Xem mẫu
- Atomic Physics
Pham Tan Thi, Ph.D.
Department of Biomedical Engineering
Faculty of Applied Sciences
Ho Chi Minh University of Technology
- History of Atomic Model
- History of Atomic Model
• Proposed an Atomic Theory which states that
all atoms are small, hard, indivisible and
indestructible particles made of a single
material formed into different shapes and
sizes.
• Aristotle did not support his atomic theory.
Democritus
(460 BC - 370 BC)
- History of Atomic Model
• Known as the “Father of Modern Chemistry”
• Was the first person to generate a list of 23
elements in his textbook
• Devised the metric systems
• Was married to a 13-year-old Marie-Anne
Pierette Paulze, who assisted him much of his
work
• Discovered/proposed that combustion occurs
when oxygen combines with other elements
Antoine Lavoisier
(1743 - 1794)
• Discovered/proposed the Law of
Conservation of Mass (or Matter) which states
that, in a chemical reaction, matter is neither
created or destroyed
- History of Atomic Model
• In 1803, he proposed an Atomic Theory which
states that:
• All substances are made of atoms; atoms
are small particles that cannot be created,
divided or destroyed.
• Atoms of the same element are exactly
alike, and atoms of different elements are
different
• Atoms join with other atoms to make new
John Dalton
substances
(1766 - 1844)
• He calculated the atomic weights of many
various elements
• He was a teacher at a very young age
• He was color blind
- History of Atomic Model
J. J. Thomson
(1856 - 1940)
• He proved that an atom can be divided into
smaller parts
• While experimenting with cathode-ray tubes,
discovered corpuscles, which were later called
electrons
• He stated that the atom is neutral
- History of Atomic Model
J. J. Thomson
(1856 - 1940)
• He proved that an atom can be divided into
smaller parts
• While experimenting with cathode-ray tubes,
discovered corpuscles, which were later called
electrons
• He stated that the atom is neutral
• In 1897, he proposed Plum
Pudding Model which states
that atoms mostly consist of
positively charged material
with negatively charged
particles (electrons) located
throughout the positive
material
• He won the Nobel Prize
- History of Atomic Model
Ernest Rutherford
(1871 - 1937)
• In 1909, he performed Gold Foil Experiment
and suggested the characteristics of the atom:
• It consists of a small core, or nucleus, that
contains most of the mass of the atom
• This nucleus is made up of particles called
protons, which have a positive charge
• The protons are surrounded by negative
charged electrons, but most of atom is
actually empty space
• He did extensive work on radioactivity (alpha,
beta particles, gamma rays/waves) and was
referred as the “Father of Nuclear Physics”
• He won the Nobel Prize
• He was a student of J.J. Thomson
- History of Atomic Model
Niels Bohr
(1885 - 1962)
• In 1913, he proposed the Bohr Model, which suggests
that electrons travel around the nucleus of an atom in
orbits for definite paths. Additionally, the electron can
jump from a path in one level to a path in another level
(depending on their energy)
• He won the Nobel Prize
• He used to work with Ernest Rutherford
- History of Atomic Model
Erwin Schrodinger
(1887 - 1961)
• In 1913, he further explained the nature of
electrons in an atom by stating that the exact
location of an electron cannot be stated;
therefore it is more accurate to view electrons
in region called electron clouds.
• Electron clouds are places where the electrons
are likely to be found
• He did extensive work on the Wave formula ➔
Schrodinger equation.
• He won the Nobel Prize
- History of Atomic Model
James Chadwick
(1891 - 1974)
• In 1932, he realized that the atomic mass of
most elements was double the number of
protons —> discovery of the neutron
• He used to work with Ernest Rutherford
• He won the Nobel Prize
- Quantum Mechanical Atomic Theory
- Schrödinger Equation in Three Dimensions
• Electrons in an atom can move in all three dimensions of space. If a
particle of mass m moves in the presence of a potential energy
function U(x,y,z), the Schrödinger equation for the particle’s wave
function ψ(x,y,z) is
✓ ◆
~ 2
@ 2
(x, y, z) @ 2 2
(x, y, z) @ (x, y, z)
2
+ 2
+
2m @x @y @z 2
+U (x, y, z) (x, y, z) = E (x, y, z)
~ 2
r + U (x, y, z) (x, y, z) = E (x, y, z)
2m
• This is a direct extension of the one-dimensional Schrödinger equation
~2 @ 2 (x)
2
+ U (x) (x) = E (x)
2m @x
- 41.1 A particle is confined in a cubical
41.2 Particle in a Three-Dimensional Box
Particle inConsider
a Three-Dimensional
box with walls at x = 0, x = L, y = 0,
y = L, z = 0, and z = L. a particle enclosed within a cubical box of sideBox
L. This could represent
an electron that’s free to move anywhere within the interior of a solid metal cube
z
but cannot escape the cube. We’ll choose the origin to be at one corner of the box,
z5L
with• the
In x-,analogy
y-, and z-axes with
along our infinite
edges of square
the box. Then potential
the particle is confined to
(U(x)
the region 0 …=x 0 , 0 … y … U(x)
… Linside, L, 0 … z=…∞ outside),
L (Fig. 41.1). Whatlet usstation-
are the
ary states of this system?
