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Chapter 3
Introduction to Quantum Mechanics
Why Quantum Mechanics?
Classical Physics is unable to offer a satisfactory explanation of
phenomena in microworld, of the structure of even the simplest
atom, Hydrogen. It makes no sense on atomic phenomena:
potoelectronic effect, thermal radiation, optical spectra of atoms,
Compton scattering, etc.
Bohr-Sommerfeld theory can partially explain hydrogen atom,
but it is not satisfied in theory, does not work for atoms with two
or more electrons.
Solution: Quantum mechanics
First introduced by Schrödinger in 1926
Wave-particle duality
Matter wave

Correspond to a wave function r , t 
Of the system, to describe particle behaviors at
space r and changing with time t.
 dV
2
Means the probability to find the
particles in volume dV = dxdydz.
Wavefunction properties
Schrödinger equation
The Schrödinger equation plays the role of Newton's
laws and conservation of energy in classical mechanics
- i.e., it predicts the future behavior of a dynamic
system. It is a wave equation in terms of the
wavefunction which predicts analytically and precisely
the probability of events or outcome. The detailed
outcome is not strictly determined, but given a large
number of events, the schrödinger equation will predict
the distribution of results.
Schrödinger equation is a fundamental assumption in
Quantum Mechanics, which can not be derived from
other theories.
Schrödinger equation
—a harmonic oscillator example
For a 1D(x) free particle
Wavefunction:
Ψ ( x, t )  Ψ o e
i
 ( E t P x)
Ψ
i
i

  EΨ o e
  EΨ
t


Ψ
P
  2 Ψ oe
2
x

2
2
i
 ( E t P x)

2
P
 2Ψ

i
 ( Et  Px)

 Ψ
Ψ

 i
2
2m x
t
2
2
2
P
E  Ek 
2m
schrödinger equation
For a particle with 1D(x) potential field U(x,t)
2
P
E  Ek  U 
U
2m
Ψ
i
  EΨ
t

Ψ
i P
 [
 U ( x, t )]Ψ
t
 2m
2
  2Ψ
P2


Ψ
2
2m x
2m
 2Ψ
P2
 2 Ψ
2
x

 Ψ
Ψ

 U ( x, t )Ψ  i
2
2m x
t
2
2
schrödinger equation
1D → 3D
schrödinger equation:
 


Ψ
 [ 2  2  2 ]Ψ  U ( x, y, z, t )Ψ  i
2m x y z
t
2
2
2
2
2
2
2



Lapalace operator： 2 


2
2
x
y
z 2
schrödinger equation in general case:





Ψ (r , t )
2

 Ψ (r , t )  U (r , t )Ψ (r , t )  i
2m
t
2
Schrödinger equation is a fundamental dynamic equation of
nonrelativistic quantum mechanics, which plays the role of
Newton’s law in classical mechanics.

 
  
Ψ (r , t )
2
  U (r , t )Ψ (r , t )  i

t
 2m

2
H, Hamilatonian
The kinetic and potential energies are
transformed into the Hamiltonian which acts
upon the wavefunction to generate the
evolution of the wavefunction in time and
space. The Schrödinger equation gives the
quantized energies of the system and gives
the form of the wavefunction so that other
properties may be calculated.
Steady Schrödinger equation


Steady wavefunction ： Ψ (r , t )  Φ(r )e
i
 Et

2 2
Ψ

 Ψ  UΨ  i
2m
t
i
2

Et






2
2

  U (r )](r )}e
[
  U (r )]Ψ (r , t )  {[ 
2m
2m
2
Ψ
   Et
i
 EΦ(r )e
t
i
2 2

 Φ  UΦ  EΦ
2m
or
2m
 Φ  2 ( E  U )Φ  0

2
The structure of hydrogen atom
The hydrogen atom consists of a single proton surrounded by a single electron. It
is thus the simplest of all atoms. The proton may be thought to be approximately
at rest at the origin of the coordinate (the center of the hydrogen atom) because
proton is about 1836 times heavier than electron. The Coulomb attractive force
works between the proton and the electron. Its potential is written:
e2
V (r )  

40 r
1
where r is the distance between the proton and the electron. The
(nonrelativistic) Schrödinger equation describing the motion of
the electron takes the form:
 2 2
1 e2 
 
