Schrödinger equation and applications - Tran Thi Ngoc Dung

Nội dung text: Schrödinger equation and applications - Tran Thi Ngoc Dung

1. SCHRệDINGER EQUATION AND APPLICATIONS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
2. CONTENTS I. Schrửdinger equation II. Applications of Schrửdinger equation 1. Particle in a 1-D infinite potential well 2. Tunnel effect
3. I. Schrửdinger Equation i De Brogile wave function of a free (Et p.r)  particle of energy E, momentum p: (r,t) oe i Wave function of a particle moving in a (Et ) field that having potential energy U(r) is: (r,t) (r)e  2m  ( r ) satisfies the time-independent (r) (E U(r))(r) 0 Schrửdinger equation 2 Schrodinger equation Newton 2nd law in Quantum Mechanics in Classical mechanics Wave function that describes the Solving Schrodinger equation state of the particle, and the possible energy levels of the particle If 1, 2 are the solutions of Schrửdinger equation, =C11+C2 2 is also
4. Derive Schrửdinger Equation FOR A FREE PARTICLE ii (.)()Et p r Et px x p y y p z z (,)r t oo e  e i  i ()Et px x p y y p z z pe x x o 2 2 i 2 ()Et p x p y p z  i 2 x y z px 22 px o e (,) r t x 22 p2   p2 y (r , t ); z  ( r , t ) yz2 2 2 2 2   2   2  ppp222 p 2 (,)rt x y z (,)(,)r t  r t x2  y 2  z 2 2 2 p2 E p2 2 mE 2m ii ()()Et2mE Et ()()r e 2  r e 2mE 2mE (r) (r) 0 (rr )  ( ) 0 2 2 
5. Derive Schrửdinger Equation (cont.) + For a free particle E: is the Kinetic energy of the free particle 2mE (r) (r) 0 2 + For a particle in a region of potential energy U(r), E is the energy of the particle, and KE is E-U 2m (r) (E U(r))(r) 0 2
6. Schrửdinger Equation (cont.) 2 (r) U(r))(r) E(r) 2m 2m 2 d2 (x) U(r))(x) E (x) 2m dx 2   PE Total KE Energy
7. REVIEW about wave fuction The statistic meaning of de Broglie Wave of a particle 2 2 * probability of finding the o | (r,t) | (. ) particle per unit volume= probabilty density . probability of finding the dP | (r,t) |2 dV particle in a volume dV 2 probability of finding the particle P | (r,t) | dV in a volume V V probability of finding the 2 particle over all space =1 (the P | (r,t) | dV 1 particle is certainly found) Normalized Condition of the wave function / Điều kiện chuẩn húa của hàm súng
8. Constraints on Wavefunction In order to represent a physically observable system, the wavefunction must satisfy certain constraints: (x,t) - Must be a single-valued function - Must be normalizable. This implies that the wavefunction approaches zero as x approaches infinity. - Must be a continuous function of x. - the first derivative of (x,t) must be continuous
9. II. Application of Schrodinger equation 1. Particle in a 1-D infinite potential energy well U Particle in a 1-D infinite potential energy well 0 0 x a U x 0,x a O a x Particle can move freely inside the well, but it can not overcome the potential barrier to get outside. For example: Electron in the metal can move freely, but it needs energy for escaping the metal
10. 2 d 2  (x) U(x) (x) E (x) KE term 2m dx2 Total E term PE term U(x) This is a basic problem in “Nano-science”. It‟s a simplified (1D) model for an electron confined in a quantum structure (e.g., “quantum dot”), which scientists/engineers make, e.g., at the UIUC Microelectronics Laboratory! (www.micro.uiuc.edu) „Quantum 0 L dots‟ U = 0 for 0 < x < L U = everywhere else (www.kfa-juelich.de/isi/) (newt.phys.unsw.edu.au)
11. 1. Particle in a 1-D infinite potential energy well SOLUTION 2m U=0, problem is 1-D, =>(x) (r) (E U(r))(r) 0 2 d2(x) 2m 2m E(x) 0 k2 E dx 2 2 2 (x) Asin kx Bcos kx x 0,(0) B 0 x a,(a) Asin ka 0 ka n n k , n 1,2,3 a n 2 n  (x) Asin( x) n (x) sin( x) n a a a 2 a a 2 2 2m n 2 2 2 n A a k 2 E | n (x) | dx A sin ( x)dx 1  a 0 0 a 2 2 2 2 2 2 n  2 h A En 2 n 2 a 2ma 8ma
