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Ch 6. Commuting and Noncommuting
Operators and the Surprising
Consequence of Entanglement
- Applied simple quantum mechanical framework in
real experiment. (Stern-Gerlach experiment)
- Noncommuting operators concerning position and
monentum. ( Heisenberg uncertainty principle)
- Particle in a 3-D box and Quantum computers
MS310 Quantum Physical Chemistry
6.1 Commutation relations
ˆ and Bˆ
There are 2 observables a and b, corresponding operator A
We can think two cases.
1) Measurement A first, B after
2) Measurement B first, A after
1) first measuremen t : Â n ( x ), second measuremen t : B̂(Â n ( x ))  B̂Â n ( x )
If ψn(x) is eigenfunction of operator A(no state change)
ˆ  ( x)  Bˆ   ( x)   Bˆ  ( x)
Bˆ A
n
n n
n
n
Also, if ψn(x) is eigenfunction of operator B(no state change also)
ˆ  ( x)   Bˆ  ( x)     ( x)
Bˆ A
n
n
n
n n n
MS310 Quantum Physical Chemistry
ˆ ( Bˆ  ( x))  A
ˆ Bˆ  ( x)
2) first measuremen t : Bˆ  n ( x), second measuremen t : A
n
n
If ψn(x) is eigenfunction of operator B
ˆ Bˆ  ( x )  A
ˆ   ( x)   A
ˆ  ( x)
A
n
n n
n
n
Also, if ψn(x) is eigenfunction of operator B
ˆ Bˆ  ( x )   A
ˆ
A
n
n  n ( x )   n n n ( x )
ψn(x) is eigenfunction of operator A and B both
→ result is independent of the order of measurement
Otherwise, two results are different.
Two operator A and B have a common set of eigenfunction
ˆ [ Bˆ f ( x )]  Bˆ [ A
ˆ f ( x )]  0
→ must satisfy the commutation relation A
(f(x) is arbitrary function) and if it satisfied, A and B commute.
ˆ [ Bˆ f ( x )]  Bˆ [ A
ˆ f ( x )]  [ A
ˆ , Bˆ ] f ( x ), [ A
ˆ , Bˆ ] : commutator
Notation : A
MS310 Quantum Physical Chemistry
ψn(x) is eigenfunction of operator A : no change after the
measurement the observable a
ˆ [ Bˆ  ( x )]  Bˆ [ A
ˆ  ( x )]  0 satisfied : state ψ (x) is not
If A
n
n
n
change by the two measurement the observable a and b
→ ‘can measure simultaneously and exactly two observable a
and b’
Ex) 6.1
Momentum and a) kinetic energy b) the total energy can be
known simultaneously?
Sol)
ˆ [ Bˆ f ( x )]  Bˆ [ A
ˆ f ( x )]
Use the commutator A
2
2
d

d
a) pˆ   i , Eˆ
x
kinetic  
dx
2m dx 2
2
2
2
2
d

d

d
d
ˆ
ˆ
[ p x , E kinetic ] f ( x )   i ( 
) f ( x )  (
)(  i ) f ( x )  0
dx 2m dx 2
2m dx 2
dx
MS310 Quantum Physical Chemistry
Momentum and kinetic energy is commute.
Therefore, momentum and kinetic energy can be known
simultaneously.
d ˆ
2 d 2
b) pˆ x   i , H  
 V ( x)
2
dx
2m dx
2
2
2
2
d

d

d
d
[ pˆ x , Hˆ ] f ( x )   i ( 

V
(
x
))
f
(
x
)

(


V
(
x
))(

i

) f ( x)
2
2
dx 2m dx
2m dx
dx
d
d
  i (V ( x ) f ( x ))  iV ( x )
f ( x)
dx
dx
d
d
d
d
  if ( x ) V ( x )  iV ( x )
f ( x )  iV ( x )
f ( x )   if ( x ) V ( x )  0
dx
dx
dx
dx
Therefore, we cannot be known the momentum and total energy
simultaneously.
MS310 Quantum Physical Chemistry
6.2 The Stern-Gerlach experiment
Consider the dipole in the inhomogeneous magnetic field.
In this situation, dipole
orient and deflect to the
magnetic field.(parallel
and antiparallel to the
magnetic field)
MS310 Quantum Physical Chemistry
Stern and Gerlach did the experiment.
condition : external magnetic field applied to the Ag beam
Result : Ag beam split two beams.
→ 2 eigenvalues of measure the z-component of the magnetic
momentum
We write operator of measurement the z-component of the
magnetic momentum as A, wavefunction of one spin as α,
other spin as β.
c
c
  1   2  , | c1 |2  | c2 |2  1
2
2
MS310 Quantum Physical Chemistry
Cannot specify the value of c1 and c2.
However, ratio of two beam is 1 by the individual measurement.
1
2
2
| c1 |average
| c2 |average

