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 Notation Name Type Example Comments Math function
Schroedinger
“Wavefunction”
Math function
Y =f(x,y,z,t)
Great for integrating, doe not handle groups
Heisenberg
Column matrix
Matrix
Handles spin, isospin, and some flavors, can’t integrate easily
Dirac
“ket”
Both matrix and math function
Very ambiguous, sometimes too ambiguous

5. Postulate 1
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

6. Actually this is nothing new

7. Postulate 2
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

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

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

10. In the 3 notations
Schroedinger
Heisenberg
Dirac

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

12. Postulate 5

13. Postulate 6

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

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

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

25. Proof of Thm 1

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

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> |2>=|b>-|1><1|b> Geometrically

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

35. Normalizing |2>

36. |3>

37. Postulate 9
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.

38. Theorem 5
Operators representing simultaneously observable quantities commute

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

51. Need more power!
Now the absolute square of any complex number, z, can be written as
|z|2 = (Re(z))2 +(Im(z))2
Of course, |z|2  (Im(z))2

52. An Aside

53. So we can now start having fun…

54. The final slide

55. Does it work?