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Quantum Physics
Mathematics
Quantum Physics
Tools in Real Life
Reality
Quantum Physics
Tools in Physics / Quantum Physics
Mathematics
Language
Numbers
Position is 2.0 m and
velocity is 4.0 m/s
1
5
27
x
Vectors
Variables
a
I
F  ma
Functions
f ( x)  x 2
g ( x, y, u )  x 2euy
f  L2


f ( ) d
2

f H
Reality
Quantum Physics
1
Superposition
Vectors - Functions
S
1
3
4
2
Music
Heat
Pulse train
Sampling
Quantum Physics
Reality - Theory / Mathematical room
Theory / Mathematical room
Reality
Quantum Physics
Reality - Theory / Mathematical room - Classical Physics
Theory
Mathematical room
Reality
 
v  r
 
a  v


F  macm
K
1 2
mv
2
Quantum Physics
Reality - Theory / Mathematical room - Quantum Physics

Theory
Mathematical room
'
Action
Reality
A
Tilstand
Tilstand '
Quantum Physics
Theories
Quantum Physics
Dn
Relativity Theory
Micro
High velocity
Ordinary Classic Theory
Do
Low velocity
Dn
Do
Do = Domain described by old theory
Dn = Domain described by new theory
Macro

Quantum Physics
Postulates
1.
Every system is described by a state vector
that is an element of a Hilbert space.

  (t )
2.
An action or a measurement on a system
is associated with an operator.
A 1  2
2
1
2
Quantum Physics
Observation / Measurement in daily life
1
The length of the table
is independent of an observation or a measurement.
The behaviour of the class
is perhaps not independent
of an observation (making a video of the class)
2
Quantum Physics
Observation of position, changing the velocity
1
A ball with a known velocity and unknown position.
Try to determine the position.
A bit unlucky one foot hits the ball.
The position is known when the ball is touched,
but now the velocity is changing.
Just after the hit of the ball,
the position is known, but now the velocity is unknown.
2
Quantum Physics
Observation of current and voltage
Input of amperemeter and voltmeter
disturb the current and voltage.
1
2
Quantum Physics
Observation of angular momentum in one direction
influence on the angular momentum in another direction
Silver atoms going through a vertical magnetic field
dividing the beam into two new beams
dependent of the angular momentum of the atom.
Three magnetic fields: Vertical, horizontal, vertical.
Every time the beam is divided into two new beams.
No sorting mechanism.
A new vertical/horisontal measurement
disturbs/changes the horisontal/vertical beam property.
1
2
Quantum Physics
Entanglement
Quantum entanglement occurs when particles such as
photons, electrons, molecules and even small diamonds
interact physically and then become separated.
When a measurement is made on one of member of such a pair,
the other member will at any subsequent time be found
to have taken the appropriate correlated value.
According to the Copenhagen interpretation of quantum physics,
their shared state is indefinite until measured.
Entanglement is a challange in our understanding of nature
og will hopefully give us new technological applications.
1
Quantum Physics
Observation / Measurement - Classical
A car (particle) is placed behind a person.
The person with the car behind, cannot see the car.
The person turns around and observeres the car.
Classically we will say:
The car was at the same place also just before the observation.

2
Quantum Physics
Observation / Measurement - Quantum
1
A
A car is placed in the position A
behind a person.
The person with the car behind,
cannot yet observe the car.
B
The person turns around
and observeres the car
in the position B.
In quantum physics
it’s possible that
the observation of a property of the car
moves the car to another position.

Quantum Physics
Observation / Measurement - Quantum
Question:
Where was the car
before the observation?
?
Realist:
The car was at B.
If this is true,
quantum physics is incomplete.
There must be some hidden variables (Einstein).
Orthodox:
The car
wasn’t really anywhere.
It’s the act of measurement
that force the particle to ‘take a stand’.
Observations not only disturb,
but they also produce.
Agnostic:
Refuse to answer.
No sense to ask before a measurent.
B
Orthodox supported by theory (Bell 1964) and experiment (Aspect 1982).
Quantum Physics
Observation
Before the measurement
M
After the measurement
1
M
2
M
M
1
Quantum Physics
Observation
2
Superposition
1
2
k
...
Before the measurement
the position of the car
is a superposition
of infinitely many positions.
2
1
1
1   cn  n
n
M
The measurement produce
a specific position of the car.
M
A repeated measurement
on the new system
produce the same result.
2
k
2  k
2
2
Quantum Physics
1
Superposition
Fourier
Music
Heat
Pulse train
Sampling


