Download De Broglie waves

Ch3 Introduction to
quantum mechanics
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De Broglie waves
Davission-Germer experiment
Uncertainty principles
Wave function
Barrier penetration
Hydrogen atom
Lasers
Some words & terms
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Wave particle duality
Wavelength
Electron diffraction
De Broglie wave
Quantum
Uncertainty principle
Wave function
Schrodinger equation
Photon, electron
Sinusoidal, Exponential
• Probability density
• Potential well
• Barrier penetration (势垒穿
透)
• tunnel
• Scanning tunnel microscope
(STM)
• Operator
• Eigenfuction, Eigenvalue
• Laser
• Spontaneous (simulated)
emission
Wave-particle duality of light
• The dual wave-particle nature of light is
now a basic part of the theory of light and
matter.
• Wave feature: υ, λ;
• Particle feature: E, p;
• Photons
– energy: E=hυ;
– momentum: p=h/λ, k=2pi/λ;
De Broglie waves
• In1924, Louis de Broglie proposed that
matter possesses wave as well as particle
characteristics, receiving the Nobel Prize in
1929.
• A moving body behaves in certain ways as
though it has a wave nature.
– Photon wavelength:
hv h
p

c 
– De Broglie wavelength: λ  h
mv
h
λ
p
m
m0
1
v2
c2
Which property is more important?
• Problem: Find the de Broglie wavelengths of
a 46g golf ball with a speed of 30m/s and an
electron with a speed of 107m/s.
• For a 46g golf ball, v  c, m  m0 ,   h  4.8 1034 m
mv
• For an electron,
h
m  m0 ,  
 7.3 10 11 m
mv
– Radius of the hydrogen atom, 5.3x10-11m.
The diffraction of electrons
• In 1927, Davisson and Germer showed that
electron beams are diffracted when they are
scattered by the regular atomic arrays of
crystals.
• A beam of electrons from a heated filament (灯
丝) is accelerated through a potential difference
V. Passing through a small aperture (孔), the
beam strikes a single crystal of nickel (镍).
Electrons are scattered in all directions by
atoms of the crystal.
The Bragg equation
• Instead of a continuous variation of scattered electron
intensity with angle, distinct maximum and minimum
were observed with their positions depending on the
electron energy.
• The Bragg equation for maxima in the diffraction
pattern is: n  2d sin 
• In a particular case, a beam of 54eV electrons were
directed perpendicularly at the nickel target and a
sharp maximum in the electron distribution occurred at
an angle of 50o with the original beam. Here θ=650,
d=0.091nm, n=1, λ=0.166nm. 54eV0.166nm
Electron microscopes
• The wave nature of moving electrons is the
basis of the electron microscope, the first of
which was built in 1932.
• Fast electrons have wavelengths very much
shorter than those of visible light. For example,
an electron with 54eV (4.4x106m/s) has the
wavelength of 0.166nm.
• In an electron microscope, current-carrying
coils produce magnetic fields that act as lenses
to focus an electron beam on a specimen. A
magnification of over 1,000,000 can be
reached.
The uncertainty principle
• The principle describes a natural limit to the
precision of simultaneous measurements of
the position and momentum of particles.
• If an object is said be at position of x with
an uncertainty of Δx, then any simultaneous
measurement of the x component of
momentum must have an uncertainty Δpx
consistent with x px  h
• It also implies that either wave or particle
properties, but not both, may be observed in
a given experiment.
The uncertainty relations
• Heisenberg uncertainty relation for position
and momentum. x px  h
– It can be used to estimate the size and the
energy of the hydrogen atom in its ground state.
• Heisenberg
uncertainty
relation
for
time
and
t E  h
energy.
– It is used to study the lifetime of unstable
systems.
Wave function
• Each particle is represented by a wave
function, and the square of its absolute
magnitude |ψ|2 evaluated at a particular place
at a particular time is proportional to the
probability of finding the body there at that
time. Probability density.
• The probability density |ψ|2 is given by the
product of ψ and its complex conjugate ψ*.
• The wave function must be single-valued,
superposable, with finite partial derivatives.
Schrodinger equation
2 2

   V  E
2m
• The wave function is governed by
Schrodinger equation, which is a type of
equation known as eigenvalue equation.
Solutions can only be obtained for certain
values of energy known as energy
eigenvalues.
Particle in a box
0, ,0  x  L
V ( x)  
, , x  0, x  L
2

