Download Chapter 9a Introduction to Quantum Mechanics

Chapter 9
Introduction to Quantum
Mechanics
(May. 20, 2005)
Conduction (传导), convection (对流),
radiation (辐射)
Hypothesis (假设), postulate （假定，基本原
理）,
Blackbody radiation，Photoelectric effect (光
电效应)
Thermotics (热学) , thermodynamics (热力学)
ultraviolet Catastrophe (disaster) (紫外灾难)
Reinforce (加强) irradiate， shine
9.1
Blackbody
radiation
and
Planck hypothesis
• Two patches of clouds in physics sky at
the beginning of 20th century.
• The speed of light  Relativity
• The blackbody radiation  foundation of
Quantum theory
9.1.1 Blackbody radiation and
Planck’s hypothesis (假设).
Types of heat energy transmission are
conduction, convection and radiation.
•
• Conduction is transfer of heat energy by
molecular vibrations not by actual motion of
material. For example, if you hold one end of
an iron rod (铁杆) and the other end of the
rod is put on a flame, you will feel hot some
time later. You can say that the heat energy
reaches your hand by heat conduction.
• Convection is transfer of heat by actual motion
of. The hot-air furnace, the hot-water heating
system, and the flow of blood in the body are
examples.
• Radiation The heat reaching the earth from
the sun cannot be transferred either by
conduction or convection since the space
between the earth and the sun has no material
medium.
The
energy
is
carried
by
electromagnetic waves that do not require a
material medium for propagation. The kind of
heat transfer is called thermal radiation.
• Blackbody is defined as the body which
can absorb all energies that fall on it. It is
something like a black hole. No lights or
material can get away from it as long as it is
trapped. A large cavity with a small hole on
its wall can be taken as a blackbody.
•Blackbody radiation: Any radiation
that enters the hole is absorbed in the
interior of the cavity, and the radiation
emitted from the hole is called
blackbody radiation.
Fig. 9.1
Blackbody
concave.
1. Stefan and Boltzmann’s law: it is
found that the radiation energy is
proportional to the fourth power of the
associated temperature.
M (T )  T
4
M(T) is actually the area
under each curve, σ is called
Stefan’s constant and T is
absolute temperature.
Fig. 9.2 the blackbody
radiation of spectra for four
different temperatures.
2. Wien’s displacement law: the peak of the
curve shifts towards longer wavelength as
the temperature falls and it satisfies
 peakT  b
where b is called the Wein’s constant.
This law is quite useful for measuring the
temperature of a blackbody with a very high
temperature. You can see the example for
how to measure the temperature on the
surface of the sun.
• The above laws describes the blackbody
radiation very well.
• The problem exists in the relation between the
radiation power Mλ(T) and the wavelength λ.
• Blackbody radiation has nothing to do with
both the material used in the blackbody
concave wall and the shape of the concave wall.
• Two typical theoretical formulas for
blackbody radiation : One is given by Rayleigh
and Jeans and the other by Wein.
• Rayleigh and Jeans
In 1890, Rayleigh and Jeans obtained a
formula using the classical electromagnetic
(Maxwell)
theory
and
the
classical
equipartition theorem of energy (能均分定理)
in thermotics (热学). The formula is given by
M  (T )  C1 T
4
Rayleigh-Jeans formula was correct for
very long wavelength in the far infrared but
hopelessly wrong in the visible light and
ultraviolet
region.
Maxwell’s
electromagnetic
theory
and
thermodynamics are known as
correct theory. The failure in
Rayleigh-Jeans’ line
explaining
blackbody
Wein’s line
radiation puzzled physicists! It
was regarded as ultraviolet
Catastrophe (disaster).
Fig. 9.3 Blackbody radiation
• Wein’s formula:
Later on in 1896, Wein derived another
important formula using thermodynamics.
5  T3
M  (T )  C2  e
C
Unfortunately, this formula is only valid in
the region of short wavelengths.
M(T)
Rayleigh-Jeans’ line
Wein’s line
1
2
3
( m)
T=1500K
• Planck’s Magic formula
In 1900, after studying the above two formulas
carefully, Planck proposed ( 提 出 ) an empirical
formula.
M  (T )  2hc 
1
2 5
e
hc
kT
1
Where c is the speed of light, k is Boltzmann’s
constant, h is Planck’s constant and e is the base
of natural logarithm.
It is surprising that the experience formula can
describe the curve of blackbody radiation
exactly for all wavelengths.
• Other unbelievable deductions:
(1) For very large wavelength, the RayleighJeans formula can be obtained from Planck’s
formula;
hc
 1
kT
hc
 1
kT
2
hc 1  hc 
e  1
 
 
kT 2  kT 
Drop the second order and higher order
terms, and RJ formula could be obtained.
hc
kT
(2) For smaller wavelength of blackbody
radiation, the Wein’s formula can be
achieved also from Planck’s experience
formula;
1
e
hc
kT
e
hc

kT
1
Then Wein’s formula could be obtained.
(3) Integrating Planck’s formula with
respect to wavelength, the Stefan and
Boltzmann’s law can be obtained as well.

