adiabatic approximation.ppt

7/26/2019 adiabatic approximation.ppt

1/23

Physics 452

Quantum mechanics II

Winter 2012

Karine Chesnel


2/23

Homework

Phys 452

Tuesday ar 2!" assi#nment $ 1%

10&'( 10&4( 10&5( 10&!

Thursday ar 22" assi#nment $ 1)

10&1( 10&2( 10&10


3/23

*dia+atic a,,ro-imationPhys 452

Classical meaning in thermodynamic

In adiabatic process,

the system does not exchange energy

with the outside environment

Internal

,rocess .ery small / slow

ner#y e-chan#e

With outside


4/23


Adiabatic theorem

If the Hamiltonian changes SLOL! in time,a particle in the "thstate of initial Hamiltonian Hi

will be carried into

the "thstate of the final Hamiltonian Hf


5/23


#athematically$ dH Edt t

=( )H t +utCharacteristic #a,

In ener#y leels

Characteristic time

o eolution

Schr%dinger e&uation$ ( ) ( )i H t t t

=

h

( ) ( ) ( ) ( ) ( )0 n ni t i t n nt c e e t

='inal solution

The ,article stays in the same state( while the Hamiltonian slowly eoles

with ( ) ( )0

1 ' '

t

n nt E t dt = h

3ynamic ,hase

( )0

'

t

mmm

dt i dt dt =

eometric ,hase


6/23


( ) ( ) ( ) ( ) ( )0 n ni t i t n nt c e e t

='inal solution

with ( ) ( )0

1 ' '

t

n nt E t dt = h

3ynamic ,hase

( )0

'

t

mmm

dt i dt

dt

=

eometric ,hase

Internal

dynamics

3ynamics

induced +y

e-ternal chan#e


7/23

*dia+atic a,,ro-imation

Phys 452

( a

Pb 10.1: infinite square well with expanding wall

Pro,osed solution

( ), n nn

x t c =

( ) ( )2

2 / 22, sin

i

ni mvx E at

n nx t x e

=

h

1& Check that solution eriies chr6din#er e7uation ( ) ( )i H t t

t

=

h

4 terms 4 terms

2& 8ind an e-,ression or the coeicients"2

0

2sin( )sin( )

i z

nc nz z e dz

=

use ( ) ( ), 0 , 0n nc x x=


8/23


Phys 452

( a

Pb 10.1: infinite square well with expanding wall

Pro,osed solution

( ), n nn

x t c =

( ) ( )2

2 / 22, sin

i

ni mvx E at

n nx t x e

=

h

'& Internal/ e-ternal time/iEte h

Phase actor"

Internal time

4& 3ynamic ,hase actor"

( )1 2

0 0

1 '( ') '

'

t t

n

dtE t dt

a vt =

+ h

( )t a vt = +

Wall motion"

e-ternal time

*dia+atic a,,ro-

i e

T T=


9/23


Pb 10.2: Spin precession drien by !agnetic field

0 cos sin cos( ) sin sin( )B B k t i t j = + +

r

Hamiltonian

e

eH MB SB

m

= = rr r r

Hamiltonian in the s,ace o the 9s,inors ( ),z z +

Check that it eriies the chr6din#er e7uation ( )i H tt

=

h

i#ens,inors o H)t* ( ) ( ),t t +

solution ( ) ( ) ( )t c t c t + + = +


10/23


Phys 452

Pb 10.2: Spin precession drien by !agnetic field

0 cos sin cos( ) sin sin( )B B k t i t j = + +

r

:Case o adia+atic transormation

1

E E + ==h

( ) ( )2

2

sin sin2

tt t

=

:Pro+a+ility o transition u, ; down

Com,are to P+ %&20

( ) ( )2

0t t

Pro+a+ility o transition u, ; down


11/23


Phys 452

Pb 10.10: adiabatic series

( ) ( )k mim k m kk

dc c t edt

= &*lso

( ) ( ) ( ) ( )mi tm mm

t c t t e

= ( ) ( )ni tm nmc t e =with


12/23

>early adia+atic a,,ro-imation

Phys 452

Pb 10.10: adiabatic series ( ) ( ) ( ) ( )'

0

0 'n m n

ti i t

m m m nc t c e e dt

= &

"pplication to the drien oscillator

( )2 2

2 2 2

2

1

2 2

H m x m xf t

m x

= +

h

ei#enunctions ( ) ( ),n nx t x f =

( ) ( ), nn ni

x t f f pf

= =

& &&

h

aluate m np ?sin# the

ladder o,erators

( )

2

mp i a a

+ =

h

aluate m n &

3riin# orce


13/23

>early adia+atic a,,ro-imation

Phys 452

Pb 10.10: adiabatic series ( ) ( ) ( ) ( )'

0

0 'n m n

ti i t

m m m nc t c e e dt

= &

aluate ( )0 0

' '

t t

nn n nn

idt i dt p dt

dt

= = h

aluate ( ) ( ) ( ) ( )( )0

1' ' '

t

m n m nt t E t E t dt = h

Possi+ility o

Transitions @@@

tartin# in nthleel ( )1 0nc t+

( )1 0nc t

Here ( ) 21 1

2 2

nE t n m xf

= +

h


14/23

>on; holonomic ,rocess

Phys 452

* ,rocess is Anon;holonomicB

when the system does not return to the ori#inal state

ater com,letin# a closed loo,

,endulum

arth

-am,le in echanics

*ter one

Com,lete

Hysteresis

loo,

irreersibility

-am,le in ma#netism


15/23

Qui9 '0

Phys 452

*& The motion o a ehicle ater one en#ine cycle

& The daily ,recession o 8oucaultDs ,endulum

C& The circular motion o a skater on rictionless ice

3& The circular motion o cars on racin# rin#

& The circular motion o a skiin#;+oat on a lake

Which one o these ,rocess does not allunder the non;holonomic cate#oryE


16/23

8oucaultDs ,endulum

Phys 452

,endulum

arth

rotatin#

olid an#le

0 2

0 0

sin d d

=

( )02 1 cos =


17/23

erryDs ,hase

Phys 452

( ) ( ) ( ) ( ) ( )0 n ni t i t

n nt c e e t =+eneral solutionAdiabatic approx

( ) ( )0

1' '

t

n nt E t dt = h

3ynamic ,hase

( )0

'

t

mmm

t i dt t

=

eometric ,hase

with

( )Rf

mmm

Ri

t i dRR

=

( )m

t i dRm R m = r

#erry$s phase


18/23

erryDs ,hasePhys 452

( )m

t i dRm R m = r

#erry$s phase


19/23

erryDs ,hase

Phys 452

(

1 2

The well e-,ands adia+atically

rom to1 2

( )2

1

1 2

nn n

di d

d

=


20/23

erryDs ,hase

Phys 452

Pb 10.%: "pplication to the infinite square well

(

1 2


rom to1 2


21/23

erryDs ,hase

Phys 452

(

1 2


rom to and contracts +ack1 2


22/23

erryDs ,hase

Phys 452

1& Calculated

d

2& Calculated

d


23/23

erryDs ,hase

Phys 452

Pb 10.5: (haracteristics of the geo!etric phase

When erryDs ,hase is 9eroE

:Case o real n

:Case o

' ni

n ne

=

0n =eometric

,hase

( ) 0n loop =

erryDs ,hase

adiabatic approximation.ppt

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