Chapter 1

1. Chapter 1 The Wave Function

2. Textbook David Jeffrey Griffiths (born 1942)

3. Math boot camp • Taylor series: Brook Taylor (1685-1731)

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: • Integration by parts: Jean-Baptiste Joseph Fourier (1768-1830)

9. 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. 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. 1.3 Some probability theory • A room with 14 people has this age distribution: • Most probable age: • Median age: • Average age:

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

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

14. 1.3 Some probability theory • Example:

15. 1.3 Some probability theory • Example:

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

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. Albert Einstein 1879 – 1955 Max Karl Ernst Ludwig Planck 1858 – 1947 Wave-particle duality • EM waves appear to consist of particles – photons • Particle and wave parameters are linked by fundamental relationships: • h – Planck’s constant, 6.626 × 10-34 J∙s

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. James Clerk Maxwell 1831-1879 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:

36. Étienne-Louis Malus 1775 – 1812 Principle of spectral decomposition • Malus’ Law: the intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2θ, where Io is the intensity of the polarized wave incident on the analyzer

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. 1.6 Louis de Broglie 1892 – 1987 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

41. 1.6 Louis de Broglie 1892 – 1987 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

42. Clinton Joseph Davisson (1881 – 1958) and Lester Halbert Germer (1896 – 1971) 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

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 functionΨ, 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. 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. 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. 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. 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