Asconsider
for the particleainthree-dimensional
a one-dimensional box that we region
considered space
in Section
40.2, (box)
we’ll say of
that equal sides
the potential ofzero
energy is inside theL,
length boxwith theoutside.
but infinite same
Hence the spatial wave function c1x, y, z2 must be zero outside the box in order
x5L potential
that the z2c1x, y, z2 in =
term U1x, y, (U(x,y,z) the0 inside, U(x,y,z)
time-independent Schrödinger=equation,
∞
x O Eq. (41.5), not be infinite. Hence the probability distribution function ƒ c1x, y, z2 ƒ 2
y5L
y
outside).
is zero outside the box, and there is zero probability that the particle will be found
• We will consider the wave function as separable, that is can be written
as a product of the three independent dimensions x, y and z:
(x, y, z) = X(x)Y (y)Z(z)
• The Schrödinger equation inside the box becomes
~2 @ 2 X(x) @ 2 Y (y) @ 2 Z(z)
Y (y)Z(z) + Z(z)X(x) + X(x)Y (y) = EX(x)Y (y)Z(z)
2m @x2 @y 2 @z 2
• OR dividing by X(x)Y(y)Z(z), we have:
~ 2
1 @ 2 X(x) 1 @ 2 Y (y) 1 @ 2 Z(z)
2
+ 2
+ 2
=E
2m X(x) @x Y (y) @y Z(z) @z
- Particle in a Three-Dimensional Box
• This says that the total energy is contributed by three terms on the
left, each depending separately on x, y and z. Let us write E = Ex +
Ey + Ez. Then this equation can be separated into three equations:
~2 @ 2 X(x) ~2 @ 2 Y (y) ~2 @ 2 Z(z)
= Ex X(x) 2
= Ey Y (y) = Ez Z(z)
2m @x2 2m @y 2m @z 2
• These obviously have the same solutions separately as our original
particle in an infinite square well, and corresponding energies
⇣ n ⇡x ⌘
x n2x ⇡ 2 h2
Xnx = Cx sin Ex = (nx = 1,2,3,…)
L 2mL2
⇣ n ⇡y ⌘
y n2y ⇡ 2 h2
Xny = Cy sin Ey = (ny = 1,2,3,…)
L 2mL 2
⇣ n ⇡z ⌘
z n2z ⇡ 2 h2
Xnz = Cz sin Ez = (nz = 1,2,3,…)
L 2mL2
- Particle in a Three-Dimensional Box
• A particle’s wave function is the product of these three solutions:
⇣ n ⇡x ⌘ ⇣ n ⇡y ⌘ ⇣ n ⇡z ⌘
x y z
(x, y, z) = X(x)Y (y)Z(z) = Csin sin sin
L L L
• We can use the three quantum numbers nx, ny and nz to label the
stationary states (state of definite energy). Here is an example of a
particle in three possible states (nx, ny, nz) = (2, 1, 1); (1, 2, 1) or (1, 1, 2)
• The three states shown here are degenerate: although they have
different values of nx, ny, nz, they have the same total energy E:
41.2 Particle in a Three-Dimensional Box 1369
2 2 2 2 2 2 2 2
41.2E
4⇡ h ⇡ h ⇡ h 3⇡ hThe value
2 = Ex to+the density
Probability
of ƒ c ƒ is proportional
Ey +
distribution function ƒ Ez =
c 1x,
nX,nY,nZ
of dots. The
y, z2 ƒ 2
for 1n , nX+
, nY2 equal
Z to (a)
wave function is2zero on the walls of2
(2,+1, 1), (b) (1, 2, 1),
2
the box and on the midplane
= of the box,2so
and (c) (1, 1, 2).