V   cn en
Quantum Physics
n
Classical:
Vector expanded in an orthonormal basis - I





V  c1i  c2 j  c3k   cn en
n
 
em  en   mn


V   cn en

 
cn  en  V

n


  
V   en  V en
n
n
n

n
 
k  e3
 
i  e1 

j  e2

c3e3
n
  

 
em  V  em   cn en   cn em  en   cn mn  cm

  
en V en

c1e1

V

V

c2e2


V   cn en
Quantum Physics
n

Classical:
Vector expanded in an orthonormal basis - II
V  c1 i  c2 j  c3 k   cn en
k  e3
em en   mn
V
i  e1
j  e2
V   cn en
em V  em
c3 e3
 cn en   cn em en   cn mn  cm
n
n

n
n
n

  
en V en
n

V
c1 e1
cn  en V
c2 e2
Complex coefficients
V   en V en
n
 c1 
v  c2 
c3 

v  c1
*
c2
*
*
c3



v  c1
*
c2
*

* 
3
c
 c1 
 c2   v
c3 
   cn  n
n
Quantum Physics
State vector expanded in an orthonormal basis
  n  n
n
   cn n
m n   mn
3
n
   cn  n
m n  n m
1
*

2
n
m   m
c
n
n
n
  cn  m  n   cn mn  cm
n
n
cn   n 
   n  n
n
c3 3
c1 1

c2 2
Quantum Physics
Bra
Space - Dual space
Ket
Dual space
Space


 A
A
2
1
1  2   2 1
*
1 A  2  1 A 2  1 A  2
   cn n
Quantum Physics
Probability amplitude
c
n
n
 
n
1      cm  m
*
m
c
n
1
n
 m  n   mn
   cn n
2
3
1
1
n

2
n
  cm cn  m  n   cm cn mn   cn
*
m,n
*
m,n
2
n
Same state
c
n
n
2
1
cn
cn2
: Probability amplitude
: Probability
Normalization of a state vector
don’t change
the probability distributions.
Therefore we postulate c and 
to represent the same state.

Quantum Physics
Projection Operator


n
n
Pn  n n
c2 2
n    n n 
Projection operator
n n
 n  n
projects the part of

in the direction of
n
n     n n
 c
k
k
k   ck n n k   ck n  nk  cn n
k
k
Pn   n  n
Pm Pn   m  m  n  n   m  mn  n   mn  m
 Pn m  n
n  
0 m  n

Quantum Physics
Unit Operator
n
   n  n
3
n


   n  n     n  n  
n
n

 n n  I
n
The operator
n  I
n

n
is an unit operator
n
n  I
1
2

Pn   n  n
Quantum Physics

Orthonormality - Completeness
n
n  I
n
   cn n
n
m n   mn
Pn  n n
 n n  I
n
Orthonormality
Projection Operator
Completeness
3
1
2

Pn   n  n
Quantum Physics
Orthonormality - Completeness
Discrete - Continuous
Discrete
State
   cn  n    n   n
n
n
  n n 

n
n  I
n
Continuous
   d c( )    d   
  d      d   ( )
n
m n   mn
  '   (   ' )
Projection Operator
Pn  n n
P   
Completeness

 d 
Orthonormality
n
n
n  I
 I
A 1  ak k
Quantum Physics
Operator
Eigenvectors - Eigenvalues
Eigenvectors are of special interest since experimentally we always observe
that subsequent measurements of a system return the same result (collapse of wave function).
A
1   cn  n
A
1
A
2  k
2  k
A
1  2   k   k
Consequence of Spectral Theorem:
The only allowed physical results of measurements of the observable A
are the elements of the spectrum of the operator which corresponds to A.
Measured quantity
A 1  ak k
Quantum Physics
A  A
Operator
Self-adjoint operator
Def: Self-adjoint operator:
Def: Hermitian operator:
A  A

