The Schrodinger Equation

Transcription

1 The Schrodinger quation Dae Yong JONG Inha University nd week

2 Review lectron device (electrical energy traducing) Understand the electron properties energy point of view To know the energy of electron Wave eq. lectron (Particle Wave Duality) From wave viewpoint phase velocity From wavelike particle viewpoint group velocity What can we do with wave equation? * dxdydz nergy Operator onto Wave q. Solve Schrodinger q. Possible energy level which electron can possess. * d The probability of finding an electron in a volume element dτ

3 The time-independent Schrodinger q. Let s derive the Wave-q for electron in materials. The condition The property of the surroundings of the electron does not change with time. Separate the Spatial and temporal part in the wave eq. Don t need to consider the temporal part ( x, y, z, t) ( x, y, z) iwt e e o jkx jk 0e x x jkx jk jkx jk e k Then, we can get the following relation 0 e t je jt jt j j t For time dependant j t 1 k for 3- D

4 (cont) The time-independent Schrodinger q V m V m x V m k V m h V m p V mv pot kin tot Operator (Refer to the Quantum mechanics): From the classical mechanics 1 k To calculate the of electron in material. V should be defined!! How can we define the potential energy V?

5 Ch. 4 The Schrodinger quation

6 Review and Contents q. for lectron Wave with energy operatior (Schrodinger q.) To solve the Schrodinger q., potential energy, V should be defined. We will solve the Schrodinger q. for xtreme case for easy calculation and understanding Free electron (without potential) Bound electron (trapped in potential box) But sometimes, electron escapes by tunneling effect More realistic case (periodic potential in Crystal)

7 4.1: Free lectron For easy calculation, Free electron Potential-free space in the positive x-direction No wall, no potential barrier (V) lectron was not confined in a certain potential. 0 m m Ae x x j where ) ( Sol. q. t j x j e Ae x ) ( Wave quation, k m k p m p m Pure kinetic energy!! nergy of electron was express with the wave number (K). Wave number (vector) has information on direction.

8 4.1: Free lectron m k nergy quation of electron!! k (wave number, vector) k k ( k x, k y, k ):direction : magnitude z Learn the wave vector in crystallography class in detail. Or refer to the solid state physics book. Related concept: reciprocal lattice Question : Free electron condition is realistic? In real case, electron could not travel infinitely. Specify the condition in detail to mimic the real situation step by step!!

9 For similarity Inside the earth orbital: Gravitational potential + Kinetic Outside arth orbital: No gravitational force, Kinetic

10 lectron in a Potential Well (Bound lectron) V lectron bound to its atomic nucleus lectron can move freely between two infinitely high potential barrier. Potential barrier do not allow the electron to escape from the potential well. lectron could not be found outside the potential well. No wave outside (mathematically) ψ = 0 for x < = 0 and x > = a - - m 0 lectron back and forth 0 nucleus a x ( x) Ae where jx Be m jx Back and forth Propagation and reflection Determine the A and B by means of boundary condition Boundary condition At x = 0, ψ = 0 and x = a, ψ = 0

11 lectron in a Potential Well (Bound lectron) ( x) 0 0 a A A(e n a Ae B ja sin a 0 jx e Be ja A is not Zero. n, where n m ) ma jx Aj sina 0,1,, 3... n = 0 ψ = 0, no physical meaning n, n 1,, 3... nergy quation of lectron is determined with integer n. (for example, n = 0.4 is not allowed.) Certain (nergy Level) is allowed. energy Quantization (discrete energy level) Assume the lowest energy as Zero-point energy. is related with the width of potential well a.

12 A particle in a box Separation between adjacent energy levels Characteristic wavelength n1 n ( n 1) h n h 8mL 8mL h (n 1) 8mL

13 4.:lectron in a Potential Well (Bound lectron) Understanding meaning of wave function, ψ Wave quation jasinx * jainx Probability * 4A sin x Wave in atom (circle) Rutherford Model r r n n By Niels Bohr Standing wave 0 a * dx 4A 4A A n classical nergy value 1 x sin x cosx 1 a 0 a sin ( x) dx a 0 1 From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

14 1, -D Visualization Wave q. probability (a) n 1 =1, n =1, (b) n 1 =1, n =, (b) n 1 =, n =1, (d) n 1 =, n =

15 4.: lectron in a Potential Well (Bound lectron) lectron in a 3-D potential wall (lectron in a box) Actually, material has a finite size. n ma n x n y n z Smallest energy (n x = n y = n z = 1) For the same energy level, for example, 3-fold degenerate energy state (n x, n y, n z ) = (1, 1, ), (1,, 1), (, 1, 1) degenerate states: the same energy but different quantum numbers

