Ch1 - SE

Ch1 - SE - Chapter 1 Schrdinger wave equation Wave particle...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 1 Schrödinger wave equation
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Wave particle duality Particles ( λ is too small to be resolved) Waves de Broglie wavelength Pictures from “Physics 2000” h mv λ= Louis de Broglie 1929 Nobel prize
Background image of page 2
Young’s double-slit experiment Picture from http://en.wikipedia.org/wiki/Double-slit_experiment () ab ik x ik x x Ae Be Ψ= + G G G G G Superposition of two matter waves 22 2 | ||| || x AB + G 2 || cos( ) AB k x ϕ + Δ⋅ + G G
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
• Wave function of a particle: (, ) x t Ψ 2 |( , ) | xt Ψ = probability density of the particle at position x at time t • Wave function of two particles: , ) A B x Ψ 2 , , ) | A B xx t Ψ = probability density of the particle A at position x and particle B at position x at the time t A B e.g. • The wave functions must be normalized to unity (exception such as momentum wave functions are normalized to Dirac delta function).
Background image of page 4
From Ketterle’s webpage
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
How to demonstrate BEC as matter waves: Interference M.R. Andrews, et al ., Science 275 , 637-641 (1997).
Background image of page 6
Construction of a wave equation of a free particle 2 h p p π λ == = 2 2 p E m ω = 2 () (,) e x p 2 ik x t px p t xt Ae A i m ψ ⎛⎞ ⎜⎟ ⎝⎠ From the assumptions: The wave function with a definite momentum 22 2 2 x ti x t mt x ψψ ∂∂ −= = = In 3D: 2 2 2 x x t −∇ = = G G = What is A ?
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Box normalization of wave function with definite momentum 1 () ik r k re V ψ = G G G 23 |( ) | 1 V rd r = G V = box volume *3 '' () () ? kk k k V rr d r ψψ δ = G G Yes, if periodic boundary condition Finite volume V discrete k ’s. 1D: 1 ik x k x e L = 3D:
Background image of page 8
One-dimension problem in free space Plane wave solution: 22 /2 2 2 ikx i k t m ikx i k t m Ae i Ae mt x −− ∂∂ ⎤⎡ −= ⎦⎣ == = = 2 (,) 2 x ti x t x −Ψ = Ψ = = 2 ikx i k t m xt Ae Ψ= = Check: Remark: 2 ikx i k t m e × = A plane matter wave with momentum (eigenfunction of KE and momentum operator) ikx e = 2 ikt m e = = Time-dependent phase factor for each energy eigenfunction k = Schrödinger equation:
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 2 1 (,) 2 () k it ikx m k x te e d k π ϕ −∞ Ψ= = One-dimension problem in free space 22 2 2 x ti x t mt x ∂∂ −Ψ = Ψ = = General solution is a superposition of plane waves: 2 |( ) | 1 kd k −∞ = Normalization condition: 2 | | k = probability density of the particle with the momentum k = k can be determined by the initial wave function (,0 ) x Ψ What is the average energy of the particle ?
Background image of page 10
How to obtain from ? () k ϕ (,0 ) x Ψ 1 ) 2 ikx x ke dk π −∞ Ψ= Fourier transform 1 ) 2 ikx kx e d x −∞ Inverse Fourier Transform • A useful relation: (' ) 2( ' ) ik k x ed x k k πδ −− −∞ = ) 3 3 3 (2 ) ( ') whole space ik k r r k k = G G G G G
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Discuss why the following wave function is not physical: x Ψ = 0 Ψ = h How about the triangular shape?
Background image of page 12
() 2 (1) Re 0 x ed x α π −∞ => Some integral formulae for Gaussian functions: 2 *2 2 (2) exp Re 0 4| | xx x αβ πα β −+ −∞ ⎛⎞ = > ⎜⎟ ⎝⎠ 2 Calculate nx x x −∞ (a) (b) Show that
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1 (,0 ) () 2 ikx xk e d k ϕ π −∞ Ψ= 0 2 0 2 e x p 2 ikx K kk kA e σ ⎡⎤
Background image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

Page1 / 60

Ch1 - SE - Chapter 1 Schrdinger wave equation Wave particle...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online