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W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
1
→ Elaborate & successful classical (non-quantal) theories exist (Mechanics, Thermodynamics, Electro-weak Interactions, Gravitation)
→ Sophisticated quantal theories exist for most, but not all, physical phenomena, expands ‘understanding’ to micro-cosmos not accessible to classical theory.
Are cl & qu theories equivalent and incomplete model views of an existing physical reality?
Does quantum theory negate all, most, some…concepts and analytical tools?
Are there isomorphisms in basic structures of cl & qu theory?→ Can one utilize classical models to guide quantal
simulations?→ Are there methods to translate (“quantize”) classical
experience?
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
2
Classical Phase Space and Functions
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
3
Ole Steuernagel, University of Hertfordshire, UK
( )
( )1 2
1 2
2
( ). :1
( ) 3
, ,...., ;
and , ,....,
: :
:
n
n
pq q p
pq
Consider system with n degrees of
freedom dof Example free particle
mass m with dofs
Canonical coordinates q q q q
momenta p p p p
Phase space
dimension n
=
=
=
→
( ); ( ) .qpSystem trajectory q t p t is confined to at all times =
( , , )
: andi ii i
Dynamics is determined by Hamiltonian H q p t
d H d HEquations of motion q p
dt p dt q
= = −
, : ( , ; )Any dynamic function f can be described in terms of q p f q p t
Restricted Dynamics: Poisson Brackets
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
4 1 1
( , ; )
( .):
N N
i ii ii i i i i i
Arbitrary function f q p t representing aspects of a dynamic system
evolves according to Hamiltonian constrain chaits
df f f f f f H f Hq p
n rule diff
dt t q p t q p p q= =
= + + = + −
: ,df f
f Hdt t
= +
, :
Poisson Bracket of any two dynamic
functions f g coupled via H
1
, :N
i i i i i
f g f gf g
q p p q=
= −
, ,: ,;i j i j ij i jq q 0Specific PB B t qup pp = = =
Required !
,i j ijq p =
1: ,i j ijQuantum Mechanics p q
i =
Poisson Brackets Properties
W. Udo Schröder, 2018
Cla
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Qua
ntum
5
, :
Poisson Bracket of any two dynamic
functions f g coupled via H
1
, :N
i i i i i
f g f gf g
q p p q=
= −
Properties of Poisson Brackets
▪ Linearity in both arguments: {af+bg,h}=a{f,h}+b {g,h}, etc.
▪ Anti-commutative: {f, g}=-{g,f}
▪ Product rule: {f·g,h}= f·{g,h}+ g·{f,h} (from rules of
differentiation)
▪ Jacobi identity: {f,{g,h}}+{g,{h,f}}+{h,{f,g}}=0
Hamiltonian Dynamics Enforced
Example of classical Equations of Motion using Poisson bracket formalism
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
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6
: q , , ( )
:p , , ( )
dq qq H q H no explicit dependence on time
dt t
dp pp H p H no explicit dependence on time
dt t
= = + =
= = + =
( )
( )
( )
( ) 11
1 1
:
, ,, ,,
, , , ,
NNi j ij
N N
Canonica of phase space coordinates
Q Q Qq q qmust fulfill Q P
p p p
l transformation
P P P
== → =
= =
q
p
t
Liouville Theorem
( )
( )
( ): ( ), ( )
( ) and ( )
,
,
i j ij
System trajectory in
Multi dimensional R t q t p t
q t p t correlated via H q
phase s
p
Specifically q p
pace
=
− =
→ ( )R t
(0)R
Hamiltonian Dynamics: Liouville Theorem
Example of classical EoM using Poisson bracket formalism
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
7
( ) ( )( )
: , 0 ( !)
