Operators

1. Operators Postulates

2. 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

3. n times same coefficients same coefficients Functions of Operators Postulates

4. Presence of i in p important !!! Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates

5. Symmetries of Matrix Elements Postulates

6. y3 ++ y2 + y2 , y3, (y2·y3) - -- position x 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) Postulates weighted average <E>

7. Instant Problem: Calculate P(p) Particle in a box: Postulates

8. Instant Problem: Calculate P(p) Particle in a box: Postulates

9. Presence of i in p important !!! Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates

10. Symmetries of Matrix Elements Postulates

11. Commutators Postulates

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

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

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

15. Presence of i in p important !!! Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates

16. Postulates

17. Gaussian Wave Packet (discrete) k0=20, Nk=40 Postulates

18. Gaussian Wave Packets Wave traveling to x>0 Normalization Postulates

19. y3 ++ y2 + y2 , y3, (y2·y3) - -- position x Eigen Functions of Hermitian Operators 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:

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

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

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

23. z y 4 x 3 2 Instant Problem: Normalize a Vector z’ Independent unit basis vectors Example: Calculate N such that Postulates

24. y x Instant Problem: Find Orthonormal to a Vector Independent unit basis vectors x Postulates

25. y3 ++ y2 + y2 , y3, (y2·y3) - -- position x PiB Wave Functions as Superpositions of Normal Modes General (all possible) solutions to PiB problem: LC of Ĥ-eigen functions {yn} (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 Yto be found in yn

26. 3 2 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 j2 3 different observables j1 Postulates All representations are equally valid, for any true observable.

27. Symmetries of Matrix Elements Postulates

28. Commutators Postulates

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

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