1. Chapter 3 Postulates of Quantum Mechanics

2. Questions QM answers 1) How is the state of a system described mathematically? (In CM – via generalized coordinates and momenta) 2) For a given state how can one predict results of measurements of various physical quantities (In CM – unambiguously, via the calculated trajectory in a phase space) 3) For a given state of the system known at time t 0 how can one find a state of this system at an arbitrary time t? (In CM – using Hamilton’s equations) Answers to these questions are given by the postulates of QM 3.A

3. State of a system 1 st postulate: At certain time t 0 a state of this system is defined by a ket belonging to the state space E 3.B.1

4. Physical quantities 2 nd postulate: Every measurable physical quantity is described by an observable operator acting in E 3.B.2

5. Measurement 3 rd postulate: Measurements of a physical quantity result only in (real) eigenvalues of a corresponding observable 3.B.3

6. Measurement 3 rd postulate: Measurements of a physical quantity result only in (real) eigenvalues of a corresponding observable It is not obvious a priori whether the spectrum of the measured quantity is continuous or discrete (e.g., a system consisting of a proton and an electron) 3.C.2

7. Spectral decomposition If Then the state of the system 4 th postulate: The probability of measuring an eigenvalue a n of an observable A in a certain state of the system is: 3.B.3

8. Spectral decomposition 3.B.3

9. Spectral decomposition The mean value of an observable: 3.C.4

10. Spectral decomposition If Then Therefore these two vectors represent the same physical state 3.B.3

11. Spectral decomposition If Then the state of the system 4 th postulate: The probability of measuring an eigenvalue of an observable A between α and α+dα in a certain state of the system is: ρ – probability density 3.B.3

12. Spectral decomposition The probability of measuring a position of a particle between x and x+dx in a certain state of the system is: The probability of measuring a momentum of a particle between p x and p x +dp x in a certain state of the system is: 3.C.1

13. Spectral decomposition The mean value of an observable: 3.C.4

14. Spectral decomposition Examples: 3.C.4

15. Reduction via measurement When the measurement is performed only one possible result is obtained Then the state of the system after the measurement of a non-degenerate eigenvalue is: For a degenerate case: 3.B.3

16. Reduction via measurement 5 th postulate: If measurement of a physical quantity in a given state of the system yields a certain eigenvalue, the state of the system immediately after the measurement is the normalized projection of the initial state onto an eigensubspace associated with that eigenvalue The state of the system after the measurement is the eigenvector corresponding to that eigenvlaue 3.B.3

17. Reduction via measurement We shall consider only ideal measurements This means that the perturbations the measurement devices produce are only due to the quantum- mechanical aspect of measurement We will consider the studied system and the measurement device together as a whole 3.C.3

18. Time evolution of the system 6 th postulate: The time evolution of the state vector of the system is determined by the Schrödinger equation: H – is the Hamiltonian operator, observable associated with the total energy of the system 3.B.4 Sir William Rowan Hamilton (1805 – 1865)

19. Quantization To transition from a classical to the quantum description of the system (e.g. in the Hamiltonian operator) one has to associate operator R with the position of the system, and operator P with the momentum of the system: Such replacement should contain appropriately symmetrized functions of R and P operators E.g. 3.B.5

20. Quantization For the Hamiltonian operator no symmetrization is necessary For a particle in an electromagnetic field: 3.B.5

21. RMS deviation How can one quantify the dispersion of the measurements around the mean value? Averaging a deviation from the average is not adequate: Instead, the RMS deviation is used: 3.C.5

22. RMS deviation How can one quantify the dispersion of the measurements around the mean value? Averaging a deviation from the average is not adequate: Instead, the RMS deviation is used: 3.C.5

23. Compatibility of observables If two (observable) operators commute, there exists a basis common to both operators There is at least one state that will simultaneously yield specific eigenvalues for these two operators, thereby these two observable can be measured simultaneously Such operators are called compatible with each other If, on the other hand, the operators do not commute, a state cannot in general be an eigenvector of both observables, thus these operators are called incompatible 3.C.6

24. Compatibility of observables When two observables are compatible, the measurement of the second does not produce any loss of the information obtained from the measurement of the first When two observables are incompatible, the measurement of the second does produces a loss of the information obtained from the measurement of the first 3.C.6

25. Compatibility of observables For the state of the system to be completely defined uniquely following a series of measurements, such measurement should be performed using a CSCO This is called preparation of a state 3.C.6

26. Properties of the Schrödinger equation The Schrödinger equation is of the first order in time, therefore there is no indeterminacy in the time evolution of the state of the system, unperturbed by measurements The Schrödinger equation is linear and homogeneous, therefore its solutions are linearly suporposable 3.D.1

