1. Chapter 1
The Wave Function

2. David Jeffrey Griffiths
Textbook
David Jeffrey Griffiths
(born 1942)

3. Brook Taylor
( )
Math boot camp
Taylor series:

4. Math boot camp
Some examples:

5. Math boot camp
Some examples:

6. Math boot camp
Complex numbers:
Complex conjugate:

7. Math boot camp
Partial derivatives:
Chain rule:

8. Math boot camp Fourier transform: Inverse Fourier transform:
Jean-Baptiste
Joseph Fourier
( )
Math boot camp
Fourier transform:
Inverse Fourier transform:
Integration by parts:

9. Some probability theory
1.3
Some probability theory
A room with 14 people has this age distribution:
The number of people of age j:
Then, e.g.
Total number of people in the room:

10. Some probability theory
1.3
Some probability theory
A room with 14 people has this age distribution:
Probability that person’s age is 15:
Probability that person’s age is j:
Sum of all probabilities:

11. Some probability theory
1.3
Some probability theory
A room with 14 people has this age distribution:
Most probable age:
Median age:
Average age:

12. Some probability theory
1.3
Some probability theory
Average value of j:
Average of the squares:
Average of a function:

13. Some probability theory
1.3
Some probability theory
Deviation from the average:
Variance:
Standard deviation:

14. Some probability theory
1.3
Some probability theory
Example:

15. Some probability theory
1.3
Some probability theory
Example:

16. Some probability theory
1.3
Some probability theory
These results can be generalized for the case of continuous distributions
Probability that certain quantity has a value between x and x + dx:
Probability density:
Probability that certain quantity has a value between a and b:
Obviously:

17. Some probability theory
1.3
Some probability theory
Average value of x:
Average of the x2:
Average of a function:
Variance:

18. Quantum physics
Problems at the end of XIX century that classical physics couldn’t explain:
Blackbody radiation – electromagnetic radiation emitted by a heated object
Photoelectric effect – emission of electrons by an illuminated metal
Spectral lines – emission of sharp spectral lines by gas atoms in an electric discharge tube

19. Quantum physics
Phenomena occurring on atomic and subatomic scales cannot be explained outside the framework of quantum physics
There are many phenomena revealing quantum behavior on a macroscopic scale, e.g. quantum physics enables one to understand the very existence of a solid body and parameters associated with it (density, elasticity, etc.)
However, as of today, there is no satisfactory theory unifying quantum physics and relativistic mechanics
In this course we will discuss non-relativistic quantum mechanics

20. Kindergarten stuff Is light a wave or a flux of particles?
Newton vs. Young
Isaac Newton
(1642 – 1727)
Thomas Young
(1773 – 1829)

21. Kindergarten stuff
Is light a wave or a flux of particles?

22. Kindergarten stuff
Is light a wave or a flux of particles?

23. Kindergarten stuff
Is light a wave or a flux of particles?

24. Kindergarten stuff Is light a wave or a flux of particles? However:
1) Blackbody radiation
2) Photoelectric effect
3) Spectral lines
4) Etc.

25. Wave-particle duality
EM waves appear to consist of particles – photons
Particle and wave parameters are linked by fundamental relationships:
h – Planck’s constant, × J∙s
Max Karl Ernst Ludwig Planck
1858 – 1947
Albert Einstein
1879 – 1955

26. Wave-particle duality

27. Wave-particle duality

28. Wave-particle duality

29. Wave-particle duality

30. Wave-particle duality

31. Wave-particle duality

32. Wave-particle duality

33. Wave-particle duality
The results of this experiment lead to a paradox:
Since the interference pattern disappears when one of the slits is covered, why then this phenomena changes so drastically?
Crucial: the process of measurement
When one performs a measurement on a microscopic system, one disturbs it in a fundamental fashion
It is impossible to observe the interference pattern and to know at the same time through which slit each photon has passed

34. Wave-particle duality
Light behaves simultaneously as a wave and a flux of particles
The wave enables calculation of particle-related probabilities; e. g., when the photon is emitted, the probability of its striking the screen is proportional to light intensity, which in turn is proportional to the square of the field amplitude

35. Wave-particle duality
Predictions of the behavior of a photon can be only probabilistic: information about the photon at time t is given by the electric field, which is a solution of the Maxwell’s equations – the field is interpreted as a probability amplitude of a photon appearing at time t at a certain location:
James Clerk Maxwell

36. Principle of spectral decomposition
Malus’ Law: the intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2 θ, where Io is the intensity of the polarized wave incident on the analyzer
Étienne-Louis Malus
1775 – 1812

37. Principle of spectral decomposition
What will happen, when intensity is low enough for the photons to reach the analyzer one by one?
NB: the detector does not register “a fraction of a photon”)
We cannot predict which photon can pass the analyzer