2
ƒ c ƒ = 0 at these locations. 2mL 2mL 2mL mL
(a) !c2, 1, 1!2 z (b) !c1, 2, 1!2 z (c) !c1, 1, 2!2 z
x x x
y y y
spots” where there is zero probability of finding the particle. As an example, con-
- Energy Degeneracy
• For a particle in a three-dimensional box, the allowed energy levels are
surprisingly complex. To find them, just count up the different possible
states.
41.2 Partic
• Here are the first six states for an equal-side box:
E 4
in
3⇡ 2 h2 (3, 2, 1), (3, 1, 2), (1, 3, 2),
6-fold degenerate 14
E1, 1, 1 la
E1,1,1 = (2, 3, 1), (1, 2, 3), (2, 1, 3) 3
o
2mL2 (2, 2, 2)
not degenerate
4E1, 1, 1
S
3-fold degenerate 11 th
(3, 1, 1), (1, 3, 1), (1, 1, 3)
3
E1, 1, 1 lo
3-fold degenerate 1
(2, 2, 1), (2, 1, 2), (1, 2, 2) 3E1, 1, 1 1
s
• If the length of sides of m
box are different: (2, 1, 1), (1, 2, 1), (1, 1, 2)
3-fold degenerate 2E1, 1, 1
!
2 2
nx ny n z ⇡ 2 h2
2
not degenerate
E= 2
+ 2 + 2 (1, 1, 1) E1, 1, 1
Lx Ly Lz 2m
(break the degeneracy) E50
Since degeneracy is a consequence of symmetry, we can remove the degener-
acy by making the box asymmetric. We do this by giving the three sides of the
- 41.5 The Schrödinger equation for the The hydro
Hydrogen Atom hydrogen atom can be solved most readily
using spherical coordinates.
1r, u, f2,
tion depen
• For hydrogen atom, the three-dimensional potential z
Electron, charge 2e,
potential-e
energy depends only on the electron’s distance from at coordinates (r, u, f) of familiar
important f
the proton: 1 e2 First, w
U (r) = that we em
4⇡"0 r
wave funct
• As before, we will seek separable wave functions, only one o
but this time, due to the spherical symmetry we Nucleus, r
charge 1e, u
will use spherical coordinates (r, θ, 𝜙) rather than at the origin That is, the
(x,y,z). Then: y depends on
substitute E
(r, ✓, ) = R(r)⇥(✓) ( ) f
nary differ
involves on
• Following the same procedure as before, we can x
separate the problem into three separate U2
-
equations • Energies are 2m rr
✓ ◆ ✓ ◆ 1 me4
En = 1
~2 d 2 dR(r) ~2 l(l + 1) (4⇡" ) 2 2n2 h2
– r + 2
+ U (r) R(r) = ER(r) 0 sin u d
2m dr dr 2m r
✓ ◆ ✓ ◆ 13.6 eV
1 d d⇥(✓) m2l En =
sin✓ + l(l + 1) 2 ⇥(✓) = E⇥(✓) n2
sin✓ d✓ d✓ sin ✓
d2 ( ) Quantum numbers: In Eqs. (41
2 that we’ll
2
+ ml ( )=E ( ) n, l, ml
d reduced ma
We won
- Hydrogen Atom: Quantum States
• Table below summarizes the quantum states of the hydrogen atom. For each value
of the quantum number n, there are n possible values of the quantum number l.
For each value of l, there are (2l + 1) values of the quantum number ml.
• Question: How many distinct states of the hydrogen atom (n, l, ml) are there for
n = 3 state? What are their energies?
- Hydrogen Atom: Quantum States
• Table below summarizes the quantum states of the hydrogen atom. For each value
of the quantum number n, there are n possible values of the quantum number l.
For each value of l, there are (2l + 1) values of the quantum number ml.
• Question: How many distinct states of the hydrogen atom (n, l, ml) are there for
n = 3 state? What are their energies?
The n = 3 state has possible l values 0,1 or 2. Each l value has ml possible values
of (0); (-1,0,1); or (-2,-1,0,1,2). The total number of states is then 1 + 3 + 5 = 9.
Each of these states have the same n, so they all have the same energy.
We will see later there is another quantum number s, for electron spin (±1/2), so
there actually 18 possible states for n = 3.
nguon tai.lieu . vn