27. Properties of the Schrödinger equation The norm evolution: The norm of the state vector remains constant 3.D.1

28. Properties of the Schrödinger equation For normalized The probability density is The probability density of finding the particle in an infinitesimal volume is: Schrödinger equation in r-representation: 3.D.1

29. Properties of the Schrödinger equation 3.D.1 +

30. Properties of the Schrödinger equation 3.D.1 +

31. Properties of the Schrödinger equation 3.D.1

32. Properties of the Schrödinger equation Introducing probability current We obtain the continuity equation The continuity equation states the local conservation of probability 3.D.1

33. Properties of the Schrödinger equation Introducing operator: We can calculate: 3.D.1

34. Properties of the Schrödinger equation It can be shown that For the Hamiltonian containing a vector potential, similar calculations lead to: 3.D.1

35. Properties of the Schrödinger equation How does the mean value of an observable evolve? Recall the CM result: 3.D.1

36. Properties of the Schrödinger equation For the Hamiltonian The Ehrenfest’s theorem: 3.D.1 Paul Ehrenfest (1880 – 1933)

37. Conservative systems When the Hamiltonian of the system does not depend explicitly on time the system is called conservative Eigenvalue equation for the Hamiltonian (τ denotes indices other than n): Since H does not depend explicitly on time, neither do eigenvalues and eigenvectors Any state can be expanded via the basis of H: 3.D.2

38. Conservative systems Using Schrödinger equation Solution: Therefore: For a continuous spectrum: 3.D.2

39. Conservative systems For an arbitrary observable B: Therefore: For a continuous spectrum: 3.D.2

40. Conservative systems If a certain state of the system is an eigenstate of H: Then: The two states at t 0 and t are physically indistinguishable, thus the physical properties of the system do not vary over time Such states are called stationary states 3.D.2

41. Constants of motion A constant of motion is an observable that satisfies the following: For a conservative system, H is a constant of motion There is always a system of common eigenvectors for H and A; these eigenvectors are stationary states: The eigenvalues of this constant of motion are called good quantum numbers 3.D.2

42. Constants of motion If a certain state of the system is an eigenstate of H: Then: The probability of finding a p when A is measured: Such probability is time-independent 3.D.2

43. Time-energy uncertainty relation Similar to the case of position-linear momentum uncertainty, it can be shown that For a conservative system, the greater the energy uncertainty, the more rapid is the time evolution 3.D.2

44. Superposition and interference Considering two orthogonal normalized states And a non-degenerate eigenket of an observable A: The probabilities of finding a n in two different states: Considering a normalized linear superposition of the two states: Then: 3.E.1

45. Superposition and interference Therefore the state of the system is not a statistical mixture of two states; interference effects are in play 3.E.1

46. Superposition and interference Therefore the state of the system is not a statistical mixture of two states; interference effects are in play 3.E.1

47. Superposition and interference Another example – light polarization Assuming that before the analyzer the state is: This is light linearly polarized at 45° with respect to axes x and y If we place an analyzer with the polarization axis perpendicular to e, none of the photons will pass 3.E.1

48. Superposition and interference Another example – light polarization Assuming that before the analyzer the state is: This is light linearly polarized at 45° with respect to axes x and y If the state was a statistical mix of two linearly polarized waves, then half of the photons would pass 3.E.1

49. Superposition and interference Considering an experiment for non-commuting observables A, B, and C: Then: 3.E.1

50. Superposition and interference Thereby: 1) Probabilistic predictions in QM are obtained by squaring the modulus of the probability amplitude 2) Linear superposition of the states of the system means that the probability amplitude could be a sum of partial amplitudes, and the probability is the square of the modulus of a sum of terms, and the partial amplitudes interfere with each other 3) When in a certain experiment we don’t know the result of an intermediate measurement, not only the probabilities of intermediate measurements should be considered, but also their probability amplitudes 3.E.1

51. Superposition and interference A somewhat different situation take places when the measurement is associated with a degenerate eigenvalue Recall the 4 th postulate: In this case the probability is calculated as a sum of squares This is not contradictory but complementary to the rule of squaring the modulus of the sum The rule: add amplitudes corresponding to the same final state and then probabilities corresponding to orthogonal final states 3.E.2

52. Superposition and interference A somewhat different situation take places when the measurement is associated with a degenerate eigenvalue Recall the 4 th postulate: If Then The rule: add amplitudes corresponding to the same final state and then probabilities corresponding to orthogonal final states 3.E.2

53. Superposition and interference This matter can be generalized onto the case of measured observables with continuous spectra E.g., a probability of finding a particle between x 1 and x 2 (in 3D): P is the projector onto the subspace of x values in the interval between x 1 and x 2 3.E.2