Canonically Conjugate Variables and Uncertainty Principle

1. Welcome
Uncertainty relation in some common canonically
conjugate pairs & Applications.
Dr. Meenu S.
Assignment 1 submitted on 12/8/2020
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2. 2
A bit mathematics,
 Conjugate variables are pairs of variables mathematically defined in such
a way that they become Fourier transform duals.
 This dual relation lead naturally to an uncertainty relation.
 More mathematically, conjugate variables are related by Noether's
theorem,
 Statement – “if the laws of physics are invariant with respect to a change in
one of the conjugate variables, then the other conjugate variable will not
change with time”.
Canonically Conjugate Variables

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 Amalie Emmy Noether (German mathematician)
 the most important woman in the history of mathematics.
 In physics, Noether's theorem explains the connection between symmetry and
conservation laws (every differentiable symmetry of the action of a physical system
has a corresponding conservation law)
 Translational symmetry  P
 Rotational symmetry  L
 Translational symmetry in time  E
Canonically Conjugate Variables …contd

4.  Are Fourier transform duals
 Several types of conjugate variables (depending on the type of work the system is doing or is
being subjected to)
 Definition ???
 Works according to the HUP in Physics. i.e., simultaneous and precise measurements of the
canonical conjugate variables are impossible.
 Their commutator is nonzero constant  [A,B]= AB-BA = iħ = - [B,A]
 Also known as complementary variables
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A brief note on Canonically Conjugate variables

5.  E.g.,
1. Position - Momentum
2. Energy - Time
3. Angle turned - Angular momentum
4. Period - Frequency
the longer a musical note is sustained, the more precisely we know its frequency.
5. Doppler - Range
the more we know about how far away a radar target is, the less we can know about the
exact velocity of approach or retreat, and vice versa. In radar terminology this pair is
known as radar ambiguity function.
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A brief note on Canonically Conjugate variables

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• Werner Heisenberg
• One of the most celebrated results of quantum mechanics
• “All information about a particle can not be measured precisely and simultaneously”
• Inherent uncertainty in the act of measuring a variable of a particle.
• Commonly applied to the position and momentum of a particle, where it says that
“the more precisely the position is known the more uncertain the momentum is and vice
versa”.
About the Uncertainty Principle

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4π
h
ΔpΔx 
Uncertainty relation in some common canonically conjugate pairs
1. Linear momentum, Position
2. Energy and Time
3. Angular momentum and angle turned
4π
h
ΔtΔE 
4π
h
ΔLΔθ 

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Physical Significance of Uncertainty Principle
A few points
1. Formally limits the precision to which two complementary observables can
be measured
2. Establishes that observables are not independent of the observer.
3. A phenomena can take on a range of values rather than a single, exact
value.
4. Necessitates a probabilistic interpretation of the behavior of matter on the
molecular level.

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My simple idea
helps you get an
idea about,
∆ why atoms don't implode?
∆ non-existence of electrons in the nucleus
∆ The ground state energy of H-atom
∆ Width of spectral lines of atomic emission
∆ How the Sun manages to shine?
∆ Strangely, that the vacuum of space is not actually empty.
∆ Mass of Meson
∆ Zero-point energy of harmonic oscillator
and many more
eV13.6
2h
me
E 2
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

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Thank You