38. Principle of spectral decomposition
The analyzer and detector can give only certain specific results – eigen (proper) results: either a photon passes the analyzer or not
To each of the eigen results there is an eigenstate
When the state before measurement is arbitrary, only the probabilities of obtaining the different eigen results can be predicted
To find these probabilities, the state has to be decomposed into a linear combination of eigenstates

39. Principle of spectral decomposition
The probability of an eigen result is proportional to the square of the absolute value of the coefficient of the corresponding eigenstate
The sum of all the probabilities should be equal to 1
Measurement disturbs the photons in a fundamental fashion

40. Wave properties of particles
1.6
Wave properties of particles
In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties
Furthermore, the frequency and wavelength of matter waves can be determined
The de Broglie wavelength of a particle is
The frequency of matter waves is
Louis de Broglie
1892 – 1987

41. Wave properties of particles
1.6
Wave properties of particles
The de Broglie equations show the dual nature of matter
Each contains matter concepts (energy and momentum) and wave concepts (wavelength and frequency)
The de Broglie wavelength of a particle is
The frequency of matter waves is
Louis de Broglie
1892 – 1987

42. Wave properties of particles
Davisson and Germer scattered low-energy electrons from a nickel target and followed this with extensive diffraction measurements from various materials
The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength
Clinton Joseph Davisson (1881 – 1958) and Lester Halbert Germer (1896 – 1971)

43. 1.2
The wave function
In quantum mechanics the object is described by a state (not trajectory)
The state is characterized by a complex wave function Ψ, which depends on particle’s position and the time
The wave function is interpreted as a probability amplitude of particle’s presence
The probability density (probability of finding the object at time t inside an elementary volume dxdydz):

44. The wave function Let us first consider 1D systems
1.2,1.4
The wave function
Let us first consider 1D systems
Then, the probability of finding a particle between a and b, at time t:
Since the particle should be somewhere:
This is called normalization
Wave functions normalized in this fashion describe physical quantum systems

45. Erwin Rudolf Josef Alexander Schrödinger
1.1
Schrödinger equation
In 1926 Schrödinger proposed an equation for the wave function describing the manner in which matter waves change in space and time
Schrödinger equation is a key element in quantum mechanics
V – potential energy (“potential”)
Superposition principle applies
Erwin Rudolf Josef Alexander Schrödinger
1892 – 1987

46. Erwin Rudolf Josef Alexander Schrödinger
1.1
Schrödinger equation
In 1926 Schrödinger proposed an equation for the wave function describing the manner in which matter waves change in space and time
Schrödinger equation in 1D:
Let us accept it as a postulate
Shortly we will discuss its meaning
Erwin Rudolf Josef Alexander Schrödinger
1892 – 1987

47. Operators The average value of the position of
1.5
Operators
The average value of the position of
In quantum mechanics the average value of a physical quantity is also called an expectation value
Its physical meaning: the average of repeated measurements on an ensemble of identically prepared systems
How does the expectation value of x change with time?

48. 1.4,1.5
Operators
Using Schrödinger equation: +

49. 1.4,1.5
Operators
Using Schrödinger equation: +

50. 1.4,1.5
Operators

51. 1.4,1.5
Operators

52. 1.4,1.5
Operators

53. 1.5
Operators

54. 1.5
Operators
Thus:
We can write:
Synopsizing:

55. Operators Defining operators of position and linear momentum:
1.5
Operators
Defining operators of position and linear momentum:
We can generalize the definition of an average on an operator:
Synopsizing:

56. Operators Defining operators of position and linear momentum:
1.5
Operators
Defining operators of position and linear momentum:
We can generalize the definition of an average on an operator:
For example:

57. Operators The total energy operator is called the Hamiltonian:
1.5, 2.1
Operators
The total energy operator is called the Hamiltonian:
Let’s recall the Schrödinger equation:
Thereby:
Does this remind us of anything?
Sir William Rowan
Hamilton
(1805 – 1865)

58. Euler- Lagrange equations
Joseph Louis
Lagrange
(1736 – 1813)
Leonhard Euler
(1707 – 1783)

59. Hamilton’s equations Hamiltonian: Hamilton’s equations of motion:
Sir William Rowan
Hamilton
(1805 – 1865)

60. Hamilton–Jacobi theory
Hamilton–Jacobi equation
S: Hamilton’s principal function
Partial differential equation
First order differential equation
Number of variables: M + 1
Sir William Rowan
Hamilton
(1805 – 1865)
Karl Gustav Jacob
Jacobi
(1804 – 1851)

61. Heisenberg uncertainty principle
1.6
Heisenberg uncertainty principle
In 1927 Heisenberg introduced the uncertainty principle: If a measurement of position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δpx, then the product of the two uncertainties can never be smaller than h/4
Werner Karl Heisenberg
1901 – 1976