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Uncertainty relations
Armen Allahverdyan
(Yerevan Physics Institute)
-- Introduction
-- Heisenberg-Kennard-Robertson
-- Precision-disturbance interpretastion
-- Joint measurement interpertation
∆x ∆ p ≥ 
why quantum particles do not fall into each other
∆p
∆x → 0 ⇒
→∞
2m
2
tunneling
U (x)
2
p
= E − U ( x) < 0
2m
Quantum mechanics = waves + probability theory
state: vector in complex linear space
| ψ = (ψ 1 ,...,ψ N ) , ψ |= (ψ 1*,...,ψ N *)
T
N
ψ | ψ = ∑ψ kψ k *
k =1
observable: linear, hermitean operator
N
N
k =1
k =1
A = ∑ ak | ak ak |, A | ψ = ∑ ak | ak ak |ψ
| a k | ψ |2
probability
ak | al = δ kl
Standard uncertainty relations
Cauchy-Schwartz
inequality
Kennard-Robertson uncertainty relation (1927)
impossible to prepare a quantum state with precise values of .....
Illustration for atoms in crystalls: W Jauch, Am J Phys 1993.
MnF2
F
σ ( x) ≥ σ min ( x)
295 K
11.78 ≥ 4.47 ×10 −12 m
60 K
15 K
7.10 ≥ 6.55
6.71 ≥ 6.60
≈ 10 −14 m
σ ( x) ≈ 10 −12 m
Bohr-Heisenberg
under joint measurements the precisions hold .....?
measuring one variable perturbs another ?
error-disturbance
Measurement
S
M
first stage: interaction between 2 quantum systems
correlations between M and S
M'
second stage: appearance of definite measurement results events
not decribed by quantum mechanic without additional axioms
measurement problem, unconventional quantum theories etc
Error-disturbance relation between conjugate variables
linear transformation
requires special Hamiltonian
error in measuring the coordinate
disturbance of the momentum
perfect measurement with finite disturbance
Simultaneous measurement of two non-commuting variables
initial time
uncertainties
unbiased
non-commuting variables are mapped to
commuting ones
irreducible noise
4 times larger
Final state of commuting variables is noisy
Arthurs & Goodman 1988
Conclusion
• Uncertainty relation as an intrinsic feature
of quantum states: clear.
• As a feature of joint measurements: more is
to be done (unbiasedness?)
• As error-disturbance relation: does not hold
in its literal form. Error-disturbance is a
more complex issue.
Quantum mechanics
state: vector in complex linear space
observable: linear, hermitean operator
Heisenberg dynamics
Hamiltonian
Schroedinger dynamics
average
Warning: non-commuting operators can be sometimes measured jointly