L  n n  1,2,3...
 2 2 2
En 
n ,  ( x) 
2mL
2
nx
sin
L
L
• The solution to Schrodinger
equation for a particle
trapped in a box is just a
series of standing de
Broglie waves.
Barrier penetration (tunneling)
• In quantum mechanics, particles can sometimes be
found in or pass through regions (E<V) that are
forbidden by energy conservation. This is called
barrier penetration.
• Wave function in this barrier potential takes the
form:
– Sinusoidal oscillation in the region x<0;
– Exponentials in the region 0<=x<=a,
– Sinusoidal oscillation in the region x>a;
V
Pe
a

2
2 m (V  E )a

Scanning tunneling microscope
• It is a remarkable application of tunneling.
• The instrument mechanically tracks the electronic
wave functions outside the surface of a material,
producing an extremely accurate representation of
the atoms at the surface.
• When a thin metal probe is brought very close to
the surface of the material. The gap between two
surfaces forms the barrier. The electrons from the
metal tunnel through the barrier into the probe and
creates a current.
• The probe moves horizontally along the
surface to maintain the same current. The
map of vertical positions of the probe forms
the surface structure.
material
E
probe
Hydrogen atom (I)
• In a hydrogen atom, the electron’s motion is
restricted by the inverse square electric field
of nucleus. The electron is free to move in
three dimensions.
• Since the electrical potential is a function of r
rather than of x,y,z, Schrődinger’s equation is
expressed in terms of spherical polar
coordinates r,θ,φ.
– r: the length of radius vector from origin O to point P;
– θ: zenith angle, angle between radius vector and +z axis,
– φ: azimuth angle, angle between the projection of the radius
vector in xy plane and the +x axis,
Hydrogen atom (II)
• The wave function can be divided into three
different functions which may be governed by
three independent equations.
  R(r )( ) ( )
• There are three quantum numbers:
– Magnetic quantum number m: the direction of the angular
momentum,
Lz  m
– Orbital quantum number l: the magnitude of the electron’s
angular momentum, L  l (l  1)
– Principle quantum number n: the total energy of the
E1
electron,
En 
n2
How atomic states are denoted?
• It is customary to specify electron angular
momentum states by a letter.
l=
0
1
2
3
4
5
6
s
p
d
f
g
h
i
….
• Sharp, principal, diffuse, fundamental,…
• Atomic states are denoted by a combination
of the principal quantum number with the
letter representing orbital angular
momentum, such as 2s (n=2, l=0),
4d(n=4,l=2).
Electron probability density
• The probability of finding the electron in a
hydrogen atom at a distance between r and
r+dr from the nucleus is:
P(r )dr  r 2 | R |2 dr
– For different states, P(r) changes with r differently.
• Zenithal probability density has angular
preference except an s state (l=0, m=0).
Azimuthal probability density is a constant.
• Thus, the electron’s probability density is
symmetrical about the z axis. It can be called
electron cloud.
Lasers
• Laser is an acronym for light amplification by
simulated emission of radiation.
• Three ways of radiation interacting with atomic
energies:
– Spontaneous emission, high E to lower E;
– Induced absorption, lower E to high E,
– Induced (stimulated) emission: A passing photon
with the right energy induces the atom to emit a
photon and makes a transition to the lower state,
producing two photons in phase.
• Population inversion is essential for a laser, that
is, a higher state has a greater population than a
lower state.
He-Ne laser
• A mixture of He and Ne is in the ratio of 5:1
to 10:1.
• The purpose of the He atoms is to help
achieve a population inversion.
– An electrical current excites He to 20.61eV,
– Excited He atoms colliding with Ne atoms to
make them to reach 20.66eV, a metastable state,
• The laser transition in Ne from the
metastable state at 20.66eV to an excited
state at 18.70eV, with the emission of a
632.8nm photon.
Usage of lasers
• There are many different types of lasers, such
as the ruby laser, the carbon dioxide laser etc.
• As lasers produce coherent monochromatic
light that can be confined to an intense
narrow beam.
• They are used to replay music in CD players,
to weld parts, to measure distance accurately
and to study molecular structure. It is also
used in medical treatment and surgery.