1. ???
The Uncertainty
Principle
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W. Udo Schröder, 2004

2. Incommensurable Observables
Wave-mechanical effect, example: Position & Momentum
Definitions of momentum and position complementary:
Narrow momentum distribution  broad spatial distr.
Broad momentum distribution  narrow spatial distr.
Observables cannot be measured simultaneously with arbitrary accuracy  incommensurable
Is this due to experimenters’ lack of skill, ability, or are observables actually not well defined?
Uncertainty Rel
W. Udo Schröder, 2004

3. Simplistic Rationalizations
Folklore: “Both momentum and position are sharp, in principle, but impossible to measure simultaneously”.
“If one measures position by shining photons on a system, the momentum is disturbed, depending on the mass of the system (Ocean liner vs. e-)”.
“The measurement disturbs the system to be measured, introducing uncertainty. Uncertainty is large for microscopic particles because of momentum transfer from scattered photon”.
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Use monochromatic photons  hn, pn determined  determine Ee, pe
W. Udo Schröder, 2004

4. Classical Relation t  n: Fourier Transforms
Fourier transform of f(t)
Uncertainty Rel
0.25
Long t pulse  narrow frequency band
W. Udo Schröder, 2004

5. Classical Relation t  n: Fourier Transforms
Fourier transform of f(t)
Uncertainty Rel
1.25
Short t pulse  broad frequency band
W. Udo Schröder, 2004

6. Gaussian Fourier Transforms
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W. Udo Schröder, 2004

7. Uncertainty Time vs Frequency
Sample frequency fs: N0=64/N= 400
switched off
time
Short pulses have no well defined n
time
Uncertainty Rel
W. Udo Schröder, 2004

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

9. Symmetries of Matrix Elements
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W. Udo Schröder, 2004

10. Commutators
Uncertainty Rel
W. Udo Schröder, 2004

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

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

13. Anti-Correlated x, px Spreads
Measurement: Catch particle in a box (detector)
Heisenberg, 1924
Heisenberg Uncertainty Relation
a=10 fm
a=30 fm
SimpSys
Probability distributions to find a particle at position x and momentum px have anti-correlated width.
 {x},{px}= conjugate spaces, like n and t in Fourier analysis
probability density
position x
momentum px
W. Udo Schröder, 2004

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

15. Uncertainty Rel
W. Udo Schröder, 2004