on one function and leads t

2、o another function in result. 26.1 Operator and physical quantityvuF Example: differential operatorvdxduudxd )(vux coordinate operator.vuF wwF C The eigen equation of operator Feigen functioneigen value.In quantum mechanics, all the physical quantities are expressed by operators. coordinate operator

3、xx xhipx2 yhipy2 zhipz2 momentum operator.If there is a classical expression of quantity F: F=F(x,p), then the corresponding operator expression in quantum mechanics can be written as: ) , (pxFF example: classical kinetic energy 22mpEkkinetic energy operator 2222222228)4(21 2dxdmhdxdhmmpEk.classical

4、 energy)( 22xUmpEEEPkEnergy operator in quantum mechanics is usually called Hamiltonian operator: )(8) (2222xUdxdmhxUEHk.)(8) (2222xUdxdmhxUEHk2222222dddH() U(x,y,z)2m dxdydz 2hin 3 dimensional coordinates:where:.26.2 Schrdinger equationThe eigen equation for energy operator is called Schrdinger equ

5、ation (Sch. eq.): EHHamiltonianwavefunction .1) derive Hamiltonian for the system according to different conditions;2) set up Schrdinger equation;3) solve Schrdinger equation to get energies and wavefunctions; 4) all other physical informations can easily derived after wavefunction is obtained. EHSt

6、eps to solve a physical system in quantum mechanics:.1dV22is the probability of finding particle in the unit volume about the certain point.The probability of finding particle in all possible space should be one. , , , , , , , 4321)(4)(3)(2)(1PPPPPPPP PP. , , , , , , , 4321)(4)(3)(2)(1xxxxxxxx xx.26

7、.3 One dimensional potential wellOne particle is confined in an one dimensional box with a width “a”. It can move free within the box, but can NOT go outside. 0 x aU(x)mU(x) (x0, xa)potential inside boxpotential outside boxU(x) = 0 (0 xa).)(8) (2222xUdxdmhxUEHk EH0)(8dd2222xUEhmx 0 x aU(x)mSchrdinge

8、r equation.0)(8dd2222xUEhmxoutside box region:U(x) (x0, xa)0The probability of finding particle outside box is ZERO.0)(8dd2222xUEhmxinside box region:U(x) = 0 (0 xa)08dd2222 Ehmx0dd222 kx2228kEhm with.0dd222 kxBcoskxAsinkx(x)solution:where constant A and B should be determined by boundary conditions

9、.0(0) 0(a) 2228kEhm withnka Asinkx(x) B=0n = 1,2,3,ank.2228kEhm ank2228nmahE Energy quantization !Energy level: E1 , E2 , E3 , E4 , n - quantum number.1d)(sin022 axxanA1dV2Asinkx(x) ankxanaxsin2)( aA2 . EH 0 x aU(x)mA series of En and are obtained, with n = 1,2,3,nxansina2(x)n1Enn8mahE2222n.xansina2

10、(x)n1Enn8mahE2222n.26.4 Hydrogen atom r4eU(r)02+e-erv2222222dddH() U(r)2m dxdydz .2222222dddH() U(r)2m dxdydz Cartesian coordinatesspherical coordinates)(r,+e-ervz)y,(x,.2222222dddH() U(r)2m dxdydz ),(r,H2()()222222201111rsin2m rrrrsinsine 4r .26.5 Quantum numbers and atomic orbitalsE),(r,H ),(r,H2(

11、)()222222201111rsin2m rrrrsinsine 4r - How to solve this complicated Sch. eq.?.E),(r,H The trick to solve this Sch. eq. :(1) variables separation(2) solve 3 equations successivelyequation for requation for equation for )()(R(r),(r,separated into 3 equations3 quantum numbers appearquantum numbernquan

12、tum numberlquantum numberlm.E),(r,H equation for requation for quantum numbernllmequation for nE energy R(r)()(quantum numberquantum numberwavefunctionwavefunctionwavefunctionorbital angular momentum 1)l(lLZ-component of angular momentum lzmL.In addition, electrons have spin angular momentum, and th

13、e z-component of which has two possible values:szmS21smwithto as spin up and spin down.which are often referred.4 quantum numbers are required to mark each orbital.slmmln . lmln.Exe -1-An electron is contained in a one-dimensional box of length 0.100nm. (a) Draw an energy-level diagram for the electron for levels up to n=4.(b) Find the wavelengths of all photons that can be emitted by the electron in making downward transitions that could eventually carry it from the n=4 state to the n=1 state. .Exe -2-How many sets of quantum numbers are possib