, ( ), 0q
i ii i qp qp
ii i
p q
i
p
i
q pAlso q p v incompressible
p q q p
dR t from continuity equatiR on
tt v
d
= = − → = + = →
=→ =
q
p
t
Liouville Theorem
( )R t
(0)R
( )
( ) ( )
( )
( ) ( ) ( )( )
2
( ), ( ) ,
( ): ( ), ( ) ; 1
: ( ), ( )
:
, , , 0
, 0N
qp
qp qp
Phase space distribut
Continuity Equation
no points created or
ion q t p t R t dR
R t q t p t dq dp
Generalized velocity v q t p t
dR t R t v R t
d
annihilated
R t
t t
=
= =
=
= +
=
=
0t
→ =
( ) ( ) ( )( ), , ,H ,qp qp
dR t R t R t v
dt → = =
( ),
0
qp
qp qp
qp
Flux density
R t v
div
=
=
Dynamic Correlations: Angular Momentum L
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
8
1 2 3
1 2 3
1 2 3
ˆ ˆ ˆx x x
Angular momentum L r p L x x x
p p p
= → =
1 2 3 3 2
2 3 1 1 3
3 1 2 2 1
L x p x p
L r p L x p x p
L x p x p
−
→ = = = − −
3
1
2q
f
r
1 1 2 3 1 2 3 2 3( , , ; , , ), (...), (...)Dynamic functions L x x x p p p L L
( )3
1 2 1 21 2 2
12 11 3: 0 0
cossin
tan
sin: cos
ta
,
n
i i i i i
L L L Lx p x p and cyclic
x p p x
p p
Also true in spherical coordinat
L L L
es L r p p
p
q f
q f
f
ff
q
ff
q
=
= − = + + − =
− −
= −
1 2 3
1: ,Quantum Mechanics L L L
i=
2
3 .
Simultaneous rotation about axes
affect motion about the rd
QM Preserves Interdependence of Observables
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
9
1
ˆ ˆ,
.
.
.
, 0i j i j
Classical mechanics enforces correlations between canonical observables
Canonical observables are interrelated via differential equations of motion
Character of phase space trajectories is preserv d
qi
q q
e
q =
1 2 3 1 2 3
1 1ˆ ˆ, 0 ; ,
: ,
ˆ ˆ0; , 0 ,
1 ˆ ˆ ˆ, ,
ˆ ˆ 1 ˆ ˆ: ,
,
:
i j i j iji j i j ijp p q p
df ff H
d
q q q pi i
L L L L L Li
dA AA H
dt t i
For cons
Time e
istency with classical mechanics
viable operators have to com
volutit t
n
p
o
= = = = =
= =
= +
= +
.ly with above quantum commutation relations
Classical mechanics is the limit of quantum mechanics for large qu. #s
Ehrenfest Theorem
Relation of classical phase space trajectory to quantum expectation values . → SCl approx
W. Udo Schröder, 2018
Cla
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al 2
Qua
ntum
10
ˆ ˆ,t t
q p
Ԧ𝑞 𝑡 , Ԧ𝑝 𝑡 𝑐𝑙
22ˆˆ ˆ, ( ), ,
2 2
1ˆ ˆ ˆ ˆ ˆ, ,
2
xx
x
x xx x x x
pi i iH x V x x p x
m m
pip
d
p
xdt
v classicax p x p pm m m
l
= + = =
+ = = =
=
2ˆˆ ˆ ˆ, ( ),2
ˆ( ), ,
xx xx
x
da
dp
dt
dV
pi i
lmo
H p V x pm
iV x st classical but V
dd xxp
= +
= −=
=
−
( )
2 2ˆˆ ˆ1 ( ) ( )2 2
0
x xp pConsider D single particle H V x H V x
m m
Assume wave function of system x known at t
− = + = +
=Paul Ehrenfest1880-1933
Ehrenfest Theorem Extended
Almost classical limit, but <dV/dx>≠d<V>/dx → quantum correction!
