Matter Waves
1,History of Quantum Mechanics
The concept of the wave nature of atomic
particles,This is the foundation of the
mathematical discipline,(1) From wave
mechanics we can understand and predict the
properties of molecules as individual entities
(the so-called microscopic state) ;(2) The
properties of molecules (the macroscopic
state) can be obtained by applying statistical
techniques to these microscopic results.
The birth of wave mechanics,(1) 1924,de
Broglie postulated that material particles
would show wave-like characteristics;(2)1926,
Schr?dinger introduced an equation to define
these characteristics,(3)1900,First proposed
the conception of ‘quanta’.
Quantum development,(1) The ‘old’
quantum mechanics,Associate with the Bohr
model of the atom; (2) The ‘new’ quantum
mechanics,Associate mainly with the work
of Heisenberg,(What is the aim of quantum
mechanics proposed?)
de Broglie postulation:1927,Verified by
Davisson and Germer,They showed that
mono-energetic electrons scattered from
crystalline nickel foil gave a diffraction
pattern analogous to that showed by X-Rays,
Similar experiments were carried out
independently by G,P,Thomson,and later
Stern showed that beams of heavier particles
(H,He,etc.) showed diffraction patterns
when reflected from the surfaces of crystals,
de Broglie’s expression for the wavelength of
these matter waves is of high accuracy,
2,The wave-particle duality of light
(1) Particle Character:The fact that light
travels in straight lines,is reflected and
refracted and has the ability to impart
momentum to anything it strikes,-- Particulate
model.
(2) Wave Character,The phenomena of
diffraction and interference.--Wave model.
When quantum theory was proposed the wave
model was dominant.
The electromagnetic spectrum:Fig.1
(The visible light is all band of electromagnetic)
(The speed of light in vacuum is dependence
with frequency and wavelength?)
The photoelectric effect:Light is able to cause
electrons to be ejected from the surface of
metals,
1902:Lenard’s study,Investigation of the
relationship between the frequency and the
intensity of light on the one hand and the
number and kinetic energy of ejected
electrons on the other,Fig.2 shows the
relationship between the frequency and the
kinetic energy per electron,
1905,Einstein extended Plank’s Hypothesis
that atomic oscillators could not only take up
or give out energy in discrete quanta,by
regarding the radiation itself as consisting of
indivisible quanta or photons,
E
n
e
r
g
y
F r e q u e n c y ( ? )
K
i
n
e
t
i
c
E
n
e
r
g
y
p
e
r
e
l
e
c
t
r
o
n
F i g, 2 T h e r e l a t i o n s h i p b e t w e e n
t h e k i n e t i c e n e r g y o f t h e e l e c t r o n s
e m i t t e d f r o m a m e t a l s u r f a c e a n d
f r e q u e n c y o f t h e l i g h t i n c i d e n t u p o n i t
Plank-Einstein relationship
E = h? (1)
The proportionality constant h relating the
energy of a photon to its frequency turned out
to be the one introduced by Plank in his
theory,So-called as Plank constant,The
particulate interpretation of the photoelectric
effect is straightforward,Each photon
absorbed by a metal can lead to emission of
one electron providing that the energy of the
photon,when transferred to the electron,is
sufficient to enable the electron to escape
from the surface of the metal,Increasing the
intensity of the light increases the number of
electrons but not their energy and so will lead
to an increase in the number of electrons
escaping but not to an increase in their energy,
The kinetic energy of the ejected electron is
mv2/2,where m is the mass of the electron and
v is the velocity,we may write an equation for
the energy balance in the experiment:
h? = A + mv2/2 (2) where A is an energy
characteristic of the metal surface,
This interpretation of the photoelectric effect
restored the balance between the wave and
particle models for light,and the position
adopted today is that light has a wave-
particle duality.
An important aspect of the treatment of the
Compton effect is the conservation of the
momentum of the colliding particles,But
how can a photon,which has no mass,have
momentun? Similarly,in writing the energy
of the photon in equation (2) as h? we
avoided any discussion of the form of this
energy,If a photon has momentum can it
not also have kinetic energy? The fact that
the photon has momentum but no mass can
be understood within the framework of
relativity,In general,for a particle of rest
mass m0,the momentum is
Therefor,for the photon with zero rest mass
When de Broglie postulated that material
particles would show wave-like
characteristics,expression (4),which relates
the momentum to the wavelength,was taken
to apply not only to photon but to matter
waves also,However,the matter has mass
and the photon does not,This is the essential
difference of the matter and the light.
