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Chapter 41
Quantum Mechanics
PHY 1371
Dr. Jie Zou
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Outline
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The double-slit experiment revisited
The uncertainty principle
PHY 1371
Dr. Jie Zou
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The double-slit experiment
produced by electrons
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Such an interference pattern cannot occur if
electrons behave as classical particles, and
hence electrons are behaving as waves.
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Wave-particle duality of
material particles
Consider the diffraction of
electrons passing through a
double slit.
Results: If the detector
detects electrons at
different positions for a
sufficiently long period of
time, one finds an
interference pattern
representing the number of
electrons arriving at any
positions along the detector
line.
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A detailed look at the doubleslit experiment
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(a)-(c): Computer simulation. (d): Real
photograph of a double-slit interference
pattern produced by electrons
The experiment is carried out at a low
beam intensity over long exposure.
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After a short exposure (after 28 electrons ):
Individual blips hitting in an apparently
random pattern.
After long exposure (after 10000eelctrons):
Interference pattern becomes clearer.
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Significance of the double-slit
experiment of electrons
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The wave-particle dual nature of electrons is clearly
shown in the experiment: Although the electrons are
detected as particles at a localized spot at some
instant of time, the probability of arrival at that spot
is determined by the intensity of two interfering
matter waves.
Interpretation of matter waves (first suggested
by Max Born in 1928): In quantum mechanics,
matter waves are described by the complex-valued
wave function .
The absolute square ||2 = * : ||2 gives the
probability of finding a particle at a given point at
some instant. The wave function  contains all the
information that can be known about the particle.
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Dr. Jie Zou
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Interference of matter waves
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With only slit 1 open, the
probability of detecting the
electron at the detector is given
by |1|2 (similarly for |2|2 ).
With both slits open, the electron
is in a superposition state: 
= 1 + 2.
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(a): Blue curve
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(b): Blue curve
(c): Red curve
Probability of detecting the electron
at the detector: ||2 = |1 + 2|2 =
|1|2 + |2|2 +2 |1||2| cos.
 = the relative phase difference
between 1 and 2 at the detector.
An electron’s wave property
interacts with both slits
simultaneously. The electron
passes
through both slits.
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Example Problem #1
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Neutrons traveling at 0.400 m/s are directed
through a double slit having a 1.00-mm
separation. An array of detectors is placed
10.0 m from the slit. Neutron mass mn =
1.675 x 10-27 kg.
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(a) What is the de Broglie wavelength of the
neutrons?
(b) How far off axis is the first zero-intensity point
on the detector array?
(c) When a neutron reaches a detector, can we
say which slit the neutron passed through?
Explain.
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Dr. Jie Zou
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The uncertainty principle
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Heisenberg uncertainty principle: If a
measurement of position is made with precision x
and a simultaneous measurement of linear
momentum is made with precision px, then the
product of the two uncertainties can never be smaller
than ħ/2: x px >= ħ/2, where ħ = h/2.
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It is impossible to measure simultaneously the exact position
and exact linear momentum of a particle.
The inescapable uncertainties x and px do not arise from
the imperfections in measuring instruments. Rather, they
arise from the quantum structure of matter.
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Dr. Jie Zou
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Example 41.2 Locating the
electron
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The speed of an electron is measured
to be 5.00 x 103 m/s to an accuracy of
0.00300%. Find the minimum
uncertainty in determining the position
of this electron.
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Dr. Jie Zou
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Homework
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Ch. 40, P. 1317, Problems: #46, 51.
PHY 1371
Dr. Jie Zou
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