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
11
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2
2 2
D.J. Tannor(4.1): :
1 1:
2 2
:
1
2
Expand aboutFollowing x
x
At x x V x V x
dV
V x
x Vdx
V x V x x x V x x x V
x x V x V x
=
+
→ + = +
+ − + −
−
( ) ( ) ( )
( ) ( ) ( ) 4
2
x
x V x
V
x p m
p V x t
m x t
=
= − −
= −
( ) ( ), 0 (0), (0), ;x t get t from semi classical trajectory = → −
Approximate semi-cl trajectory: Solve set of ODEs from initial conditions
Semi-classical trajectory accounts for some quantum corrections.
( ) ( )
22
: .2 2
x xp pV x V x const
m m
= + − +
:
Assume energy
difference qu cl−
Alternative (Hydrodynamic) Formulation TDSE
TDSE is re-written as equations for real profile function A(x,t) and real phase function S(x,t). → probability density (x,t), velocity S/x
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
12
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( )
* 2,
( ) ( )
, : , , ,
ˆ,
,
,
,
iS x t
x t A
General trial wave function for single particle m
x t x t x t A x t
i x t H x t
e
t
x t → = =
= →
=
2 2 2
2
2
2
1
2 2
10
2
S S AV
t m x mA x
A A S A S
t m x x m x
+ + =
+ + =
2
2
( , )
( , ) 0
(
( , )!
1 ( , )0
( , )
" " ( ))
( , )
, :
x
x
S S x t
t m x x m t xx
x t
t x
Flux
S x t
m x
x
current dens
d x tconserved p
ity velocity
moment
robdt
S x tm
x
t
x
u
t
m
= + + = +
→→ = +
→
=
=
=
3d r
2A
From Im
From Re
Geometrical Optics Limit
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
13
( )( )
( )
2 2 2
2
2 2 2
2
2 22 2
2
1
2 2
2
: 1
S AV
m x mA x
S Am E V x
x A x
x r
t
AA
S
S
+ + =
→ = − +
→ =
+
( ) ( )( )
( )
( )
,, ,
,
: ,
i iS x t E t
E x t A x t e A xStationa t e
S x t E
ry stat
S
tE
es
t
−
= =
= →
= −
( )( )( )1
2deBroglie wave length m E V r−
= −
( ) ( )
2 2
2 22 2 2
22 2
22
: ( , ):
1 ; ( )
" " .
oo t o tp p pt
Classical limit small deBroglie large m high energy A A
S A n r index of refractionA
rays wave fronts S const
S n
Cl velocity S
= + →
⊥ =
→
S(0)=aS(0)=b
S(0)=c S(0)=d
D.J. Tannor
. " "Cl action
S E t=
Principle of Least Action
Classical trajectories of particle (mass m)can be defined by minimum “action” S along path → Lagrange EoM
Analogy: Fermat Principle for optical rays. Media of different index of refraction n.
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
14
( ) ( ) ( ) ( )( )
( )
( ) ( )2
2
2
1
2
0
1
(
, : ( ): ,
2
( )
0 ): 2 ( )T
t
t
x
particle m
x t x t acti
Action S T m x V
on S t t L x t x t dt
Lagrangian L T V m V for
Least action extremum Lagrange E
x dt
oM
→ = −
→ =
=
− = −
( ), 2
Phase difference D d
Constructive interference if n
= − →
=
c
c/n
d
D
Fermat Principle
x1
t
x
Least Action
0
V(x)
T
( )
( )211
1
: ; 1,...,N
(t 0 )2 2
n
Nn nn n
n
Discretize trajectory x x n t n
x xx xmS t T V t
t
++
=
= =
+− = → = −
Quantal Trajectories: Path
Consider particle (mass m) in 1D:
W. Udo Schröder, 2018
Cla
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al 2
Qua
ntum
15
21ˆ ˆ ( )2
xH p V xm
= +
x1
t
x
t
V(x)
t’
x
x'
( )( )
( ) ( )
( ) ( )
ˆ
, ,,
,,, ,
iH t t
x t x e t x t x t
x t
dx t
x t
x x
x t
dx
d x
x
tx
+ +
−
−
+
−
−
−
= = =
=
→
, ,
,
Evaluate quantal transition amplitude x t x t in p representation
See Townsend Tannor
−
( )
( ) ( )
( )( )
( ) ( ) ( ) ( )
ˆ
:
, : , ,
ˆ
ˆ,
iH t t
For given end point x t x t dx x t
TDSE unitary time transformation
Position repr
U t t e
x t x t x
esen
U t t
tatio
dx
x
x x
x
t
n
− −
+
−
→ −
=
= = −
Free-Particle Propagator
W. Udo Schröder, 2018
Cla
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Qua
ntum
16
( ) ( )
( )
( )
2ˆ ˆ
2
2
2
2
2
2
ˆ
1ˆ ˆ:
ˆ
2
ˆ1 1H px
x
i it t t t
m
it t
m
it
m
x
p
p
t
Simplest case free particle H pm
x e x e x
x e x
x x
x
dpdp p p p p
dp p ep
− − − −
− −
− −
→
=
=
=
x1
t
x
t
V(x)
t’
x
x'
, ,propagatorEvaluate tran x t xsition amplitude in p representationt −=
( ) ( ) ( )
( )
( )( )
2
ˆ22
2
1:
2
1...