M (T )   M  (T )d  T
0
 Is called Stefen constant.
4
(4) Finally, according to the basic mathematical
theory and differentiating the Planck’s formula
with respect to wavelength, Wien’s displacement
law can also be derived!
dM  (T )
0
d
 max T  b
Planck’s empirical formula matched all the
different classical physics results obtained by
the
Maxwell
electromagnetic
theory,
thermodynamics and statistics! However, no
one knew why at that time. This phenomenon
seemed unbelievable, incredible and even
impossible, but is true!
In order to derive this formula theoretically,
Planck proposed a brave hypothesis which is
also incredible.
Planck’s Hypotheses:
• The molecules and atoms composing the
blackbody concave can be regarded as the
linear harmonic oscillator with electrical
charge;
• The oscillators can only be in a special
energy state. All these energies must be the
integer multiples of a smallest energy (ε0 =
hν). Therefore the energies of the
oscillators are E = n hν with n = 1, 2, 3, …
Using the hypothesis and classical physics,
Planck arrived at his experience formula in
two months later. The correct result shows
that Planck’s hypothesis is correct!
Quantum theory and modern physics was
founded by these hypotheses!
• Planck-Einstein Energy Quantization Law:
E  h 
Quantum energy
hc

Frequency
h  6.626 10
Planck constant
1eV  1.602 10
19
15
J s
J  4.136 10 eV  s
1J  6.242 10 eV
18
34
It is pity that Planck himself did not believe
his such a wonderful hypothesis and he spent
about ten years to solve the same problem
using classical physics, but obviously in vain
(徒劳的). Planck’s case is similar to Newton’s
experience as he spent the rest of his life to
show the existence of God.
Example: Calculate the photon energies for
the following types of electromagnetic
radiation: (a) a 600kHz radio wave; (b) the
500nm (wavelength of) green light; (c) a 0.1
nm (wavelength of) X-rays.
Solution: (a) for the radio wave, we can use the
Planck-Einstein law directly
E  h  4.136 10
9
 2.48 10 eV
15
eV  s  600 10 Hz
3
(b) The light wave is specified by wavelength,
we can use the law explained in wavelength:
6
1.24110 eV  m
E

