Quantum Mechanics: Free Particle Dynamics and Spectrum

One Dimensional Quantum Mechanics: The Free Particle LUKE CORCOS WITH JACKY CHONG DIRECTED READING PROGRAM

Books Lectures on Quantum Mechanics for Mathematics Students by Faddev and Yakubovskii Modern Quantum Mechanics by Sakurai

Introduction Quantum mechanical behavior is described by the Schr dinger equation: ( ) d t = ( ) ih H t Time Dependent: dt ( = ) ( ) H t E t Time Independent: E=Energy; h=Plancks constant

Introduction 2 P ) Q = + ( H V Hamiltonian: 2 m Coordinate Representation Momentum Representation Q = ( ) ( ) Q x x x = ( ) ( ) Q p ih p p P h = ( ) ( ) = P x x ( ) ( ) P p p p i x Q= Position Operator P= Momentum Operator

Free Particle (V=0) 2 P 2 H When V=0: = m Let s first look at solution of the Time-Independent Schr dinger Equation and solve for the spectrum of the Hamiltonian Operator 2 P 2 = E = E H m

Spectrum of Free Particle h = In the coordinate representation: ( ) ( ) P x x i x 2 2 2 2 P 2 h 2 d So = E = E m m dx For simplicity, define units such that h=1 and m=1/2 k = 2 Define the wave number k>0 s.t. E

Spectrum of Free Particle + k = 2 ' ' 0 So = ikx ( ) x Ce which has solutions: k 2 ( = Now Normalize: * ' ikx ik x ) ( ) x x dx C e e dx ' k k Using the property of Fourier Transforms 1 2 2 = 2 = ' ikx ik x ( ) ' k C e e dx C k C 2

Free Particle Dynamics To understand how the Free Particle develops with time, we now shift to the Time-Dependent Schr dinger Equation ) ( dt d t = ( ) ih H t For simplicity, let s start with the momentum representation: P = ( ) ( ) p p p

Free Particle Dynamics ( dt , ) d p t = 2 ( , ) i p p t 2dp = =1 ( ) 0 , p ( ) p ( ) p Where and 2 = ip t ( , ) p t Ce The Solution: = = ( ) 0 , p ( ) C p 2 = ip t ( , ) ( ) p t p e

Free Particle Dynamics Now we put the solution into the coordinate representation using the Inverse Fourier Transform 2 2 = = 1 1 1 ip t ip t ( , ) [ ( ) ] [ ( )] [ ] x t F p e F p F e 2 x 4 1 i 2 = = 1 1 ip t [ ( )] ( ) F p x [ ] F e e t 2 it

The General Solution ( ) 2 4 x y 1 = ( , ) ( ) x t y e dy it 2 it

Approximation of Long Term Dynamics 2 = 1 ip t ( , ) [ ( ) ] x t F p e 1 2 = ( ) i px p t ( , ) ( ) x t p e dp 2 px 1 2 ( ) it p = ( , ) ( ) x t p e dp t 2 Let s now use the stationary phase approximation to look at the dynamics as t

Stationary Phase Approximation This method approximates integrals of the form b a N = ( ) iNf x ( ) ( ) I N g x e dx This has the solution 1 2 2 1 i ~ ~ ~ = + + ( ) ( ) exp[ ( ) sgn ( ' ' )] I N g x iNf x f x O ~ ( ' ' ) 4 N x~ f x N = Where is the point where ( ' ) 0 f x

Approximation of Long Term Dynamics px 1 2 ( ) it p = ( , ) ( ) x t p e dp t 2 px x 2 ~= N = = 2 t ( ) f p p p t t 2 x 4 1 1 x 2 as ( ) i t = + ( , ) ( ) x t e O 4 t t t 2 t

Consequences 2 x 4 1 1 x 2 ( ) i = + ( , ) ( ) x t e O 4 t t t 2 t x 2 = p 1. The point of stationary phase occurs at So recalling that m=1/2, the classical momentum equation is realized. 2. so ) ( ) , ( t , ( t x t p = mv 1 C = x t O ( , ) x t t 2 t ) 0 This implies as

Source: https://en.wikipedia.org/wiki/Wave_packet#Free_propagator

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