2
Measure the direction of x-component of magnetic momentum
of the beam of state α : ‘split’ 2 beams!
(in this case, write the operator : B and wavefunction : γ, δ)
c3
c4
2
2


 , | c3 |2  | c4 |2  1, | c3 |average
 | c4 |average
1
2
2
MS310 Quantum Physical Chemistry
Then, operator A and B commute? No.
If two operator commute → eigenfunctions of 2 operators same
→ result of second measurement is only 1 state. Why?
‘after’ the first measurement, wave function collapse to only 1
measured state. Second measurement measures the ‘collapsed’
state, one of the eigenfunctions of the first measurement. If two
operator commute, second measurement measures the
eigenfunction of operator B, and result must be one state.
However, result of second measurement also split to 2 beams.
Therefore, measurement of z-component of magnetic moment
and measurement of x-component of magnetic moment do not
commute.
Result : Ag atom doesn’t have well-defined values for both μz
and μx simultaneously.
MS310 Quantum Physical Chemistry
6.2.1 The history of the Stern-Gerlach experiment
Experiment did in 1921
Ag beam generation : oven in a vacuum chamber was
collimated by 2 narrow slits of 0.03mm width
Beam passed into inhomogeneous magnetic field 3.5cm and
impinged on a glass plate.
1 hr operation in this experiment.
How can see the Ag?
→ ‘sulfur’
Sulfur reacts to Ag and makes Ag2S.
Ag2S : black, and it can see less than 10-7 mol of Ag
→ reason of successful experiment
MS310 Quantum Physical Chemistry
6.3 The Heisenberg uncertainty principle
Heisenberg uncertainty principle : ‘cannot know simultaneously
position and momentum of particle’
It starts that position and momentum do not commute.
Wavefunction of free particle : Ψ(x,t)=Aexp[i(kx – ωt – φ)]
Set φ=0 and t=0 : focus on spatial variation of ψ(x)
We normalized wavefunction into finite interval [-L,L]
L
L
L
L
* *
*
 ikx ikx
A

(
x
)
A

(
x
)
dx

1
,
A
A
e

 e dx  1, | A |
1
2L
Probability of x=x0 : P(x0)dx=ψ*(x0)ψ(x0)dx
L → ∞ : probability approaches to 0! → no data of position
It gives this result : if we know momentum exactly, position is
completely unknown
Similarly, if we know position exactly, momentum is completely
unknown
MS310 Quantum Physical Chemistry
Consider the superposition of plane waves of very similar wave
vectors
1
1 n  m i ( k 0  n k ) x
ik0 x
 ( x )  Ae  A  e
, k  k0
2
2 n  m
See the case of m=10(21 waves superposition)
MS310 Quantum Physical Chemistry
Wave vector k0 : 7.00 x 10-10 m
Case of 21 waves, peak of the probability : 0 , 3.14 x 10-10 m
→ range of probability exist decrease(wave packet) : probability
localized into finite interval → uncertainty of position increase.
Superposition of a lot of plane wave : cannot know exactly the
wave vector of particle → ‘uncertainty’ of momentum
More wave superposition occurs, uncertainty of particle
decrease, but uncertainty of momentum increase!
Consider the ∆k << k0, momentum of wavefunction is given by
(k0  mk )  p  (k0  mk )
It means, range of momentum increase when m increase.
Finally, we can obtain Heisenberg uncertainty principle
px 

2
MS310 Quantum Physical Chemistry
Text p.88
MS310 Quantum Physical Chemistry
Text p.89
MS310 Quantum Physical Chemistry
Text p.89
MS310 Quantum Physical Chemistry
6.4 The Heisenberg uncertainty principle expressed
in terms of standard deviation
Heisenberg uncertainty principle can be written in the form
 x p 

2
σx,σp : standard deviation of position and momentum
 2p  p2    p  2 ,  x2  x 2    x  2
This 4 values are defined by postulate 4.
 p 2    * ( x ) pˆ 2 ( x )dx
 p  2  (   * ( x ) pˆ  ( x )dx )2
 x 2    * ( x ) xˆ 2 ( x )dx
 x  2  (   * ( x ) xˆ  ( x )dx )2
MS310 Quantum Physical Chemistry
Consider the particle in a box
xˆ  x , pˆ   i

, n ( x ) 
x
2
nx
sin
a
a
2
nx
2
nx
2
nx
1
 x   (
sin
) x(
sin
)dx   x sin 2
dx  a
a
a
a
a
a0
a
2
0
a
a
2
nx 2 2
nx
2
nx
1
1
 x 2   (
sin
)x (
sin
)dx   x 2 sin 2
dx  a 2 (  2 2 )
a
a
a
a
a0
a
3 2 n
0
a
a
2
nx