 2.26eV
9

550 10 m
hc
(c). For X-rays, we have
1.24110 6 eV  m
4
E


1
.
24

10
eV  12.4keV
9

0.110 m
hc
Therefore you can see that the higher frequency
corresponds to the higher energy. The X-rays have
quite high energy, so they have high power of
penetration.
Here we emphasize that the particle properties of light
and the photon will be defined. As we know the light is
electromagnetic waves and it has the properties of
waves.
Planck associated the energy quanta only with the
light emission in the cavity walls and Einstein extended
them to the absorption of radiation in his explanation
of the photoelectric effect.
9.2 Photoelectric Effect
The quantum nature of light had its origin
in the theory of thermal radiation and was
strongly reinforced by the discovery of the
photoelectric effect.
9.2.1 Photoelectric Effect
In figure 9.4, a glass tube
contains two electrodes (电极) of
the same material, one of which
is irradiated (被照射) by light.
The electrodes are connected to
a battery and a sensitive current
detector measures the current
flow between them.
Fig. 9.4 Apparatus
to investigate the
photoelectric
effect that was
first found in 1887
by Hertz.
The current flow is a direct measure of the rate of
emission of electrons from the irradiated electrode.
The electrons in the electrodes can be ejected
by light and have a certain amount of
kinetic energy.Now we change:
(1) the frequency and intensity of light,
(2) the electromotive force (e.m.f. 电动势 or
voltage),
(3) the nature of electrode surface.
It is found that:
(1). For a given electrode material, no
photoemission exists at all below a certain
frequency of the incident light. When the
frequency increases, the emission begins at a
certain frequency. The frequency is called
threshold frequency of the material. The
threshold frequency has to be measured in the
existence of e.m.f. (electromotive force) as at
such a case the photoelectrons have no kinetic
energy to move from the cathode (阴极) to
anode (阳极). Different electrode material has
different threshold frequency.
(2). The rate of electron emission is directly
proportional to the intensity of the incident light.
Photoelectric current ∝ The intensity of light
(3). Increasing the intensity of the incident light does
not increase the kinetic energy of the photoelectrons.
Intensity of light ∝ kinetic energy of
photoelectron
However increasing the frequency of light does
increase the kinetic energy of photoelectrons even for
very low intensity levels.
Frequency of light ∝ kinetic energy of
photoelectron
(4). There is no measurable time delay between
irradiating the electrode and the emission of
photoelectrons, even when the light is of
very low intensity. As soon as the electrode is
irradiated, photoelectrons are ejected.
(5) The photoelectric current is deeply affected
by the nature of the electrodes and chemical
contamination of their surface.
It is found that the second and the fifth
conclusions can be explained easily by
classical theory of physics —
Maxwell’s electromagnetic theory of
light, but the other three cases conflict
with any reasonable interpretation of
the classical theory of physics.
Let’s have a look at the three trouble cases:
(a) The existence of a threshold frequency:
Classical theory cannot explain the
phenomenon as the light energy does not
depend on the frequency of light. Light
energies should depend on its intensity
and the irradiating time.
(b) In third case, it is unbelievable that the
kinetic energies of photoelectrons do not
depend on the intensity of incident light as
the intensity indicates the light energy!
(c). No time delay in the photoelectric effect
Since the rate of energy supply to the electrode
surface is proportional to the intensity of the
light, we would expect to find a time delay in
photoelectron emission for a very low intensity
light beam. The delay would allow the light to
deliver adequate energy to the electrode surface
to cause the emission. Therefore “the no time
delay phenomenon” puzzled us.
In 1905, Einstein solved the photoelectric
effect problem by applying the Planck’s
hypothesis. He pointed out that Planck’s
quantization hypothesis applied not only to
the emission of radiation by a material object
but also to its transmission and its absorption
by another material object. The light is not
only electromagnetic waves but also a
quantum. All the effects of photoelectric
emission can be readily explained from the
following assumptions:
(1) The photoemission of an electron from a
cathode occurs when an electron absorbs a
photon of the incident light;
(2) The photon energy is calculated by the
Planck’s quantum relationship: E = hν.
(3) The minimum energy is required to release
an electron from the surface of the cathode.
The minimum energy is the characteristic
of the cathode material and the nature of its
surface. It is called work function.
The equation for the photoelectric emission
can be written out by supposing the photon
energy is completely absorbed by the electron.
After this absorption, the kinetic energy of the
electron should have the energy of the photon.
If this energy is greater than the work function
of the material, the electron should become a
photoelectron and jumps out of the material
and probably have some kinetic energy.
Therefore we have the equation of photoelectric
effect:
1 2
h  A  mv
2
Photon energy
Photoelectron kinetic energy
Work function
Using this equation and Einstein’s assumption, you could readily
explain all the results in the photoelectric effect: why does
threshold frequency exist (problem)? why is the number of
photoelectrons proportional to the light intensity? why does high
intensity not mean high photoelectron energy (problem)? why is
there no time delay (problem)?
Example: Ultraviolet light of wavelength 150nm falls on
a chromium electrode. Calculate the maximum kinetic
energy and the corresponding velocity of the
photoelectrons (the work function of chromium is
4.37eV).
Solution: using the equation of the photoelectric effect, it
is convenient to express the energy in electron volts. The
photon energy is
1.241106 eV  m
E  h 

 8.27eV
9

150 10 m
and
1 2
h  A  mv
2
1 2
 mv  (8.27  4.37)eV  3.90eV
2
hc
1eV  1.602 1019 J  1.602 1019 N  m  1.602 1019 kg  m2  s 2
∴
1 2
19
2
2
mv  3.90eV  3.90 1.602 10 kg  m  s
2
∴
2  3.90eV
12.496 1019
6
v


