1. Operators
Postulates
W. Udo Schröder, 2004

2. This is actually true for all wf’s
Quantum Operators
Quantum mechanical operators must be linear and Hermitian.
For any linear combination of solutions y1 and y2 of Schrödinger Equation  Effect of  should be linear combination of individual effects
Â(ay1+by2) = a Ây1+ b Ây2
Classical observables have real values  operators must have real eigen values (a* = a, Hermitian)  (Â-EF ya)
same value
Postulates
Hermitian operator  “matrix element”
Check this out for p
This is actually true for all wf’s
W. Udo Schröder, 2004

3. Functions of Operators
n times
same coefficients
Postulates
W. Udo Schröder, 2004

4. Hermitian and Anti-Hermitian Operators
Transposed and complex conjugate ME
Hermitian
Postulates
Presence of i in p important !!!
W. Udo Schröder, 2004

5. Symmetries of Matrix Elements
Postulates
W. Udo Schröder, 2004

6. Expectation Values in Component Representation
Solutions to PiB problem: a) LC of p-eigen functions
Y generally not EF to p-operator  Observable not sharp (s ≠0)
Solutions to PiB problem: b) LC of Ĥ-eigen functions
y generally not EF to Ĥ-operator  Observable not sharp (s ≠0)
Example:
|cn|2= Probability(state yn)
position x
y2 , y3, (y2·y3) + ++ -- - y2 y3
Postulates
weighted average <E>
W. Udo Schröder, 2004

7. Instant Problem: Calculate P(p)
Particle in a box:
Postulates
W. Udo Schröder, 2004

8. Instant Problem: Calculate P(p)
Particle in a box:
Postulates
W. Udo Schröder, 2004

9. Hermitian and Anti-Hermitian Operators
Transposed and complex conjugate ME
Hermitian
Postulates
Presence of i in p important !!!
W. Udo Schröder, 2004

10. Symmetries of Matrix Elements
Postulates
W. Udo Schröder, 2004

11. Commutators
Postulates
W. Udo Schröder, 2004

12. Heisenberg’s Uncertainty Relation
Observed for PiB model:
Is this general, for which observables A,B ?
Postulates
W. Udo Schröder, 2004

13. Heisenberg Uncertainty Relation Example:  already derived for PiB ≥0
anti-Hermitian Hermitian <>=imaginary <>=real ≥0
Postulates
Heisenberg
Uncertainty Relation
Example:
 already derived for PiB
W. Udo Schröder, 2004

14. The End -- of this Section
Now,
that was fun,
wasn’t it ?!
Postulates
W. Udo Schröder, 2004

15. Hermitian and Anti-Hermitian Operators
Transposed and complex conjugate ME
Hermitian
Postulates
Presence of i in p important !!!
W. Udo Schröder, 2004

16. Postulates
W. Udo Schröder, 2004

17. Gaussian Wave Packet (discrete)
k0=20, Nk=40
Postulates
W. Udo Schröder, 2004

18. Gaussian Wave Packets Wave traveling to x>0 Normalization
Postulates
W. Udo Schröder, 2004

19. Eigen Functions of Hermitian Operators
position x
y2 , y3, (y2·y3) + ++ -- - y2 y3
Set of all eigen functions {ya} of Hermitian  form a complete set of orthogonal basis “vectors”
Integral over overlap vanishes
identical integrals (Hermitian)
Postulates
{|ya>}=complete: must cover all possible outcomes of measurements of A
normalized ya:
W. Udo Schröder, 2004

20. Particle-in-a-Box Ĥ-Eigen Functions
-a/ Position x +a/2
Wave Function.
Normal Modes
All PiB energy eigen functions = orthonormal set
Scalar product (Overlap)
Integral over overlap vanishes
j,c ≠ Ĥ-EF
Representation of Y (PiB)
= math. solutions of PiB problem
Postulates
All physical solutions can be represented by LC of set {yn} or {|yn>}
W. Udo Schröder, 2004

21. Illustration: Representations of Ordinary Vectorsz’ 2 3 x y z
Normal vector spaces: coordinate system defined by set of independent unit, orthogonal basis vectors 4
Scalar Product
Components:Projections
Example
Postulates
Representation of r in basis {x,y,z}
Representation of r in basis {x’,y’,z’}
LC of basis vectors
W. Udo Schröder, 2004

22. Instant Problem: Find Components of a Vectorz’ 4 2 3 x y z
Independent unit basis vectors
Example: Calculate cx, cy, cz of
Postulates
W. Udo Schröder, 2004

23. Instant Problem: Normalize a Vector4 2 3 x y z
Independent unit basis vectors
Example: Calculate N such that
Postulates
W. Udo Schröder, 2004

24. Instant Problem: Find Orthonormal to a Vectory
Independent unit basis vectors x x
Postulates
W. Udo Schröder, 2004

25. PiB Wave Functions as Superpositions of Normal Modes
General (all possible) solutions to PiB problem: LC of Ĥ-eigen functions {yn}
position x
y2 , y3, (y2·y3) + ++ -- - y2 y3
(Y ≠ Ĥ-EF)
Orthogonality/ Normality
(<sin|cos> cross terms vanish)
Constraint on cn & cm:Normalization of Y:
Postulates
“Fourier” Coefficients cn
cn=<yn|Y>: Amplitude of yn in Y |cn|2: Probability of Y to be found in yn
W. Udo Schröder, 2004

26. Representations of Wave Functions/Kets
Normal vector spaces: coordinate system defined by set of independent unit basis vectors j3
Express Y in terms of sets of orthonormalized EFs 2 3 j2 3
different observables j1
Postulates
All representations are equally valid, for any true observable.
W. Udo Schröder, 2004

27. Symmetries of Matrix Elements
Postulates
W. Udo Schröder, 2004

28. Commutators
Postulates
W. Udo Schröder, 2004

29. Heisenberg’s Uncertainty Relation
Observed for PiB model:
Is this general, for which observables A,B ?
Postulates
W. Udo Schröder, 2004

30. Heisenberg Uncertainty Relation Example:  already derived for PiB ≥0
anti-Hermitian Hermitian <>=imaginary <>=real ≥0
Postulates
Heisenberg
Uncertainty Relation
Example:
 already derived for PiB
W. Udo Schröder, 2004