16 Question Consider an electron in an infinite potential well of size 0.1 nm (typical size of an atom). What is the ground energy of the electron? What is the energy required to put the electron an the third energy level? How can this energy be provided? solution n1, a0.1nm 18 n nx J 37. 6eV ma Frequency of electron : J 37.6eV rad / s / S n3, a0.1nm n nx 3 1 n eV ma eV c h h 4.1 nm 1 Provide the photon energy with 4.1 nm wavelength. (X-ray photon)

17 V 4.3 Finite Potential Barrier (Tunnel ffect) Finite potential energy (Vo) > tol m 0 X < 0 X < 0 ( V ) 0 o I Ae jx m Be jx At X = 0 I II I Vo 0 lectron II De x ( t kx) e j Tunneling: The penetration of a potential barrier by an electron wave. A B D I II jx jae jbe x x x 0 ja jb D jx De x II II II m Ce Ce m ( V for x, II jx x * as De II x De ) 0, so C 0 x De e o De jx m ( V ), where V o o j D A 1 D B 1 j j From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010) x II j( tkx) C D 0

18 4.4 lectron in a periodic field of a crystal (The Solid State) Consider the electron in real X-tal To solve the Schrodinger q., potential nergy should be defined. How to model the potential in X-tal? X-tal? Periodic arrangement of atom/ion Si - - Si - - Si Si - - Si - - Si Si - - Si - - Si This electron is affected by the nucleus right beside and next next next too many nuclei to be considered supercomputer is required for more realistic calculation. Si nucleus is attracting electrons. The closer to nucleus, the strong force. Model of potential for easy calculation From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

19 4.4 lectron in a periodic field of a crystal (The Solid State) Consider the electron in real X-tal To solve the Schrodinger q., potential nergy should be defined. How to model the potential in X-tal? V(x) = function with complex form m ( V o ) 0 xtremely hard to solve the wave equation. From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

20 4.4 lectron in a periodic field of a crystal (The Solid State) Model of potential for easy calculation I region m 0 m II region m ( V o ) 0 m ( V ) o Feature of potential shape Periodic potential (Period is depending on lattice constant of X-tal) Bloch function with periodic potential energy ( x) u( x) e jkx u(x) is a periodic function which possesses the periodicity of the lattice in the x-direction. u(x) is no longer a constant but changes periodically with increasing x (modulated amplitude). derivation sin a P cosa cos ka a mavob where P For P = 3π/ αa for Led line just satisfies the eq. An electron that moves in a periodically varying potential field can only occupy certain allowed energy zone (band). αa is function of nergy!! αa increase narrower forbidden band From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

21 계열

22 4.4 lectron in a periodic field of a crystal (The Solid State) Model of potential for easy calculation For P = 3π/ Band gap P mav b o m Analysis Potential barrier strength, V o b is large P is large fast decay, steep Narrower band P zero, then k cos a cos ka k m for free electron P infinite, then sina n 0, sina 0, αa nπ or n a a ma for bound electron lectron in atom Band gap: No energy allowed lectron in periodic potential: From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

23 4.4 lectron in a periodic field of a crystal (The Solid State) Band diagram with many atoms Paulli exclusion principle No electron with the same energy nergy level with Quantum number S S P6 3S 3P6 4S.. Many atoms One atom Periodic potential (~ large # of atoms and electrons) ach electron has discrete energy level. But the nergy difference is so small. Seems like continuous nergy level Band From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

24 Summary: Solution of Schrodinger q. Box in Potential Well n ma Discrete -level lectron can just exist on the specific -level. n x n y n z No Potential m k Continuous -level lectron can exist at any -level. Free movable electron Periodic Potential sin a P cosa cos ka a mavob where P Band structure lectron can exist in the certain energy range region. From lectronic properties of materials, Fourth dition, Hummel ( Springer, 010)

25 Notice!! lectron without Potential No band gap, free electron lectron in periodic potential Band gap but movable like free electron inside each band Periodic potential (~ large # of atoms and electrons) ach electron has discrete energy level. But the nergy difference is so small. Seems like continuous nergy level Band In metal, usually electrons are filled in valence band. (bound to nuclei) However, as the band is continuously connected 3- Dimensionally, electron in valence band can move into conduction band easily(freely). This movable electron is called free electron in metal. free electron without potential vs free electron in metal As the metal has periodic potential, I would like to call the free movable electron instead of free electron in metal.

26 Summary Free electron (without potential) Total energy which electron has = Kinetic nergy 1 mv p mv k k m electron in periodic potential but free movable energy Potential does not affect the magnitude of energy but adds the information of potential n ma k n :include the information on crystalstructure(a : lattice constant) a n 1first period, n nd period...