 A   A 
  ,
*
The distinction between Hermitian and self-adjoint operators
is relevant only for operators in infinite-dimensional vector spaces.
Proof:
A  A
 A    A  A 
  A

*
  A 

*
  A
*
Pn   n  n
Quantum Physics

Operator
Theorem
Theorem:
n
n  I
A   an  n  n
n
 A   A
n

*

1 A  2   2 A 1
 1 ,  2
*
A  A
Proof:
  a 1  b  2
 A
 A
1 A  2
arbitrary a, b, 1 ,  2
 a 1 A 1  b  2 A  2  a *b 1 A  2  ab*  2 A 1
2
*
2
 a 1 A 1  b  2 A  2  ab* 1 A  2  a *b  2 A 1
2
2
  2 A 1 
*
1 A  2   2 A 1
*
for a  1,b  1
i 1 A  2  i  2 A 1  i 1 A  2  i  2 A 1
*
for a  1,b  i
*
*
1 A  2   2 A 1
A  A
*
Canceling i and adding
real
*
Quantum Physics
Hermitian operator
The eigenvalues of a Hermitian operator are real
Theorem:
The eigenvalues of a Hermitian operator are real.
Proof:
A  A

*

a
a

A   a  
A  A
A  a 
0   A   A
  A    A 
 A    A


 a    a  a *    a    a *  a  
a*  a


A 1  a1 1 
  1  2  0
A  2  a2  2 

a1  a2

A  A
Quantum Physics
Hermitian operator
Eigenstates with different eigenvalues are orthogonal
Theorem:
Proof:
Eigenvalues corresponding to distinct eigenvalues
of an Hermitian operator must be orthogonal.
A A

A 1  a1 1
3
1
A  2  a2  2
2

0  1 A  2   2 A 1
*
 a2 1  2  a1  2 1
*
 a2 1  2  a1 1  2
 a2  a1  1  2

1  2  0
if
a1  a2

Quantum Physics
Operator
expanded by eigenvectors
A n  an n
   cn n
Pn   n  n

n
A   an  n  n
n  I
n
n
Eigenvectors are of special interest since
experimentally we always observe that
subsequent measurements of a system
return the same result (collapse of wave function)
The measurable quantity is associated with
the eigenvalue. This eigenvalue should be real so
A have to be a self-adjoint operator A+ = A
Operator
eigenvecto rs  n
eigenvalue s
n
A   A cn  n   cn A  n   cn an  n   an cn  n
n
n
n
  an  n   n   an  n
n
A   an  n  n
n
n
n


 n     an  n  n  
n

Every operator can be expanded
by their eigenvectors and eigenvalues
A
an
A   an cn   A 
2
Quantum Physics
Average of Operator [1/2]
n
   cn n
n
A   an cn
2
n
A   a n cn   a n  n 
2
n
2
  an   n  n 
n
n
   an  n  n     A  n  n 
n
n
  A  n
n


 n    A   n  n     A 
n

Aa(nt )cn 
)
A 
 
n cA
n (t
2
2
Quantum Physics
Average of Operator [2/2]
n
n
(t )   cn (t ) n
a c (t )

A (t ) 
 c (t )
A 
n
2
n
2
n
 an n  (t )
a c (t )
a  (t )    (t )


A (t ) 


  (t )    (t )
 c (t )
   (t )
 (t )  a    (t )
 (t )  A    (t )


 (t )     (t )
    (t )
 (t ) A    (t )
 (t ) A     (t )
 (t ) A  (t )



 (t )     (t )
 (t )     (t )
 (t )  (t )
2
n
n
2
2
n
n
n
2
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
A a 
Quantum Physics
Determiate state
Determinate state:
A state prepared so a measurement of operator A
is certain to return the same value a every time.
0  A   A  A
2

2

A a
Unless the state is an eigenstate of the actual operator,
we can never predict the result of the operator
only the probability.
  A  a 
2
  A  a   A  a 

A  a 
0
A a
A a 
The determinate state of the operator A
that return the same value a every time
is the eigenstate of A with the eigenvalue a.
1
2
2
A B    A, B 
 2i

