adiabatic approximation.ppt
Transcript of adiabatic approximation.ppt
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Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
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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
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*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
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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
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#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
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( ) ( ) ( ) ( ) ( )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
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( 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=
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( 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=
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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 + + = +
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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
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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
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>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
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>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
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>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
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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
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8oucaultDs ,endulum
Phys 452
,endulum
arth
rotatin#
olid an#le
0 2
0 0
sin d d
=
( )02 1 cos =
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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$s phase
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erryDs ,hasePhys 452
( )m
t i dRm R m = r
#erry$s phase
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erryDs ,hase
Phys 452
(
1 2
The well e-,ands adia+atically
rom to1 2
( )2
1
1 2
nn n
di d
d
=
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erryDs ,hase
Phys 452
Pb 10.%: "pplication to the infinite square well
(
1 2
The well e-,ands adia+atically
rom to1 2
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erryDs ,hase
Phys 452
(
1 2
The well e-,ands adia+atically
rom to and contracts +ack1 2
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erryDs ,hase
Phys 452
1& Calculated
d
2& Calculated
d
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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