1
.
17

10
m/ s
31
m
9.1110
9.2.2 The mass and momentum of
photon
According to relativity, the particles with zero
static mass are possibly existent. From the
relativistic equation of energy-momentum,
E  p c m c
2
when m0 = 0,
2 2
2 4
0
E = pc = mc2 = hν
∴ the mass of photon is
E h
mp  2  2
c
c
and the momentum of photon should be
E h h
p 

c
c

9.3 Compton effect
A phenomenon called Compton scattering, first observed in
1924 by Compton, provides additional direct confirmation
of the quantum nature of electromagnetic radiation. When
X-rays impinges ( 冲 击 , 撞 击 ) on matter, some of the
radiation is scattered (散射), just as the visible light falling
on a rough surface undergoes diffuse reflection (漫射).
Observation shows that some of the scattered radiation has
smaller frequency and longer wavelength than the incident
radiation, and that the change in wavelength depends on
the angle through which the radiation is scattered.
Specifically, if the scattered radiation emerges at an angle φ
with the respect to the incident direction, and if  and  are
the wavelength of the incident and scattered radiation,
respectively, it is found that
h
h
2 
1  cos    2 sin  
   
mc
mc
2
where m is the electron mass.
, p
, p
φ
In figure 8.4, the electron is
initially at rest with incident
photon of wavelength  and
momentum p; scattered
P
photon with longer
wavelength  and
Fig. 8.4 Schematic diagram of Compton
momentum p and recoiling
scattering.
(反冲, 后退) electron with
momentum P. The direction of the scattered photon makes an
angle φ with that of the incident photon, and the angle between p
and p is also φ.
h
c 
 0.00243nm called Compton wavelength.
·
·
mc
Compton scattering cannot be understood on the basis classical
electromagnetic theory. On the basis of classical principles, the
scattering mechanism is induced by motion of electrons in the
material, caused by the incident radiation. This motion must have
the same frequency as that of incident wave because of forced
vibration, and so the scattered wave radiated by the oscillating
charges should have the same frequency. There is no way the
frequency can shifted by this mechanism.
The quantum theory, by contrast, provides a beautifully simple
explanation. We imagine the scattering process as a collision of
two particles, the incident photon and an electron at rest as
shown in Fig. 8.4. The photon gives up some of its energy and
momentum to the electron, which recoils as a result of this impact;
and the final photon has less energy, smaller frequency and
longer wavelength than the initial one. The equation can be
derived as in our Chinese text book.
9.4 The wave-particle duality of light
The concept that waves carrying energy may
have a corpuscular (particle) aspect and that
particles may have a wave aspect; which of the
two models is the more appropriate will
depend
on the properties the model is seeking to explain. For example,
waves of electromagnetic radiation need to be visualized as
particles, called photons to explain the photoelectric effect. Now
you may confuse the two properties of light and ask what the
light actually is?
The fact is that the light shows the property of waves in its
interference and diffraction and performances the particle
P
property in blackbody
radiation and

photoelectric effect.
O
S0
d
Till now we can only
 =d sin = x/L
say that the light has

duality property.
We can say that light is wave when it is involved in its propagation
only like interference and diffraction. This means that light
interacts with itself. The light shows photon property when it
interact with other materials. Let’s see another example of photon.
9.5 Line spectra and Energy
quantization in atoms
The quantum hypothesis, used in the preceding section for the
analysis of the photoelectric effect, also plays an important role
in the understanding of atomic spectra.
9.4.1 Line spectra of Hydrogen atoms
It is found that hydrogen always gives a set of line spectra in the
save position, sodium another set, iron still another and so on
the line structure of the spectrum extends into both the
ultraviolet and infrared regions. It is impossible to explain such
a line spectrum phenomenon without using quantum theory. For
many years, unsuccessful attempts were made to correlate the
observed frequencies with those of a fundamental and its
overtones (denoting other lines here). Finally, in 1885, Balmer
found a simple formula which gave the frequencies of a group
lines emitted by atomic hydrogen. Since the spectrum of this
element is relatively simple, and fairly typical of a number of
others, we shall consider it in more detail.
Under the proper conditions of excitation, atomic hydrogen may
be made to emit the sequence of lines illustrated in Fig. 8.5. This
sequence is called series.
434.1
364.6
There is evidently a certain
656.3
486.1
410.2
(nm)
order in this spectrum, the
lines becoming crowded more
and more closely together as
H
H H H H
the limit of the series is
Fig. 8.5 the Balmer series of atomic hydrogen.
approached. The line of
longest wavelength or lowest frequency, in the red, is known as
H, the next, in the blue-green, as H, the third as H, and so on.
Balmer found that the wavelength of these lines were given
accurately by the simple formula
 1 1 
 R 2  2 ,

