Too Many to Count

1. Too Many to Count

2. Three Notations

3. The Three Notations of Quantum Mechanics • There are three notations (dialects if you like) commonly used in quantum mechanics • Sometimes they can be used interchangeably and sometimes not • Each has a strength and each has a weakness • They are named for the 3 “fathers” of quantum mechanics • Schroedinger • Heisenberg • Dirac

4. How they compare

5. Quantum Mechanical States are described by vectors in a linear vector space Linear vector space means a field of scalars over which the space From section 4.4 of Liboff’s text Postulate 1

6. Actually this is nothing new

7. A dual space exists with the same dimensionally as the original vector space AKA “dual continuum” the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1) Required to allow the inner product so that vectors can be normalized Postulate 2

8. Dual Spaces for the notations • To transform a vector from one space to another, a Hermitian conjugation is performed.

9. An inner product exists. Back in E&M, we called the inner product: “dot product” Inner product = dot product = scalar product Postulate 3

10. In the 3 notations

11. Postulate 4 The dual space is linear and has the following property

12. Postulate 5

13. Postulate 6

14. Multiplying a ket by a complex number (different from zero) does not change the physical state to which the ket corresponds Postulate 7

15. Postulate 7 is discussing normalization

16. It is convenient for define an orthonormal basis (and you’ve been doing it all your life!)

17. Operators • A mathematical operation on a vector which changes that vector into another • This is not mere multiplication (like Postulate 7) but we are actually changing something like its direction or perhaps other quantities. • Example: Let Q be the differential operator with respect to x Direction of operation Direction of operation

18. Physical observables (such as position or momentum) are represented by linear Hermitian operators Postulate 8

19. What does linear mean?

20. What does Hermitian mean?

21. A special case for operators Called “Eigenvector” or “Eigenfunction” or “Eigenket” Called “eigenvalue”

22. What does an eigenvalue mean in Schroedinger notation?

23. What does an eigenvalue mean in Heisenberg notation?

24. Eigenvalues of a Hermitian operator are real i.e. If Q+=Q then q*=q Theorem 1

25. Proof of Thm 1

26. Eigenvectors of a Hermitian operator are orthogonal if they belong to different eigenvalues Theorem 2

27. Proof of Thm 2 Note: An operator may have a set of eigenvalues of which 2 or more are equal; this is called degeneracy

28. Projection operators • Graphically, the inner product represents the project of a onto b or in Dirac notation |a> onto |b> |b> |a> <a|b> If |a> is considered a unit vector, then the vector which represents projection of |b> onto |a> is written <a|b>|a> or |a><a|b>

29. Theorem 3 A projection operator is idempotent i.e. Q2 =Q

30. Theorem 4

31. Proof of Thm 4

32. Creating a set of orthogonal vectors from a set of normalized linear independent kets • Let |a>, |b>, and |c> be a set of normalized linear independent kets • We are going to create a new set of kets (|1>, |2>, |3>) from these which will be orthogonal to one another i.e. <1|2>=0, <1|3>=0 and <2|3>=0 • First, pick one of the original set and build the rest of the set around it • |1>=|a>

33. Constructing |2> Geometrically |2>=|b>-|1><1|b> |b> |b>-|1><1|b> |1><1|b> |1> -|1><1|b>

34. Test that |2> is orthogonal to |1>

35. Normalizing |2>

36. |3>

37. Eigenvalues are the only possible outcome of physical measurements If physical observables are represented by Hermitian operators and these have real eigenvalues, it is reasonable to assume that there is a connection between their eigenvalues and the results of experiments. Postulate 9

38. Operators representing simultaneously observable quantities commute Theorem 5

39. Proof of Thm 5 Commutator Brackets [a,b]=(ab-ba) If [a,b]=0 then a and b commute QM analog of Poisson brackets

40. An Example of non-commuting operators

41. Postulate 10 The average value in the state |a> of an observable represented by an operator Q, is Called an “expectation value” or called the “mean”

42. In Schroedinger Notation

43. In Heisenberg Notation

44. Defining Standard Deviation • Let Q= operator • DQ= standard deviation of measurement of Q • (DQ)2= variance of that measurement • Sometimes called mean square deviation from the mean • (DQ)2 =<(Q-<Q>)2> • Or, more compactly • (DQ)2 =<Q2>-<Q>2

45. The Uncertainty Principle • If two observables are represented by commuting operators then you can measure the physical observables simultaneously • If the operators DO NOT COMMUTE then a SIMULTANEOUS measurement will NOT BE EXACTLY REPEATABLE • There will be a spread in the measurement such that the product of the standard deviations will exceed a minimum value; the size of the minimum depends on the observable • To calculate this, we first have to build some mathematical machinery.

46. Theorem 6 Schwartz’s Inequality

47. Proof of Thm 6

48. Theorem 7 Let a = A-<A> and b =B -<B> then [a,b] =[A,B]

49. Derivation of the Uncertainty Principle for any Operator

50. Derivation of the Uncertainty Principle … page 2