Lecture 7. Heisenberg Uncertainty Principle

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Lecture 7. Heisenberg Uncertainty Principle

• Outline
• Wave Packets
• Uncertainty position ? momentum (1927)
• Uncertainty energy ? time

The uncertainty principle saves us from
contradiction between the complimentary wave-like
and particle-like properties of quantons.
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Wave Packets
Plane monochromatic wave describes the particles
state with a well-defined momentum p hk
According to the statistical interpretation of
the wave function, the probability to find the
particle within ?x is proportional to
- thus, the probability is constant
(x-independent), the particle is unlocalized!
snapshot of the distribution A(x,t) at t t0
To localize the particle, we need to form a wave
packet (group), by combining plane waves with
different ks.
Because of the dispersion of dB waves, the packet
will spread with time. Again, it doesnt mean
that the particle itself spreads out with time,
rather, the probability of finding the particle
away from a moving center of the distribution
A(x,t) grows with time.
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Localization of the Wave Packet
To localize a quanton, we need to prepare a
superposition of plane waves with some non-zero
range of k values,

the result of addition
Now the momentum is now well-defined it is
somewhere within the range
Thus, for a wave packet
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Uncertainty position ? momentum
More accurate definition of ?x for an arbitrary
distribution A(x), ?x is the standard deviation
for this distribution (the root-mean-square
deviation of the values from their mean)
A quantons position and momentum cannot be
simultaneously well-defined their uncertainties
?x and ?px are inter-related
Uncertainty of the momentum
Uncertainty of the position

• The U.P. couples only incompatible
  observables (every pair of observables whose
  operators do not commute)

• The indeterminacy reflects the limits beyond
  which we can no longer use the intuitive
  classical notions of the particles trajectory.

• In quantum mechanics, the division of the total
  energy of a particle into its kinetic (p) and
  potential (x) parts is also blurred.

The complimentary wave-like and particle-like
properties of quantons can be reconciled only
within limits imposed by the uncertainty
principle.
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Electron Diffraction
Lets consider a plane dB wave that describes an
electron with a well-defined momentum p (along
y). To determine the electron position along the
x-axis, we use a screen with a slit ?x.
Because of diffraction, the electron wave spreads
within a cone 2? behind the slit
The uncertainty in the momentum along x
Thus, if we measure x with uncertainty ?x, at the
next moment we wont be able to determine px with
an accuracy better than ?px .
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Heisenberg Microscope
The diffraction-limited size of the image of a
point-like object
The scattered photons are collected within an
angle ?
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...not a bug, but a feature...
The U.P. is not caused by the measurement
process, it is an intrinsic aspect of the wave
nature of quantons.
It reflects the properties of quantum states
rather then the limitations on the accuracy of
measurements. In contrast to a widespread belief,
the U.P. does not limit accuracy of measurements.
One can perform measurements with an ensemble of
identical particles for some of them, ?x is
measured, for the other - ?p, however, the
dispersions of these measurements would satisfy
the condition (?x2)?(?p2)gth2/4 despite the fact
that only one type of measurements has been done
on each particle.
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Q.M. vs. Cl.M.
- to figure out whether one needs to use Q.M. or
Cl.M., we can estimate this product (if it is gtgt
h, the behavior is classical).
Examples
Air at T300K and normal pressure
Distance between the molecules
Cl.M. is okay
Cold atomic gases
Q.M. !

• Bose-Einstein condensation if atoms are boson
• degenerate Fermi gas if atoms are fermions

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Problem
A certain device is designed to make a
simultaneous measurement of the position and
velocity of an electron. If the device is
designed to measure the velocity to an accuracy
of one part in 106, what will be the limitation
on the accuracy of the corresponding position
measurement if the velocity measurements yields
the result (a) v2.9?106 m/s, (a) v2.9?108 m/s.
non-relativistic
(a)
(b)
relativistic
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Problem
How accurately can the position of a 2.5keV
electron be measured assuming its kinetic energy
is known to 1.
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U.P. as a tool to estimate the ground state energy
For a confined motion (bound systems), U.P. sets
a limit how small a momentum (and, thus, the
kinetic energy) could be. This is an important
tool that helps us to estimate the ground-state
energies of various quantum systems.
1. Estimate the lowest possible kinetic energy of
a neutron contained in a typical nucleus of a
radius a 1?10-15m.
non-relativistic
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Classical Atom (oxymoron!)
- unambiguous separation of the total energy E
into K and U (we ignore the rest energy)
- the potential energy is the result of
interaction between an electron and the electric
field created by a proton
electrostatic potential (the field characteristic
)
1D motion
2D motion
return point
- no limit on E and, thus, on r !
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The Uncertainty Principle applied to an H Atom
- the dependence of the classical energy on the
atoms radius
Quantum approach
minimum of Equantum corresponds to the ground
state the state with a minimum energy
?
non-relativistic motion
The speed of the electron motion in H atom
The dB wavelength of the electron is comparable
with R
semi-classical cartoon (doesnt make sense for
the low-energy states)
Thus, we need Q.M. to describe the system!
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The ground state of a quantum harmonic oscillator
(Beiser 3.39) The frequency of oscillation of a
harmonic oscillator of mass m and spring
constant C is
(x its displacement from the equilibrium
position)
The energy of the oscillator is
In classical physics the minimum energy of the
oscillator is 0. Use the U.P. to find Emin in
quantum physics (hint express E in terms of x
only and find Emin from dE/dx0).
Emin - the minimum energy (the ground state
energy)
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Zero-Point Oscillations
The zero-point motion persists even at T0.
- the smaller the mass, the larger the amplitude
of zero-point oscillations
Light atoms the zero-point oscillations are
sufficient to prevent liquid Helium-4 from
freezing at atmospheric pressure, no matter how
low the temperature.
Probing the Quantum Limit of Vibrations
The amplitude of the thermal (classical)
vibrations
Quantum oscillations should become observable at
Nanomechanical systems are already approaching
this limit (see the paper by Schwab and Roukes)
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Uncertainty Energy ? Time
For a free (non-relativistic) particle
?t the time scale at which a system changes its
energy by ?E
Incorrect to treat ?t as the measurement time
Example the lifetime of a free neutron is 15
min. How uncertain is its energy?
Example Truly monochromatic light corresponds to
an infinitely long plane wave. What is the spread
in the frequency of monochromatic light after
it passes through a fast shutter that forms
1-?s-long light pulses?
Example Calculate how long a virtual
electron-positron pair can exist.
The minimum amount of energy that needs to be
borrowed to make a pair -
By the uncertainty principle, the maximum time
for which such borrowing can go on
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HW 4
Homework 4 Beiser Ch. 3, Problems 1, 5, 12,
14, 19, 24, 32, 38