2
nx
2n
nx
nx
 p   (
sin
)(  i (
sin
))dx   i 2  sin
cos
dx  0
a
a

x
a
a
a
a
a
0
0
a
a
2
2
nx
2
nx
2 2 n 2  2
 2 n 2 2
2 
2 nx
 p  (
sin
)(  
(
sin
))dx 
sin
dx 
2
3

a
a
x
a
a
a
a
a2
0
0
a
a
2
p 
 2 n2 2
a2
n
1
1
 2n2 1

, x  a

,  p x  

a
12 2 2 n 2
12
2
n  1 :  p x  
 2n2
1
2 1

 
  0.57 
12
2
12 2
2
MS310 Quantum Physical Chemistry
n=1 : minimum → uncertainty principle satisfied for all n
Relative uncertainty in x and p
x
 x
a

1
1
 2 2
12 2 n 
a/2
p
1
2
n / a
 2 2,

1
2
3  n
n


/
a
p 
(use  p 2  , instead of  p  because  p  0)
when n→∞, uncertainty of position increases.
→ related to probability of finding particle is equal everywhere
case of large n.
However, uncertainty of momentum is independent to n.
→ uncertainty of momentum can be negligible.
But, it is not enough : there are 2 p values when p2 determined.
Solution : change the wavefunction as eigenfunctions of
momentum operator
MS310 Quantum Physical Chemistry
relative probability density of wave vector
2
nx
sin
, n  1,2,3,4..., a  x  0 and
a
a
 n ( x )  0, otherwise
 n ( x) 
 n ( x) 
k 
A e
k  
ikx
k
Momentum approaches to classical value
when n increase! → relative uncertainty of
momentum ‘decrease’ as n increase!
MS310 Quantum Physical Chemistry
6.5 A thought experiment using a particle in
a 3-dimensional box
We do thought experiment by these steps.
1) A particle in a box
→ know the wavefunction of particle
2) Insert barrier
→ tunneling probability decrease in middle region
3) Move apart : separate to 2 boxes
2
2


a


b

,
|
a
|

|
b
| 1
→ wavefunction represent by
left
right
(each function satisfies the particle in a box and can assume
a=b) : superposition state
4) Look in box(measurement one of the boxes)
→ we can see only particle is in the left or right : ‘rapidly
decay of superposition state when measurement occurs’
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
6.6 Entangled states, teleportation, and
quantum computer
Consider the case in 6.5, the particle(single particle) is in the
superposition state.
  a left  b right , | a |2  | b2 | 1
Also, this wavefunction is not an eigenfunction of position.
If two quantum particles are strongly coupled : entangled state
Beam of photons is incident on transparent cystalline BaTiO3.
→ Only 2 direction of electric field vector of photon :
Horizontal(H) and Vertical(V) : polarization state
Probability of horizontal and vertical is same by the
measurement, and ‘if polarization of first photon measured,
the other polarization will be measured exactly!’
MS310 Quantum Physical Chemistry
Wavefunction can be described by  12 
1
( 1 ( H ) 2 (V )   1 (V ) 2 ( H ))
2
This wavefunction is not an eigenfunction of single particle
operator, and measure the single particle have no meaning
because this system is ‘entangled’ state.
MS310 Quantum Physical Chemistry
How can use this result : ‘teleportation’
There are a pair of entangled photons,
Alice has photon A and Bob has
photon B.
Consider the photon A is entangled to
photon X. It means photon A and
photon X is orthogonal.
Photon B is entangled to photon A
and photon B must be orthogonal to
photon A. Therefore, state of photon
B is same as photon X, the message
of Alice : teleportation!
MS310 Quantum Physical Chemistry
More interesting application : quantum computer
Classical computer(our PC) : bit
n bit memory : 000…0 to 111…1 : 2n state
Quantum computer : qubit
Use the superposition of different quantum state.
In photon system, H and V can be correspond to 0 and 1
Superposition state → qubit
n-qubit system : entangled 2n state
In a bit, 2M state stored in length M. However, in a qubit, 2M
state stored in M-qubit, only one superposition state!
Therefore, 2M simultaneous calculation can be parallel in Mqubit quantum computer.
If M=30, 1030 calculation can be parallel, and it expects the
speed of calculation improve surprisingly.
MS310 Quantum Physical Chemistry
Summary
- The Heisenberg uncertainty principle limits the degree to
which observables of noncommuting operators
can be
known simultaneously.
- The Stern-Gerlach experiment clearly demonstrates that the
prediction of quantum mechanics is obeyed at the atomic
level.
- Entanglement is the basis of both teleportation and quantum
computing
MS310 Quantum Physical Chemistry