Review of canonical formalism in CH
Newton's E o M
ZUCH
Mii
sa
can be derived from the least action principle
action
te
S 044
t
dt L x c
L
g
th I
x ie
a
Vix
Lagrangian
kit _Kz
851am
x
ftp
xcti
o
tix
Euler Lagrange Eg
Tiff
w
I
d
2x
equivalent toNewton's E o M
Elm
on
Ee mii
au
m
o
Hamiltonian is defined by
sip Lex.si
HCx.pl
Eg
L
9 Ii
VCH
I
die
P mic
H zip
in
pie
m
Newton's
p
zm.io
pie
am
Vcu
p
2x
I
214
2P
Vcu
Hamiltonian is energy
E o M can be recast into Hamilton eq
211
I
2M
I
E Leg
w
P
Consider the differential of
del
IH
2x
ax
of poi
xSep
fix
Iip
L
own
211
2P
IE
Ifi'd.net tdep
mm
g
The time evolution of dynamical variables VK.pl is given
w
2e
where
I
v
by
H
is the Poisson bracket
e
2b
2e
I
L in time
psix
deH
1
p
die
Six
St poi
s
JX
w
w
at iii ipa ii Ei
2b 2x
ax at
W
2b AP
Srp at
IH
op
for V
In particular
EH
2T
W
1
M
2k
EH
2x
N H
IH
2x
H
H H
0
Since His energy this implies the conservation of theenergy
From the analogy between thequantum commutator and classical
Poisson bracket
air
at
we expect
ii IT
u
B
dynamics
Quantum
Suppose the system is
at a state 147 at time t O
after some time t
o
in general
the system is at a different state 1441
We can think of thetimeevolution operator Ole connecting 14 and 14kt
OH147
1414
For the probability conservation it must be 4147
1
41 CHOC4 47
4411441
Ctl
1
for any 147 with 4147 1
I
It
441 441
The operators satisfying Ut U I
i
i
t
called unitary operators
are
The time evolution of the quantum States are described by
Otherwise the probability will not be conserved
unitary operators
44,14417
1
C
044 UH
1
a
The time evolution operator must satisfy
n
n
title
n
n
Ul
t U t
o
I
n
U tz Ult
2
Tf
3
4
For infinitesimal time evolutions all of theabove properties can be
satisfied
if
i
iride
I
de
with Gt
r
ignoring Old 17 terms
o
Cdt Cdt
I
iridt
1
iridt
1
UCdt Utd 4
1
iridt
l
ird c
I
in Cdt dt
Uld 1 1 dta
z
UC delUCdt
3
U o
Using
I
1
told 5
I
Old51
Olde
Old 4
t
iridt
I
I
iridt
g
j
2
Uct It
Uct Ide
ird c UH
l
U dt Uct
VH1
it
20th
is UH
at
Ct
de
The solution to this equation is
t
e
irt
The proportionality constant must be 1
since Uco
1
So far we consider theStates evolve in time while the observables
are constant in time
This picture is called Schrodinger's picture
As oppose to this
Heisenberg thought the states were constant
in time while observables were described by time dependent operators
This picture is called
Heisenberg's picture
Let's consider how the expectation value of an observable evolves in time
4th I A 1441
Ace y
Schrodinger's picture
41Utu A UHH
541 At4147
L
Heisenberg's picture
In Heisenberg'spicture all observables evolve in time as
ALet Ute A Uk
while physical States are constant
In Heisenberg's picture At4 is time dependent so the eigenkets must
also be time dependent
la Cei
is
They are given by
Utca la
a
time dependent eigenket
constanteigenket in 5P
in H P
y
1
A
Ct 1a Iti
e
Ult Ault Uttilai
T
Aila
nm
a Utes1a
ai la ti
Schrodinger and Heisenberg pictures are physically equivalent
We can show this statement by comparing the probability distributions
of the measurement of A at time t
assuming the state and
operator are given by 147 and A at
1 0
In Schrodinger's picture the probability of observing a value A is
Rail5
p
In
Itai14Ce I
tail Uct 1431
same
Heisenberg picture it is
Pit41 p
sail41471
Caillou 1471
Therefore both pictures predict the same results
I Summary
observables
States
Schrodinger's
picture
Heisenberg's
picture
Hit
µ
eigenkets
UC4147
constant
y
A constant
lair
AHI Utc4AUH
laicD Ukula.is
ed Consider infinitesimal time evolution Uldt
observable
irdt
for an
is in Heisenberg's picture
Adt bit Olde
U etat
l tilde tercet i
inde
bit
dt
i
ga
bits
A
we see
É
I
I
r
E
Comparing this with theabove eq
where
I
It is Hamiltonian operator
or
energy
g
it
Float
3 Oct
Oct
é
t
Multiplying 147 to eq.BG
144
UH 47
and using Schrodinger state
we have
it 1144
This equation is called
A 144
Schrodinger equation
the quantum States evolve in time
and tells us how
Energy eigenstates and Solutions to the S E
There is a systematic way of solving Schrodinger equation
The key of this idea is the following 3 points
1
Any state can be expanded in ternes of the energy eigenstates
FIE CE 47
14
2
C
E 143
If the state at t o is given by an energy eigenstate IE
E IE
HIE
thestate evolves in time as
é't't Ei
Eilts
Namely
tht is a phase factor IE ti and E
If
hence called
14,141
combination
stationary States
and 14 et
are solutions to the S.E
superposition
143141
is also a solution
correspond to
Therefore theenergyeigenstates do not change
the same physical state
in time
a
EqCal is a solution to the S E
Since the factor
3
E Ci IE
C 14,61
any linear
of them
Ca 4,41
C Ca
EG
Proof 2
é't't E
IE It
it
IE H1
it
é't
it
c
E
E
é't
E
E
leaky
Proof 3
it
4,141
it
c
4,41
4 ti
t ca
laces
c
I
1414,417
H
4,41
c
Fromtheabove 3 points
1426
ca 4G
C É
with Ck
Ek 4
TE
H14341
yr
it is clearthat
if the state is given at t o by 147
14 t
ibis
H
c é
tie
t
at t 0
it is given
by
EC e É't IED
provided there is no measurement after t o
If some measurement is made the state jumps into someeigenstate of the
corresponding observable
Theorem
Consider an observable A which commutes withthe Hamiltonian
A 513 0
stays the same eigenstate of A over time
An eigenstate of A Ia
A lait
for any t
ai hilts
Proof
Theformal solution to the S E
where ett
p.to
For 14401
a
Alain
Aé
t
É
1461
1441
it
H1461
is givenby
14401
Jett ten
tI
ÉÉ a
lait
la E é
AH O
and
Alais
ai laity
T.la
The above theorem is trivial in the Heisenberg picture
From eq
above
and
the time dependent observables in the H P
evolves in time as
3
it
This equation is called
H V
Heisenberg's equation of motion
In H P it is trivial that an energyeigenstate stays the same energy
eigenstate since both state and I
It if H H
0
are independent of time
In thesameway
if
A A1 0 theeigenstate
of A stays the same eigenstate over time