SCHRÖDINGER EQUATION AND APPLICATIONS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016 CONTENTS I Schrödinger equation II Applications of Schrödinger equation 1 Particle in a 1 D infinite[.]
SCHRÖDINGER EQUATION AND APPLICATIONS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016 CONTENTS I Schrödinger equation II Applications of Schrödinger equation Particle in a 1-D infinite potential well Tunnel effect I Schrödinger Equation De Brogile wave function of a free particle of energy E, momentum p: Wave function of a particle moving in a field that having potential energy U(r) is: ( r ) satisfies the time-independent Schrödinger equation ( r , t ) o ( r , t) i ( Et p r ) e i ( Et ) ( r )e 2m ( r ) (E U( r ))( r ) Schrodinger equation in Quantum Mechanics Newton 2nd law in Classical mechanics Solving Schrodinger equation Wave function that describes the state of the particle, and the possible energy levels of the particle If 1, 2 are the solutions of Schrödinger equation, =C11+C2 is also Derive Schrödinger Equation FOR A FREE PARTICLE (r , t ) o e i ( Et p r ) oe i ( Et p x x p y y p z z ) i ( Et px x p y y p z z ) i px o e x i ( Et px x p y y p z z ) px2 2 i px o e (r , t ) x p y2 pz2 2 2 (r , t ); (r , t ) 2 y z px2 p y2 pz2 2 2 2 p2 (r , t ) (r , t ) (r , t ) x y z p2 E p 2mE 2m i i ( Et ) ( Et ) 2mE (r )e ( r )e (r ) 2mE (r ) 2mE ( r ) ( r ) Derive Schrödinger Equation (cont.) + For a free particle E: is the Kinetic energy of the free particle 2mE ( r ) ( r ) + 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 ) Schrödinger Equation (cont.) ( r ) U ( r )) ( r ) E ( r ) 2m 2m 2 d2 ( x ) U ( r )) ( x ) E (x) 2m Total dx PE KE Energy REVIEW about wave fuction The statistic meaning of de Broglie Wave of a particle probability of finding the particle per unit volume= probabilty density probability of finding the particle in a volume dV o2 probability of finding the particle in a volume V probability of finding the particle over all space =1 (the particle is certainly found) | ( r , t ) | (.* ) dP | ( r , t ) | dV P | ( r , t ) | dV V P | ( r , t ) | dV Normalized Condition of the wave function / Điều kiện chuẩn hóa hàm sóng 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 II Application of Schrodinger equation Particle in a 1-D infinite potential energy well U Particle in a 1-D infinite potential energy well U 0 O a 0xa x 0, x 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 d ( x) U ( x) ( x) E ( x) 2m dx KE term 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 dots‟ L U = for < x < L U = everywhere else (www.kfa-juelich.de/isi/) (newt.phys.unsw.edu.au) ...CONTENTS I Schrödinger equation II Applications of Schrödinger equation Particle in a 1-D infinite potential well Tunnel effect I Schrödinger Equation De Brogile wave function of... mechanics Solving Schrodinger equation Wave function that describes the state of the particle, and the possible energy levels of the particle If 1, 2 are the solutions of Schrödinger equation, =C11+C2... time-independent Schrödinger equation ( r , t ) o ( r , t) i ( Et p r ) e i ( Et ) ( r )e 2m ( r ) (E U( r ))( r ) Schrodinger equation in Quantum Mechanics