2 n 
1
(9.4.1)
where  is the wavelength, R is a constant called the Rydberg
constant, and n may have the integral values 3, 4, 5, etc.* if  is
in meters,
R  1.097 107 m 1
(9.4.2)
Substituting R and n = 3 into the above formula, one obtains the
wavelength of the H-line:
  656.3nm
For n = 4, one obtains the wavelength of the H-line, etc. for n= ,
one obtains the limit of the series, at  = 364.6nm –shortest
wavelength in the series.
Other series spectra for hydrogen have since been discovered.
These are known, after their discoveries, as Lymann, Paschen,
Brackett and Pfund series. The formulas for these are
Lymann series:
1
1 1 
 R 2  2 ,

1 n 
n  2,3,4, 
Paschen series:
1 1 
 R 2  2 ,

3 n 
n  4,5,6, 
1
(9.4.3)
1
Brackett series:
 1 1 
 R 2  2 ,

4 n 
n  4,5,6, 
Pfund series:
1 1 
 R 2  2 ,

5 n 
n  4,5,6, 
1
The Lymann series is in the ultraviolet, and the Paschen,
Brackett, and Pfund series are in the infrared. All these formulas
can be generalized into one formula which is called the general
Balmer series.
1 
 1
 R 2  2 ,

n 
k
1
n  k  1, k  2, k  3, 
(9.4.4)
All the spectra of atomic hydrogen can be described by this
simple formula. As no one can explain this formula, it was ever
called Balmer formula puzzle.
9.4.2 Bohr’s atomic theory
Bohr’s was not by any means the first attempt to understand the
internal structure of atoms. Starting in 1906, Rutherford and his
co-workers had performed experiments on the scattering of alpha
Particles by thin metallic. These experiments showed that each
atom contains a massive nucleus whose size is much smaller than
overall size of the atom. The nucleus is surrounded by a swarm (一
大群) of electrons. To account for the fact, Rutherford postulated
that the electrons revolve about the nucleus in orbits, more or less
as the planets in the solar system revolve around the sun, but with
electrical attraction providing the necessary centripetal force.
This assumption, however, has an unfortunate consequence . A
body moving in a circle is continuously accelerated toward the
center of the circle and, according to classical electromagnetic
theory, an accelerated electron radiates energy. The total energy of
the electrons would therefore gradually decrease, their orbits
would become smaller and smaller, and eventually they would
spiral into the nucleus and come to rest. Furthermore, according to
classical theory, the frequency of the electromagnetic waves
emitted by a revolving electron is equal to the frequency of
revolution. Their angular velocities would change continuously
and they would emit a continuous spectrum (a mixture of
frequencies), in contradiction to the line spectrum actually
observed.
Faced with the dilemma, Bohr concluded that , in spite of the
success of electromagnetic theory in explaining large scale
phenomenon, it could not be applied to the processes on an atomic
scale. He therefore postulated that an electron in an atom can
revolve in certain stable orbits, each having a definite associated
energy, without emitting radiation. The momentum mvr of the
electron on the stable orbits is supposed to be equal to the integer
multiple of h/2π. This condition may be stated as
h
(9.4.5)
mvr  n
n  1,2,3,
2
where n is quantum number, this is the hypothesis of stable state
And it is called the quantization condition of orbital angular
momentum.
For the transition hypothesis, Bohr postulated that the radiation
happens only at the transition of electron from one stable state to
another stable state. The radiation frequency or the energy of the
photon is equal to the difference of the energies corresponding to
the two stable states.
h  En  Ek
En  Ek

h
(9.4.6)
Another equation can be obtained by the electrostatic force of
attraction between two charges and Newton’s law:
1 e2
v2
m
2
40 r
r
(9.4.7)
Solving the simultaneous equation of (8.4.5) and (8.4.7), we have
 0h2 2
rn 
n ,
2
me
e2
vn 
2 0 hn
(n  1,2,3,)
So the total energy of the electron on the nth orbit is
1 2
 e2
me4
En  Ek  E p  mvn 

2 2
2
40 rn
8 0 h n
(9.4.8)
It is easy to see that all the energy in atoms should be discrete
not continuous. When the electron transits from nth orbit to
kth orbit, the frequency and wavelength can be calculated as
E n  Ek
me4  1
1

