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Basics in Quantum Mechanics
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Basics in Quantum Mechanics
• Dimension scaling 1: an atom has a typical dimension in
the order of 1Å=10-10m, which means an atom to a
baseball (~5cm) is like a baseball to the global earth
(~15,000km)!
• Dimension scaling 2: a nucleus has a typical dimension in
the order of 10-5Å, which means a nucleus to the atom is
like a rice grain (~1mm) to a huge building (~100m)! In
this scheme, the electron is like a dust moving around the
a rice grain with a distance in the range of the huge
building.
• Do we still believe that the motion of the electron must
follow Newton’s laws?
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Basics in Quantum Mechanics
• Experimental evidence shown inconsistency in Newton’s
theorem
–
–
–
–
–
1. Black body radiation
2. Photon-electron effect
3. Atomic line spectrum
4. Atomic stability
5. Specific heat of solids
• Mending works
– 1. Plank’s quantized photon energy E=hv (1900)
– 2. Einstein’s quantized momentum p=h/λ from p=E/c for photons
(1905)
– 3. Bohr’s quantized electron angular momentum J=nh/(2π),
n=1,2,3,… (1913)
– 4. de Broglie’s mater-wave duality 2πr=nλ, n=1,2,3,… (?)
– 5. Einstein’s quantized phonon energy and momentum (1907)
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Basics in Quantum Mechanics
• Experimental confirmations
– Compton’s photon-electron scattering experiment
supported Plank-Einstein’s theorem (1923), so photons
and phonons are particle like, or “real” waves (with
zero static mass, reflecting the interaction between
matters) can be particle like
– Davison and Germer’s electron-Nickel crystal
scattering experiment supported Bohr-de-Broglie’s
theorem (1927), so electrons are wave like, or matters
(with nonzero static mass) can be wave like
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Basics in Quantum Mechanics
A free particle in vacuum is described by:
p2
E 
2m
Following de Broglie’s matter-wave duality, we have :
E  
p  k
This free particle can be mapped to a plane traveling wave
in vacuum with dispersion relation and traveling speed given
as:
d k p
k 2

2m
vg 
dk

m

m
v
The parabolic dispersion relation suggests that the plane
traveling wave satisfy the following wave equation:

(r , t )
2 2 
 2 ( / j ) 2

j

  (r , t ) 
 (r , t )
t
2m
2m
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Basics in Quantum Mechanics
In vacuum, the particle only bears the kinetic energy.
In arbitrary space, however, energy conservation requires:

 
(r , t )  2 ( / j ) 2



j

(r , t )  V (r )(r , t )  H(r , t )
t
2m
This is SchrÖdinger’s equation (1926).
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Basics in Quantum Mechanics
• Interpretation of matter-wave
– Wave packet? No, as otherwise the dispersion would make the
particle (e.g., an electron) change its diameter, which is certainly
absurd.
– Statistical behavior of a large number of particles? No, as a single
particle exhibit the wave behavior as well.
– Born’s interpretation (1926): the intensity of the wave function at
certain location in space is the probability of finding the particle at
this location.
• Therefore, the classical mechanics (Newton’s theory)
predicts the trajectory of an object, whereas the quantum
mechanics predicts the probability of an object’s
emergence in space.
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Basics in Quantum Mechanics
• A classical view of a matter: (1) has mass, charge, etc.,
known as the particle property; they are the inherent
property that defines the matter; (2) stay at a position, or
move along a trajectory; they are not the inherent property
of a matter! It was just introduced along with Newton’s
theorem.
• A quantum mechanics view of a matter: (1) the same; (2)
stay at a position at certain time with certain probability.
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Basics in Quantum Mechanics
• Matter-wave properties:
– Presentation (value quantity and operator)
– Superposition and self-interference
– Uncertainty principle
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