Quantum Physics
Uncertainty [1/3]
A   A
 A  A     A  A    A  A  A  A   f f
B    B  B     B  B    B  B   B  B   g g
A
2

2
A B 
2
2
2
2
2
2
2
2
 f f g g  f g
2


1

2
2
2
z  Re( z )  Im( z )  Im( z )   z  z * 
 2i

2
A2 B 2 
f g
2
1

   f g  g f 
 2i

2
2
2
1
2
2
A B    A, B 
 2i

Quantum Physics
Uncertainty [2/3]
f g   A  A   B  B

  A  A B  B  
  AB  A B  B A  A B  
  AB   B  A   A  B   A B  
  AB  B  A  A  B  A B  
 AB  B A  A B  A B
 AB  A B
g f  BA  A B
2
Quantum Physics
Uncertainty [3/3]
1
A  B 
2
A, B 
f g  AB  A B
g f  BA  A B
f g  g f  AB  BA  AB  BA  A, B 
A B 
2
2
2
 f g
A  B 
1
2
A, B
2
1
 1

   f g  g f    A, B  
 2i
  2i

2
Quantum Physics
Kinematics and dynamics
Necessary with correspondence rules
that identify variables with operators.
This can be done by studying special
symmetries and transformations.
The laws of nature are believed to be invariant
under certain space-time symmetry operations,
including displacements, rotations,
and transformations between frames of reference.
Corresponding to each such space-time transformation
there must be a transformation of observables,
opetators and states.
Energy, linear and angular momentum
are closely related to space-time symmetry transformations.
Operator 
Variable
A

a
a A

T
a' A'
'
a  a'
A  A'
  '
Quantum Physics
Noether’s theorem
Noether’s (first) theorem
states that any differentiable symmetry of the action (law) of a physical system
has a corresponding conservation law.
Symmetry
Conservation
Time translation
Space translation
Rotation
Energy
Linear momentum
Angular momentum
E
p
L
Quantum Physics
U U  I
Invariant transformation
Unitary operator
'  U 
U
'

   cn  n
 '   cn '  n '
cn  cn '
2
2
cn  cn '
2
2
n   n ' '
n   n I 
 n '  '   nU  U   n U U 
The laws of nature are believed to be invariant
under certain space-time symmetry.
Therefore we are looking for a continous
transformation
that are preservering the probability distribution.
U U  I
Any mapping of the vector space onto itself
that preserves the value of |<|>|
may be implemented by an operator U
being either unitary (linear) or antiunitary (antilinear).
Only linear operators
can describe continous transformations.
d d 
d  d 
U (d )  U     U  U  
2 2
2 2
A'  UAU 1
Quantum Physics
Unitary operator
n '  U n
A  n  an  n
A'  n '  an  n '

Any mapping of the vector space onto itself
that preserves the value of |<|>|
may be implemented by an operator U
being ether unitary (linear) or antiunitary (antilinear).
Only linear operators can describe
continous transformations.
A'U  n  A'  n '  an  n '  anU  n
U 1 A'U  n  U 1anU  n  anU 1U  n  an  n  A  n
U 1 A'U  A
A'  UAU 1
Only linear operators
can describe continous transformations.
d d 
d  d 
U (d )  U     U  U  
2 2
2 2
U ( s)  eiKs
Quantum Physics
Generator of infinitesimal transformation
U ( s ) I 
dU
ds
 O( s 2 )
s 0
 dU dU  
2
I  UU  I  s 

  O( s )
ds  s 0
 ds

 dU dU 



ds
ds  s 0


dU
ds
K  K
 iK
s 0
U ( s1  s2 )  U ( s1 )U ( s2 )

