Operators Postulates W. Udo Schröder, 2004
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
Functions of Operators n times same coefficients Postulates W. Udo Schröder, 2004
Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates Presence of i in p important !!! W. Udo Schröder, 2004
Symmetries of Matrix Elements Postulates W. Udo Schröder, 2004
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
Instant Problem: Calculate P(p) Particle in a box: Postulates W. Udo Schröder, 2004
Instant Problem: Calculate P(p) Particle in a box: Postulates W. Udo Schröder, 2004
Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates Presence of i in p important !!! W. Udo Schröder, 2004
Symmetries of Matrix Elements Postulates W. Udo Schröder, 2004
Commutators Postulates W. Udo Schröder, 2004
Heisenberg’s Uncertainty Relation Observed for PiB model: Is this general, for which observables A,B ? Postulates W. Udo Schröder, 2004
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
The End -- of this Section Now, that was fun, wasn’t it ?! Postulates W. Udo Schröder, 2004
Hermitian and Anti-Hermitian Operators Transposed and complex conjugate ME Hermitian Postulates Presence of i in p important !!! W. Udo Schröder, 2004
Postulates W. Udo Schröder, 2004
Gaussian Wave Packet (discrete) k0=20, Nk=40 Postulates W. Udo Schröder, 2004
Gaussian Wave Packets Wave traveling to x>0 Normalization Postulates W. Udo Schröder, 2004
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
Particle-in-a-Box Ĥ-Eigen Functions -a/2 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
Illustration: Representations of Ordinary Vectors z’ 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
Instant Problem: Find Components of a Vector z’ 4 2 3 x y z Independent unit basis vectors Example: Calculate cx, cy, cz of Postulates W. Udo Schröder, 2004
Instant Problem: Normalize a Vector 4 2 3 x y z Independent unit basis vectors Example: Calculate N such that Postulates W. Udo Schröder, 2004
Instant Problem: Find Orthonormal to a Vector y Independent unit basis vectors x x Postulates W. Udo Schröder, 2004
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
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
Symmetries of Matrix Elements Postulates W. Udo Schröder, 2004
Commutators Postulates W. Udo Schröder, 2004
Heisenberg’s Uncertainty Relation Observed for PiB model: Is this general, for which observables A,B ? Postulates W. Udo Schröder, 2004
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