12. 1. Particle in a 1-D potential energy well Conclusion 1. Wave function depends on an 2. The energy of the particle is quantized integer n (quantum number) 2 2 2 2 n  2 h 2 n En n  (x) sin( x) 2ma 2 8ma 2 n a a 2  (x) sin( x) 1 a a 2 2  (x) sin( x) 2 a a 2 3  (x) sin( x) 3 a a
13. FINITE POTENTIAL ENERGY WELL
14. FINITE POTENTIAL WELL INFINITE POTENTIAL WELL FINITE POTENTIAL WELL
15. Finite potential well Finite potential well: In a potential well with finite depth the energy levels are lower than those for an infinitely deep well with the same width, and the number of energy levels corresponding to bound states is finite
16. + at n=3, the probability density is 2 |(x)| maximum at x= a/6.a/2,5a/6 In the state n, the probability + +the probability to find the density to find the electron has n particle of state n=2 in the interval antinode and n+1 node x=a/4 and 3a/4 is ẵ + + the probability to find the particle of state n=3 in the interval x=a/4 and 3a/4 0 a/6 a/3 a/2 2a/3 5a/6 a n=3, E2=9E1 3a / 4 3a / 4 2 2 2 3 | n 3(x) | dx sin ( x)dx 0.4 a / 4 a / 4 a a n=2, E =4E 0 a/4 a/2 3a/4 a 2 1 n=1 , E1 0 a/2 a x
17. TUNNELING EFFECT Particle has energy E, encounter a potential barrier of potential energy Uo, the energy of the particle E Tunneling effect
18. TUNNEL EFFECT Particle has energy E, encounter a potential barrier of potential energy U, E<U U 0 x 0 U o U Uo 0 x a (I) (II) (III) 0 x a E Transmission coefficient O a x 2 2 probability density behind | 3 (x) | | A3 | T the barrier 2 2 | 1(x) | | A1 | = probability density coming to the barrier
19. TUNNEL EFFECT
20. Tunnel effect 2 d 1(x) 2m 2 2m (1) E1(x) 0 ; k1 E dx 2 2 2 2 d 2 (x) 2m 2 2m (2) (E Uo )2 (x) 0 ;k2 (Uo E) dx 2 2  2 0 2 d 3(x) 2m 2 2 2m (3) E3(x) 0 ; k1 k3 E dx 2 2 2 ik 1x ik 1x (4) 1(x) A1e B1e k2x k2x (5) 2 (x) A2e B2e ik 1(x a) ik 1(x a) (6) 3(x) A3e B3e d d x 0; (0)  (0); 1 2 + wave function 1 2 continuous dx x 0 dx x 0 + the first d2 d3 x a;2 (a) 3(a); derivative is dx x a dx x a continuous
21. U Uo ik 1x ik 1x 1(x) A1e B1e (I) (III) (II) k2x k2x 2 (x) A2e B2e E ik 1(x a) ik 1(x a) 3(x) A3e B3e O a x ik 1x Incident wave: A1e ik 1x Reflection wave B1e ik (x a) Transmission wave: 1 A3e ik 1(x a) Reflection wave from the B3e infinity But there is no reflection => B3=0
22. (7) A1 B1 A2 B2 ik1x ik1x (4) 1(x) A1e B1e (8) ik1(A1 B1) k2 (A2 B2 ) k 2x k 2x (5) 2 (x) A2e B2e k2a k2a (9) A2e B2e A3 ik1 (x a) (6) 3 (x) A3e k2a k2a (10) k2 (A2e B2e ) ik1A3 d1 d2 x 0;1(0) 2 (0); k dx x 0 dx x 0 n 1 k2 d2 d3 x a;2 (a) 3 (a); dx x a dx x a (7) A1 B1 A2 B2 1 (8) (A B ) (A B ) 1 1 in 2 2 k2a k2a (9) A2e B2e A3 k2a k2a (10) (A2e B2e ) inA3 2 A3 4n k 2a 4n k 2a T e e A1 (n i)(1 in) (n i)(1 in) 16n 2 T e 2k 2a (1 n 2 )2
23. (7) A1 B1 A2 B2 1 in k2a (9) (10) A2 A3e 1 2 (8) (A1 B1) (A2 B2 ) 1 in in (9) (10) B A ek2a 2 2 3 k2a k2a (9) A2e B2e A3 1 1 1 1 (7) (8) A (1 )A (1 )B k2a k2a 1 2 2 (10) (A2e B2e ) inA3 2 in 2 in A2 B2 1 1 1 i 1 in k2a A1 (1 )B2 1 A3e 2 in 2 n 2 A 4n 3 e k2a A1 (n i)(1 in) 2 A3 4n k 2a 4n k 2a T e e A1 (n i)(1 in) (n i)(1 in) 16n 2 T e 2k 2a (1 n 2 )2
24. Transmision coefficient 2m 2 2 2a (U E) 2 o 16n 2k 2a 16n  U T e e U 2 2 2 2 o (1 n ) (1 n ) (I) (II) (III) E O a x 2m E k 2 E n 1  k 2m U E 2 (U E) o 2 o
25. Time-dependent SEQ  To explore how particle wavefunctions evolve with time, which is useful for a number of applications as we shall see, we need to consider the time-dependent SEQ: 22 This equation describes the full d(,)(,) x t d x t time- and space dependence of 2 U()(,) x  x t i 2m dx dt a quantum particle in a potential U(x), replacing the classical i2 = -1 particle dynamics law, F=ma  Important feature: Superposition Principle  The time-dependent SEQ is linear in  (a constant times  is also a solution), and so the Superposition Principle applies: If 1 and 2 are solutions to the time-dependent SEQ, then so is any linear combination of 1 and 2 (example:  0.6 1 + 0.8i2)