Physics 2 - Lecture 8: Schrodinger Equation and Applications - Huynh Quang Linh

Nội dung text: Physics 2 - Lecture 8: Schrodinger Equation and Applications - Huynh Quang Linh

1. SCHRÖDINGER EQUATION AND APPLICATIONS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
2. I. Schrödinger 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 Schrödinger 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 Schrödinger equation, =C11+C2 2 is also
3. Derive Schrödinger 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
4. 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 hóa của hàm sóng
5. 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
6. 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
7. FINITE POTENTIAL ENERGY WELL
8. 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
9. TUNNELING EFFECT Particle has energy E, encounter a potential barrier of potential energy Uo, the energy of the particle E Tunneling effect
10. TUNNEL EFFECT
11. 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
12. (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
13. 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)