quantic vs classic world what defines the granular nature of our universe? our classical...
Post on 05-Jan-2016
212 Views
Preview:
TRANSCRIPT
What defines the granular nature of our Universe?
2
346.626 10
kgm
sdeBroglieh
mv mv
-24 -10
-35
for an atom, 10
f
~1
or a person 60
0
~ 10
~
~
Kgm
s
Kgm
s
m
m
mv
mv
Our classical “behavior” vs the atomic “quantum” characteristics
are a consequence of the absolute size of hSo far, we “solved” the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and
Can we obtain Q assuming classical behavior for the H?
What do you expect to be “different” between the 2 answers?
So far, we “solved” the Q.M. problem and then count all states to
get the partition function Q. While trying to count states, we
invoke the classical limit and
Can we obtain Q assuming classical behavior for the H?
We can obtain the states energies by solving the classical Hamiltonian equations
where j=1,2,...3N
j jq p
qj j
H Hp
We count the states by analogy to the quantum treatment…
QM treatment classical treatment
e e
classq e dpdq
p,qH
2
21 conjugated pair p,q
1Harmonic Oscillator: ,
2 2 osc
pp q k q
mH
vib,class
,q
p qe dp dq
-
H
2
21
2 2 osc
pk q
mkT kTe dp dq-
2
21
2 2vib,classq
oscp
k qmkT kTe dp e dq
- -
121
2vib,class
2q . 2
osc
kTconst mkT
k
122
we use1
2
bxe dx
b0
12
. 2osc
mconst kT
k
vib,classq .
kTconst
osc
where we used 1
2
m
k
vib,QMqkT
h
1const
h
2 2 2
Translation kinetic energy: ,2
x y zp p pp q
mH
trans,class 3 conjugated variables pairs
,q x y z x y z
p qe dp dp dp dq dq dq
-
H
2 2 2
2trans,class
-
q . x y zp p p
mkTx y z x y zconst e dp dp dp dq dq dq
-
Volume
22 2
2 2 2trans,classq .
yx zpp p
mkT mkT mkTx y zconst e dp e dp e d Vp
- - -
122
we use1
2
bxe dx
b0
3
2trans,classq . 2const mkT V
32
trans,QM 2
2q
mkTV
h
1const 3h
22
2
1Rotations ,
2 sin
pp q p
IH
rot,class 2 conjugated variables pairs
,q
p qe dp dp d d
H
22
22
rot,class
0 0
12 sin
q
pp
kTe dp dp d d
-
I
2rot,classq .8const kTI
2
rot,QM
8q
kT2
Ih
1
const 2h
vib
3trans
2rot
q 1 conjugated variables pairs
q 3 conjugated variables pairs
q 2 conjugated variables pairs
h
h
h
Comparing the high T limit of the QM Stat.Mech. with classical
Stat. Mech. we infer the constants
We are establishing a “correspondence principle” between
QM and Classical Statistical Mechanics
classical treatmentQM treatment
,
1
1 p q
j jj
sq e e dp dq
H
sh
# of conjugated pairs p,qs
,
1
for a system with N indistinguishable partic
1
les !
!
N
p q
class j jj
N
s
dp dq
N
Q eN
H
sh
Consider N interacting particles:
Each particle has s degrees of freedom
j
j
coordinates q
momenta p
s N
s N
EVERYTHINGknowing and we know about the syst.
we can predict a trajectory
j jq p
Construct a space with 2 dimensions,
2 axes (one for each
aSP CE
nd )A
j j
s N
sPHASE
N q p
in fully desc rone point ph ibes the syase s stem p e ac at t
evolution of the complete system = evolution phase pof oint
where j=1,2,...sN
j jq p
qj j
H Hp
j jintegration of these eqs. q t t p
a setMicro of canonical identi Ense cal sm ystems ble: A
each identical system one phase point
Equal a priori probability principle
each point is equally probable
there must be a point for each and every set of ,
consistent with N,V,E
p q
in a surface of constant E, density of points is uniform
j j
evolution of each system is independent of the others
in phase space, each point's trajectory is independent
q t t evolves independently p
for a given instant in time density number , ,
, , = # systems with between
between q
t f p q t
f p q t p p and p dp
q q and dq
integrating over the whole phase space
gives the total number of systems, , f p q t dpdq A
Ensemble average of property ,
, , , 1, , , ,
, ,
p q
p q f p q t dpdqp q p q f p q t dpdq
f p q t dpdq A
Gibbs postulate:
Ensemble average of property hermodynamic value prop.T
Liouville equation
1 2 1 2
1 2 1 2
consider a small volume defined by
V
around the phase point , ,... , , ,...n n
n n
p p p q q q
p p p q q q
#points inside VN
N N’
#pts entering through one face # pts exiting through another face?
