Quantum Mechanics: Deviation, Time-Independent Equation, and Energy Bands, Slides of Computer Science

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Schrödinger Equation

1. Wave Equation

2. Time Independent Schrodinger Equation

3. Transfer Matrix Approach and the

PCPBT Tool

4. Periodic Potentials and the Periodic

Potentials Tool

► Statistical interpretation

Born’s statistical interpretation : { probability of finding the particle between x and ( x + dx ) at time t }

► Probability

is probability density

 x t dx

  ^

2 The probability of infinite interval : ( , ) Docsity.com

► Operator and expectation value (average / mean)

expectation value of position x :

expectation value of momentum :

x  x x t dx  x dx

 

  (^)      ( , )^2 *

operator x represent position;

operator represent momentum in x -direction. ■ all physics quantities can be written in terms of position and momentum

► Heisenberg uncertainty principle

(proof ref. chap 3)

standard deviation the variance of distribution, where individual physics quantity

2 2

2 2

2 2

2 2 2

2

( 2 )

( ) ( )

j j

j j j j

j j j j

j j j

 

  

  

    

2. The Time Independent

Schrodinger Equation

► 2.1 Stationary state

Assume V is independent of t , use separation of variables

Deduce from equation (2.1) , then

time-independent Schrödinger equation

thus equation (2.2)

(iii) Linear combination of separable solution

► 2.2 Infinite square well( one dimensional box)

and boundary conditions:

normalize

■ first three states and probability density of infinite square well

solve equation (2.3) by use ladder operator

rewrite equation (2.3) by ladder operator :

compare equation(2.3)

similarly

discussion (i)

and

(iii) there must exist a min state with

and from

and

so the ladder of stationary states can illustrate :