2 2
ip x
m x xii i iH t t p x x t t t tp
m
Insert momentum wf in position representation x p e
mx e x edp e e
i t t
− − − − − − −
=
=
→ = = −
( )
( )
2
22
x xmAction gain S t E t
t t
− = −
,: , i SStructure of propagat x t x t eor Gaussian integral
Coherent State Propagator
W. Udo Schröder, 2018
Cla
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Qua
ntum
17
( )
( )
2
2ˆ
2
ˆ2
: ( ) 0,
1
2
1ˆ ˆ: ( )2
(
:
ˆ ..
)
)
, 0
. (
0
1
pH t
m
H t
i i ip x x t
x
i
For free particle V x
x e x e e
Now add potential H p V xm
V x much more difficult except for t
iNon linear op
t t t
dp
H t O te
− − −
−
→
=
= −
=
+
→
→ − +−
( ) ( )ˆ ,
:
1
2
( ) , 0, 0
i i iH t p x x E p x t
approximation Evaluate evolution
x x x x e x e e
by coherent sum integrals over small intervals
Quan
t
dp
ta
x
l
− − −
= = + →
→
( ) ( )( ) ( )
( ),1
,0 ,0,2
,, 0
i ip x x E p x t
x tx t dx x d xx e exdp
− −+
−
= =
1i it t t+ = +
xi+
1=
xi+
x
Integration over all intermediate x-values and all p values. Extend to 3D.
Following Townsend, MAQM
Coherent State Propagator
W. Udo Schröder, 2018
Cla
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Qua
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18
1i it t t+ = +
xi+
1=
xi+
x
ˆ ˆ ˆ
0 0 0 0, , , ,
i i iH t H t H t
x t x t x t e e xe t− − −
=
N identical terms
1̂ i i idx x x+
−
= ,..,i 1 N=
:
0, 0
Small intervals
t x
ˆ
ˆ
1 1
1 1 2
0
2
0, , ,
,
,
,
N N
iH t
iH t
N N N N
x t x t x t
x x
e
t t
t
e
x −
−
−
−
−
− − −
→ =
2 2 3 3 1 1 0
ˆ
0
ˆ
, , , ,
i iH t H
N N
t
N Nx t x t te x xet− −
− − −
−
( )( )1
0 0 11
11
1 exp2
,2
, ,N
i ii
iN
Ni i
x xdp dp ip Edxx t p tx x
tdx t
−−
=−
− −
=
0 0 1 11
1
ˆ
1, ,, , ; , ,Ni i
N
Ni
H t
i N
i
ixx t x t dx dx wie tx xh x tt tt−
=−
− − = =
Gaussian momentum integralsSpace integrals
Coherent State Propagator
W. Udo Schröder, 2018
Cla
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al 2
Qua
ntum
19
1i it t t+ = +
xi+
1=
xi+
x
( ) ( ) ( ) ( )0
0
0
00
, , exp , expx x
x
t
x t
i idD xx t t L x x S xt D tt tx x
=
=
( )
( )
( )( )
11
21
1
2
1
exp ,
exp2
exp
2
exp2 2
2
2
i
ii
i iii i ii
i ii ii i
i i
dp
x xdp ip E p x t
t
x xdp pip V x t
t m
x xm i m it t V x
i t t
+
−
+−
−−
+−
−−
−
→
− = −
− = − −
− = −
( )1i−
( )1
0 0
12
:
Nx
NN x
x
x
m
iSpace integrals D x t over all patLim dx dx h
ts
−→
=
( )2
, 00
exp exp2
t
N t t
i mLim dt x V x
→ →
→ = −
( ) ( )2
0
exp , ( , )2
t
t
dp wi m
dt L x x Lagrangian L x x x V x T Vith
= − = −
=
Around the Classical Path
W. Udo Schröder, 2018
Cla
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al 2
Qua
ntum
2
0
( )
20
2
20 0 0
2
0 :
1...;with
V(x
02
2
) 0
:
.