3,The Schr?dinger equation① The most important of all is some of same
characteristics between the seeking wave
equation and more familiar wave equations,
② The wave equation will be time-
independent,③ All of wave equation which
would have included in a classical treatment
– the kinetic energy of the particles,the
repulsions between particles of like electrical
charge and the attraction between those of
opposite charge,④ de Broglie relationship (4)
will be in some way involved.
(1) The wave motion along only one
coordinate axis,(2) The wave will be a
stationary one,that is,the given wave motion
the nodes will not move with time,A suitable
expression is
Where x is the coordinate axis along which
the wave motion occurs with wavelength ?,
and A represents the maximum amplitude of
vibration,The relative amplitude of the wave
at any point along x is represented by the
function y(x),and as the maximum value of
sin(2?x/?) is unity,y(x) called wavefunction.
We derive a differential equation for y by
twice differentiating each side of (5) with
respect to x,obtaining
This equation has an infinite number of
solutions like (5),Firstly,equation (7) in no
way place any limitations on A so that an
infinite number of acceptable values exist,
Secondly,the equation does not limit ? in
any way,Thirdly,we note that a solution of
equation (7) more general than (5) is
where ? is a phase angle which can have any
value,However,we can remove this freedom
by the specification that y(0) = 0 so that
It follows that we may set ? equal to 0 as in
the function (5),
We can restrict the infinite number of value
of ? which are acceptable as solutions in a
similar manner by specifying the relative
amplitude at some other point along the x
axis,The simplest way of doing this to
require there to be a node at another specified
value of x,say at l,so that y(l) = 0.
A general solution of
Where n is an integer,It follows that
We refer to conditions which constrain the
form of the wave function as boundary
conditions,
We have gone into some detail over the
from of equation (7), The derivation is a
simple one,we merely replace the
wavelength,?,by the momentum,p,in
equation (7) using the de Broglie relationship
(4),For a single particle moving in a one-
dimensional space (x) with momentum px,we
therefore have wave function [which is
traditionally represented by the Greek letter
psi(?)] which satisfies the equation
Equation (13) can not give directly the energy
levels of the system because it does not
contain any description of the forces acting on
the particle,Force is defined by the
derivative of the potential energy and hence
we need to introduce into (13) some function
of the potential energy,The momentum of
the particle is related to its kinetic energy (T)
by the expression
where m is the mass of the particle,The
kinetic energy is the difference between the
total energy (E) and the potential energy (V),
T = E – V (15)
hence the momentum in (13) can be replace
according to the expression
to give the equation
which on re-arrangement give
Equation (18) is the equation proposed by
Schr?dinger for a particle moving in one
dimension in a potential V,The extension to
three-dimension is made by letting ? be a
function of the three Cartesian coordinates
and writing the partial derivatives instead of
the complete derivatives in (17) as follows
Equation (19) can be written in a more
compact form as
and is called the Hamiltonian of the system,
4,The HamiltonianThe function which represents the total
energy of a system in classical mechanics
expressed in terms of the coordinates and
momenta of all the particles is called
Hamilton’s function,For a particle having
mass m and moving under the influence of a
potential V which is a function of the position
of the particle,Hamilton’s function is,
The first term being the kinetic energy
according to expression (14),We can set up a
correspondence between Hamilton’s
function (22) and the operator,defined as
the Hamiltonian in (22),if we make the
substitution
for a set of i particles of mass mi interacting
with a potential energy V which is a function
of the relative position of the particles,
Hamilton’s function will be
The total kinetic energy being a sum of the
kinetic energies of each particle,The
appropriates many-particle Hamiltonian is
obtained be making the substitution in
accord with (24)
The Schr?dinger equation for such a system
is then simply
The solutions,?,of such an equation are
called the eigenfunctions of H and the
associated energies,E,are the eigenvalues.
5,The physicl significance of
the wavefunction
Born suggested that ?2 is interpreted as a
probability distribution for the particle such
that the probability of finding the particle in
a small element of space dv is proportion to
?2dv,The proportionality constant can be
evaluated when we recognize that here is unit
probability of finding the particle somewhere
in space,That is,the integral of the
probability density over all space must be
unity
Expression (28) is called the normalization
condition for the wavefunction,It can be seen
equation (27) that the energy of the particle is
unaffected by this normalization,If ? is a
solution of (27) so too is k? where k is any
constant,If we have any solution of (27)
which is not normalized,it is easy to
normalize it by multiplying by the number N