q1q1q1 q1+q1
1
2 1 2
1# pts entering at q
at a given time , ,n nt
dqf p q t q q p p p
dt
1 1 1 2
1 1 2 2
1 1# pts exiting at
at a given time , , ,.., , ,., , n
n n
q q
t dq p q q q qf p q q q q t q q p
dt
1 1 1# pts entering at q # pts exiting at
at a given time at a given time the net flow is given by - q q
t t
1
2 1 2
1 1 1 2
1 1 2 2
, ,
, , ,.., , ,., ,
n n
n
n n
dqf p q t q q p p p
dt
dq p q q q qf p q q q q t q q p
dt
1 1 1 2 1 1 1 2 2
, , , , ,., , , , ,..n n n
f p q t q f p q q q q t q p q q q q q q p
1
1
1
1 1 2 1
11 1 2 1 1 1
11
, ,., ,, , ,., , , ,., ,
for small q,
, , ,., , , ,., ,
n
n n
f p q q tq
qf p q q q q t f p q q t
qq p q q q q t q p q q t q
n n q
1 1 1 2 1 1 1 2
1 1
1 1 1 1 1 1 1
1 1 1 1
f=f(p,q,t)
, , , , ,., , , , ,..
n n
where
f p q t q f p q q q q t q p q q q q
q f f qfq fq f q q q q q
q q q q
1
1 1 1
1 1
q ff q q q
q q
1 1 1 2 1 1 1
2
2 2
1
1 1
1 1
1 this is the net flow in q d
, , , , ,., , , , ,.
irection
.
n n n
n
f p q t q f p q q q q t q p q q
q p
q
q
q q q p
q ff q q
q q
2
1
1
1
1 1
1 1 this is the net flow in q direction
in a similar way we can obtain the net flow in p direction
n
p p qp f
f pp p
p
Net flow in all directions
since V is the volume in phase space (generated by all )
the net flow in all directions is
qp axes
d N
dt
we can simplify the equation by recalling the eqs. of motion
2 1
1 1
1 1
1 1 1 1
11
n n
n
j
p p q qp f q f
f f q pp p q q
d Np
dt
j jq p
p qj j
H H
j
j
jq
q
p
pj
2 2
and j
j
j
j
q
q
p
q p p p qj j j j
H H
V
1
, ,j
j j
n
jj
f fq
p q
f p q tp
t
Nd
Vdt
the change in density with time
around point (p,q) is
Full derivative of f(p,q,t)
1
it is a partial derivative
because ( , , )
, , j
j j
n
jj t
f fq
p p qq f
f p q tp
t
on the other hand, the full derivative is
, , j
j j
j jj
f f fq
t p qdf p q t dt dp dq
, ,j j
j
j jj
dp dqf f fq
t p dt q dt
df p q t
dt
j
j j
j jj
f f fq
t p qp q
1j
j j
j
j j
n
j j jj j
f fq
f
p qpq
q
fpp q
, ,0
df p q t
dt
Constant density conceptthe density around any moving phase point
is constant for that trajectory
ot the point has coordinates ,
t the point has coordinates ,o op q
p q , ; , ;o o of p q t f p q t
, ;
, ; and are functions of ,o o
o o
o o
p p p q t
q q p q tp q p q
Volume around p,q
NN
Volume around po,qo
with N in the surface
o at t , V around and has phase points on its surfaceo op q N
o at t +dt, points propagated, to a new V around and
(different shape).
in initial volume final volume
p q
N N
no trajectory can cross the surface of that volume because
if 2 trajectories cross (they are at the same in the same
phase space position), then they must remain the same.
Since the process is govern
t
by which cannot yield two
different evolutions trajectories do not cross
H
, ; , ; and # pts inside volume is constant
shape changes, but volume is constanto o of p q t f p q t
o o p q p q at all times
top related