2x x x
Consider quantal and classical paths x x
S S S and
Ins gradpect quantal pathways around c
Class Least S S
lass path at
x t const o
or const
t t
S
xm
Lr
0
= →
= + + +
=
= = = →
( ) ( )
( ) ( )0
0
0 0
2
exp
, ;
)
, ,
( 2 ?
:
t
t
x
x
N
D x t
Classical
Result of quantal calc
x t x
Lagrangian L T V
All pa
Phase Action
contr
iS x t
S x t dt L
ibute to proths
m i t no
pagator x
ne pre
integ
t
r
ferr
x x
with equal weight
als
ed
=
=
−
=
→
=
But classical particles follow specific path of minimum action=Proven experience!Is quantal treatment consistent with these observations? By what mechanism?
Around the Classical Path
W. Udo Schröder, 2018
Cla
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Qua
ntum
2
1
0
2 2
20
2 (1)
(0)
2
2
1) Path following bottom of potential V(x) V(x )
.
2 2
2)
2
cl x x
t
cl
x x t x t const
m x m xdt Least action
tt
Path on PES side wall const for
x
S S
ce
mconst x L xx
tSt
→
= = → = =
= =
→
→ → →
=
=
→=
( )
22
2(2) 2
0
1 11 1
2 2 3 3
cl
t
cl
x x tVariations about x x t x t t t
t t t
m m xS dt x t S No term linear in
t
= = → = + −
= = + = +
2(1)
2 2 22 2
4 40
22 2
2
3
tm x t m xL x m d
xS mt t
t t t
= = → =
=
( )2
2 2
2(3) 2 (1)
0
...2 1 1
1 12
. ....3 2 2
cl
t
x x tVariations about x x t x t t t
tt t
m m xS dt x t S
t
= = → = + −
= + += + = +
Interference Off the Classical Path (min S)
W. Udo Schröder, 2018
Cla
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Qua
ntum
2
2
(1) 2 22
2
15
10
2 2 210
3 3 3
2 0.51110
3 197 10
1.7 10 1.7
S mc x mc x mcx x
c t c t c
MeVx
MeV m
x x
m A
−
−
−
=
(3) (1) ....1
12
Use S S for qualitative estimates
++=
( )(3)1
( )
0.01 (scale )
iS
Calculate T eNorm
Vary path by Angstrom
=
=
( ): 0.01 26
?
xEx electron at speed c x t eV
Interference of phase factors around classical path
=
Math Help
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
2
3
2
2 22 2
24 4
( )
( , ) :2 4 2
( , )
x
x x
x x
Gaussian integrals I dxe
I dxe x x x x x
I e dxe e
a
a b
b ba ba a
a
a
b b ba b a b a
a a a
a b
a
+−
−
+− +
−
++ +− +
−
= =
= → − = − − → = −
→ = =
End
W. Udo Schröder, 2018
Cla
ssic
al 2
Qua
ntum
2
4
H2+F2→2HF