 2 3 2  2
h
8 0 h  k
n 

nk
me4  1 1 
 1 1 
  2 3  2  2   R 2  2 
 c 8 0 h c  k n 
n 
k
1
(9.4.9)
where
me4
R 2 3
8 0 h c
is Rydberg constant. It is found that
the value of R is matched with experimental data very well.
Till then, the 30-years puzzle of line spectra of atoms was
solved by Bohr since equation (8.4.9) is exactly the general
Balmer formula.
When Bohr’s theory met problems in explaining a little bit
more complex atoms (He) or molecules (H2), Bohr realized
that his theory is full of contradictions as he used both quantum
and classical theories. The problem was solved completely
after De Broglie proposed that electron also had the waveparticle duality. Since then, the proper theory describing the
motion of the micro-particles , quantum mechanics, have been
gradually established by many scientists.
9.6 De Broglie Wave
9.6.1 De Broglie
In the previous sections we traced the development of the
quantum character of electromagnetic waves. Now we will
turn to the consequences of the discovery that particles of
classical physics also possess a wave nature. The first person
to propose this idea was the French scientist Louis De
Broglie.
De Broglie’s result came from the study of relativity. He
noted that the formula for the photon momentum can also be
written in terms of wavelength
h
h h
P  mpc  2 c 

c
c

(9.5.1)
If the relationship is true for massive particles as well as for
photons, the view of matter and light would be much more
unified.
In certain circumstances each could behave as a wave, and in
other instances each could behave as a particles.
De Broglie’s point was the assumption that momentumwavelength relation is true for both photons and massive
particles.
So De Broglie wave equations are
P
h

h
h
 
p mv
E  h
(9.5.2)
Where P is the momentum of particles, λis the wavelength of
particles. At first sight, to claim that a particle such as an electron
has a wavelength seems somewhat absurd. The classical concept
of an electron is a point particle of definite mass and charge, but
De Broglie argued that the wavelength of the wave associated with
an electron might be so small that it had not been previously
noticed. If we wish to prove that an electron has a wave nature, we
must perform an experiment in which electrons behave as waves.
Electron diffractions
In order to show the wave nature of electrons, we must
demonstrate interference and diffraction for beams of electrons. At
this point, recall that interference and diffraction of light become
noticeable when light travels through slits whose width and
separation are comparable with the wavelength of the light. So let
us first look at an example to determine the magnitude of the
expected wavelength for some representative objects. Therefore,
we use de Broglie wavelength for an electron whose kinetic energy
is 600 eV is 0.0501nm. The de Broglie wavelength for a golf ball
of mass 45g traveling at 40m/s is: 3.68 ×10-34 m. Such a short
wave is hardly observed.
We recall that diffraction of light waves is obvious when the slit
width is about the same as or smaller than the wavelength. We
must mow consider whether we could observe diffraction of
electrons whose wavelength is a small fraction of a nanometer.
For a grating to show observable diffraction, the slit separation
should be comparable to the wavelength, but we cannot rule a
series of lines that are only a small fraction of a nanometer apart,
as such a length is less than the separation of the atoms in solid
materials.
When electrons pass through a thin gold or other metal foils (箔),
we can get diffraction patterns. So it indicates the wave nature of
electrons. Look at the pictures on page 265 and 231 in your
Chinese and English text book respectively.
Introduce: Electron single and double slits experiments;
electronic microscope, nuclear reactor etc. (see Ch. book)
9.7 The Heisenberg Uncertainty
principle
Heisenberg proposed a principle that has come to be regarded as
basic to the theory of quantum mechanics. It is called
"uncertainty principle", and it limits the extent to which we can
possess accurate knowledge about certain pairs of dynamical
variables.
Both momentum and position are vectors. When dealing with a
real three dimensional situation, we take the uncertainties of the
components of each vector in the same direction.
Our sample calculation is restricted to the simplest interpretation
of what we mean by uncertainty. A more elaborate (详细阐述的)
statistical interpretation gives the lower
limit of the uncertainty product as:
h 
h
p x  p  sin   

 x x
Consider other order
diffractions, we have:
x  p x  h
But precise derivation is
h
x  p x 
4
p
θ
Δx
Δpx
We can briefly review how quantum dynamics differs from
classical dynamics. Classically, both the momentum and position
of a point particle can determined to whatever degree of
accuracy that the measuring apparatus permits. However, from
the quantum viewpoint, the product of the momentum and
position uncertainties must be at least as great as h/4/π.
See the examples in our Ch. Text book on page 266.
Another pair of important uncertainty is the time and energy. It is
found by E = p^2/2m + Ep that
E  x  h
From these examples and Heisenberg principle, we know that
classical theory is still useful in the macro-cases. However you
have to use quantum theory in the micro-world.