 
   E
4 0 r 
 2m
The Schrödinger equation and its solution
The polar coordinate (r,,) shown in the following is more
convenient than the Cartesian coordinate (x,y,z).
The Schrödinger equation and its solution
We solve the Schrödinger equation by setting the boundary
condition that the wave function should be smoothly continuous
at every point of the coordinate space and should converge to 0
at the infinitely long distance. Then we have a set of discrete
energy eigenvalues and the corresponding eigenstates. The
details of the method to solve it is omitted here. If you want to
study them, please refer to some other textbooks of quantum
mechanics.
The wave functions of the eigenstates is expressed as
 (r, , )  Rnl (r )Ylm ( , )
 (r, , )  Rnl (r )Ylm ( , )
the part Rnl(r) is the radial wave function which is
specified by a set of integers, n and l. n and l are
quantum numbers, which characterize the eigenstates.
n = 1,2,3,…; l = 0,1,2,…; l  n-1
The part Ylm(,) denotes the angular wave function. It
describes the revolving state of the electron around the
coordinate origin (proton), which is specified by a set of
quantum numbers (integers), l and m:
m l, i.e., m = -l, -l +1, …, l - 1, l
Quantum numbers
n: principal quantum number,
n = 1, 2, 3, …;
l: orbital quantum number,
l = 0, 1, 2, …, n-1;
m: magnetic quantum number,
m = -l, -1+1, …, l-1, l.
The energy eigenvalues of hydrogen atom are determined
only by the quantum number n
The ground state:
The abscissa denotes the position coordinate of the
electron (the distance between the proton and electron), r ,
in units of the Bohr radius , where
4 0 2
10
a0 

0
.
529

10
m
2
me
The energy is quantised; En is continues when n  
The orbital quantum number l expresses the speed of the
revolution of the electron, i.e. the magnitude of the
angular momentum of the electron; and the magnetic
quantum number m represents the orientation (direction)
of angular momentum vector.
The angular momentum of the electron is quantised:

L 
h
l (l  1)

2
l (l  1) 
The orientation of the angular momentum is also
quantised, i.e., the component in z direction is quantised:
h
LZ  m
 m
2
For example： l  2
B(z)
L  l (l 1)   2(2 1)   6 
LZ  m
m  0 ,  1,  2 ,    ,  l
L 6
Lz  2

m=1
0
m=0

m = -1
 2
LZ  0,  ,  2
m=2
m = -2
These result implies that not only energy but also
angular momentum and its orientation are
quantized in quantum mechanics. This was
confirmed by the Stern-Gerlach experiment
(1922). Needless to say, this also originates from
the particle-wave duality of electrons. And this can
never understood by the classical theory.
The probability to find an electron
at the position r from the center—
the probability density in the space:
r Rnl r 
2

2
2

0
r Rnl r  dr  1
2
The atomic models
Plum-pudding model
by Thomson
Electron cloud model
Planet model by Rutherford
Bohr’s model
Atomic orbitals
n=1,l=0 n=2,l=0 n=2,l=1 n=3,l=0 n=3,l=1 n=3,l=2
m=0
m=1
m=2
Atomic orbitals
n=4,l=0 n=4,l=1 n=4,l=2 n=4,l=3
m=0
m=1
m=2
m=3
The probability density of the electrons of H atom
n=1
l=0
m=0
l=0
m=0
n=2
l=1
l=0
n=3
l=1
l=2
n=6
l=3
n = 11 l = 6
m=0
& m = ±1
m=0
m=0
& m = ±1
m=0
& m = ±1 & m = ±2
m=0
m = ±3
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 1, l = 0, m = 0,
spherically symmetrical distributions
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 2, l = 0, m = 0,
spherically symmetrical distributions
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 2, l = 1, m = 0,
Dumbbell shaped distribution
along one axis
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 2, l = 1, m = ±1,
Dumbbell shaped distribution
along one axis
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 0, m = 0,
spherically symmetrical distributions
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 1, m = 0,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 1, m = ±1,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 2, m = 0,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 2, m = ±1,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 3, l = 2, m = ±2,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 6, l = 3, m = 0,
The colors in the plots of the probability
distributions vary from blue to red
corresponding to the increase of the
probability from small (zero) to large
values.
n = 11, l = 6, m = ±3,