U ( s1  s2 )
s2
s
 U ( s1 )
2 0
dU ( s )
 U ( s1 )iK
ds s  s1
U ( s )  eiKs
d
U ( s2 )
ds2
s
2 0
Any mapping of the vector space onto itself
that preserves the value of |<|>|
may be implemented by an operator U
being ether unitary (linear) or antiunitary (antilinear).
Only linear operators can describe
continous transformations.
Quantum Physics
Time operator
- Time dependent Schrödinger equation
Energy operator - Generator for time displacement
 (t  T )  DT  (t )  e
i
 H
h

ih
 H 
t
 (t )
i
i


  I  H  O(a 2 )   (t )   (t )  H  (t )  O( H 2 )
h
h


i
 (t  T )   (t )   H  (t )  O( H 2 )
h

i
 (t )   H  (t )
Energy operator
t
h
Generator for time displacement

ih
 (t )  H  (t )
t
p2
H
V
2m

ih  (t )  H  (t )
t


H (r )  E (r )
Quantum Physics
Schrödinger equation - Time independent
Stationary states

ih  (t )  H  (t )
t
 

r ih  (t )  r H  (t )
t

 
ih
r  (t )  H r  (t )
t

ih  ( x, t )  H ( x, t )
t
 (t )  e
i
Et
h
 (t0 )
H  (t0 )  E  (t0 )


r H  (t0 )  r E  (t0 )


H r  (t0 )  E r  (t0 )


H (r )  E (r )

  0
t
Quantum Physics
Normalization is time-independent



  
  
t
t
t
1
1
H   
H
ih
ih
1
1

H  
 H
ih
ih
1
1

 H  
 H
ih
ih
1
1

 H 
 H
ih
ih
0

ih

 H 
t
H  H
h
p̂  ih  
i
Quantum Physics
Moment operator in position space



A (r )  r A 

i
 a  pˆ 

 
Da r  e h r  r  a
 
 
 (r  a )  r  a 


   hi a pˆ
  hi a pˆ

i 
 ( r  a )e
  re
  r I  a  pˆ  O(a 2 ) 
h
 i   
  (r )   a  pˆ  (r )  O(a 2 )
h

 
 

 ( r  a )   ( r )  a   ( r )  O ( a 2 )

pˆ  ih 
h

i

 (r )  r 
   
rˆ (r )  r  (r )
d
ih
(t )  H (t )
dt


H (r )  E (r )
Quantum Physics
Schrödinger equation - Time independent


p2
h2 2
H
 V (r )  
  V (r )
2m
2m

ih
 (t )  H  (t )
t
 

r ih
 (t )  r H  (t )
t

 
ih
r  (t )  H r  (t )
t

ih  ( x, t )  H ( x, t )
t
 (t )  e
i
Et
h
 (t0 )
H  (t0 )  E  (t0 )


r H  (t0 )  r E  (t0 )


H r  (t0 )  E r  (t0 )


H (r )  E (r )
 h2 2
  




V
(
r
)

(
r
)

E

(
r
)


2
m


Quantum Physics
A 1  2
Operators
Position space
Momentum space
Position
r
r

Momentum
p
h

i
p
Potensial energy
U
U (r )
Kinetic energy
p2
T
2m
Total energy
E
h2 2


2m
h 

i t
h2 2

  U ( x)
2m
h 
i p
 h  

U  
 i p 
p2
2m
h 

i t
 h  
p2

 U  
2m
i

p


Quantum Physics
A  B 
Uncertainty
1
A, B
2
Position / Momentum - Energy / Time
xˆ  x
pˆ x 
h 
i x
Position
Momentum
xˆ, pˆ   xˆpˆ  pˆ xˆ   ( xˆpˆ )  ( pˆ xˆ )
tˆ  t
h 
Eˆ  
i t
Eˆ , tˆ  Eˆtˆ  tˆEˆ   ( Eˆtˆ)  (tˆEˆ )
 xˆ ( pˆ  )  pˆ ( xˆ )
h   h 
x 
 xˆ 
 
 i x  i x
h
h
 xˆ '   x '
i
i
h
h
  x '  x '  
i
i
xˆ, pˆ   h
i
1
1 h h
x  p  xˆ, pˆ  

2
2 i 2
h
2
h
E  t 
2
x  p 
 Eˆ (tˆ )  tˆ( Eˆ  )
h 
t   tˆ  h   
i t
 i t 
h
h
    t '  t '
i
i
h
h
    t 't '   
i
i
h
Eˆ , tˆ  
i
1 ˆ ˆ
1 h h
E  t 
E, t   
2
2 i 2

 
 
Energy
Time