1. Quantum Physics Mathematics

2. Quantum Physics Tools in Real Life
Reality

3. Quantum Physics Tools in Physics / Quantum Physics Real Number – Vector - Statevector
Speed represented by a real number
80 km/h
Velocity represented by a vector
80 km/h NorthEast
State representered by a statevector

4. Quantum Physics Tools in Physics / Quantum Physics
Mathematics
Language
Numbers
Variables
Functions
Position is 2.0 m and
velocity is 4.0 m/s
Vectors
Reality

5. Quantum Physics Superposition Vectors - Functions1 3 2 4
Music
Pulse train
Heat
Sampling

6. Quantum Physics Reality - Theory / Mathematical room

7. Quantum Physics Reality - Theory / Mathematical room - Classical Physics

8. Quantum Physics Reality - Theory / Mathematical room - Quantum Physics

9. Quantum Physics Postulate 11.
Every system is described by a state vector
that is an element of a Hilbert space.

10. Quantum Physics Postulate 22.
An action or a measurement on a system
is associated with an operator.

11. Quantum Physics Observation / Measurement in daily life
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)

12. Quantum Physics Observation of position, changing the velocity
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.

13. Quantum Physics Observation of current and voltage
Input of amperemeter and voltmeter
disturb the current and voltage.

14. Quantum Physics Stern-Gerlach experiment [1/2] Observation of angular momentum in one direction influence on the angular momentum in another direction V1
Silver atoms going through a vertical magnetic field
dividing the beam into two new beams
dependent of the angular momentum of the atom. B1 B B2 V2 V1
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. H1
B11
B121 B1
B12 B
B122 B2

15. Quantum Physics Stern-Gerlach experiment [2/2] Observation of angular momentum in one direction influence on the angular momentum in another direction z+ z+
S-G
z-axis
S-G
z-axis 1 B z-
No z- z+ x+
S-G
z-axis
S-G
x-axis 2 B z- x- z+ x+ z+
S-G
z-axis
S-G
x-axis
S-G
z-axis 3 B z- x- z-

16. 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.

17. 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.

18. Quantum Physics Observation / Measurement - Quantum
A car is placed in the position A
behind a person.
The person with the car behind,
cannot yet observe the car.
The person turns around
and observeres the car
in the position B. B
In quantum physics
it’s possible that
the observation of a property of the car
moves the car to another position.

19. 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. B
Agnostic:
Refuse to answer.
No sense to ask before a measurent.
Orthodox supported by theory (Bell 1964) and experiment (Aspect 1982).

20. Quantum Physics Observation
Before the measurement M
After the measurement

21. Quantum Physics Observation Superposition
Before the measurement
the position of the car
is a superposition
of infinitely many positions. M
The measurement produce
a specific position of the car. M
A repeated measurement
on the new system
produce the same result.

22. Quantum Physics Superposition Fourier
Music
Pulse train
Heat
Sampling

23. Quantum Physics Classical: Vector expanded in an orthonormal basis - I

24. Quantum Physics Classical: Vector expanded in an orthonormal basis - II
Complex coefficients

25. Quantum Physics State vector expanded in an orthonormal basis

26. Quantum Physics Space - Dual space Introduction
Ket
Bra
Dual space
Space

27. Quantum Physics Space - Dual space Example - Real Elements
Ket
Bra
Dual space
Space

28. Quantum Physics Space - Dual space Example - Complex Elements
Ket
Bra
Bra
Ket
Dual space
Space

29. Quantum Physics Space - Dual space Example - Scalar Product
Ket
Bra
Dual space
Space

30. Quantum Physics Space - Dual space Example - Operator I
Ket
Bra
Dual space
Space

31. Quantum Physics Space - Dual space Example - Operator II
Ket
Bra
Dual space
Space

32. Quantum Physics Space - Dual space Example - Operator III
Ket
Bra
Dual space
Space

33. Quantum Physics Probability amplitude
Same state
Normalization of a state vector
don’t change
the probability distributions.
Therefore we postulate c and 
to represent the same state.
cn : Probability amplitude
cn2 : Probability

34. Quantum Physics Projection Operator Theory

35. Quantum Physics Projection Operator Example - Projection to basisfunction

36. Quantum Physics Projection Operator Example - Projection to subroom

37. Quantum Physics Unit Operator Theory

38. Quantum Physics Unit Operator Example

39. Quantum Physics Orthonormality - Completeness
Projection Operator
Completeness

40. Quantum Physics Orthonormality - Completeness Discrete - Continuous
State
Orthonormality
Projection Operator
Completeness

41. 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 A
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

42. Quantum Physics Operator Self-adjoint operator
Def: Self-adjoint operator:
Def: Hermitian operator:
The distinction between Hermitian and self-adjoint operators
is relevant only for operators in infinite-dimensional vector spaces.
Proof:

43. Quantum Physics Operator Theorem
Proof:
Canceling i and adding

44. Quantum Physics Hermitian operator The eigenvalues of a Hermitian operator are real
Theorem:
The eigenvalues of a Hermitian operator are real.
Proof:

45. Quantum Physics Hermitian operator Eigenstates with different eigenvalues are orthogonal
Theorem:
Eigenstates corresponding to distinct eigenvalues
of an Hermitian operator must be orthogonal.
Proof:

46. Quantum Physics Operator expanded by eigenvectors
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
Every operator can be expanded
by their eigenvectors and eigenvalues

47. Quantum Physics Average of Operator [1/2]

48. Quantum Physics Average of Operator [2/2]

49. 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.
Unless the state is an eigenstate of the actual operator,
we can never predict the result of the operator
only the probability.
The determinate state of the operator A
that return the same value a every time
is the eigenstate of A with the eigenvalue a.

50. Quantum Physics Uncertainty [1/3]

51. Quantum Physics Uncertainty [2/3]

52. Quantum Physics Uncertainty [3/3]

53. Quantum Physics Kinematics and dynamics
Necessary with correspondence rules
that identify variables with operators.
This can be done by studying special
symmetries and transformations.
Operator  Variable
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.

54. 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 Energy E
Space translation Linear momentum p
Rotation Angular momentum L

55. Quantum Physics Invariant transformation Unitary operator
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.
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.

56. Quantum Physics Unitary operator
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.
Only linear operators
can describe continous transformations.

57. Quantum Physics Generator of infinitesimal transformation
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.

58. Quantum Physics Time operator Time dependent Schrödinger equation Energy operator Generator for time displacement
Energy operator
Generator for time displacement

59. Quantum Physics Schrödinger Equation Time dependent / Time independent
potensial
Studier av svingninger (spesielt resonans)
for å hindre at f.eks. bruer kollapser under påvirkning av vindkast.
Time independent
probability 59

60. Quantum Physics Schrödinger equation - Time independent
Stationary states

61. Quantum Physics Normalization is time-independent

62. Quantum Physics Moment operator in position space

63. Quantum Physics Schrödinger equation - Time independent

64. Quantum Physics Operators
Position space
Momentum space
Position
Momentum
Potensial energy
Kinetic energy
Total energy

65. Quantum Physics Uncertainty Position / Momentum - Energy / Time