Transcript Chapter 8 The quantum theory of motion

Chapter 8
The quantum theory of motion
Confined motion in an infinite square well in one dimension
Schrodinger equation
 2 d 2ψ(x)

 Eψ ( x)
2
2m dx
(x)  A sin (kx),
General solution
Boundary conditions
2L

, n  1,2,3,  
n
Normalized wave functions
 n(x) 
2
 nx 
sin 
,
L
L


n  1,2,3,  
2
2mE
k



• A wave function is specified
by the quantum number n.
• The wave function n(x) has
n-1 nodes where the
probability of finding the
particle is zero.
• In a state of wave function
with more nodes, the particle
has higher kinetic energy.
• The probability density to
find the particle is nonuniform within the well and
is separated by the nodes.
Energy levels of an infinite square well
•
•
•
•
•
•
•
The allowed energy levels are
quantized.
The allowed lowest energy is not zero
and this level is the ground state.
Energy level
 2k 2
n2h2
E

2m
8mL2
n  1,2,3,  
The smaller the mass of the particle is,
the larger E is.
Quantum effect is important for
particles of very small mass.
The larger the width of the square
well, the smaller E is.
Quantum effect is important for
particles in highly confining regions.
E is proportional to h2. This is why
quantum effect is not observed by
objects in daily life but by electrons in
atoms and molecules.
Energy difference
between two
adjacent levels
E  En 1  En
h2
 ( 2n  1)
8mL2
Confined motion in an infinite potential well in two dimensions
• Position operator
xˆ  x, yˆ  y
if 0  x  L1

 0, and 0  y  L
2
V ( x, y )  

,
otherwise.
• Momentum operator
 
 
pˆ x 
, pˆ y 
i x
i y
• Energy operator

Hˆ  i
t
L1 and L2 are the lengths of the box in
the x and y directions, respectively.
Time-independent Schrodinger’s equation
 2   2ψ  2ψ 
 2  2   Eψ

2m  x
y 
The wave function  is a function of x and y.
Two-dimensional wave functions
Separation of variables
ψ ( x, y )  X ( x)Y ( y )
(n1,n2)=(1,1)
(n1,n2)=(2,1)
Solution
 n x 
2
sin  1 , n1  1,2,3,  
L1  L1 
 n2 y 
2
, n2  1,2,3,  
Yn2 ( y ) 
sin 
L2
 L2 
X n1 ( x) 
(n1,n2)=(1,2)
 n1x   n2 y 
4
 sin 
,
ψ n1 ,n2 ( x, y) 
sin 
L1L2  L1   L2 
n1  1,2,3,  , n2  1,2,3,  
A wave function is specified by two
quantum numbers n1 and n2.
(n1,n2)=(2,2)
Energy levels and degeneracy
• Energy levels
kx 
• For a potential well in a square region,
the ground state is (n1,n2)=(1,1) with
the lowest energy
n1
n
, ky  2
L1
L2
En1 , n2 

 2 k x2  k y2
E1,1

• The states of (n1,n2)=(2,1) and
(n1,n2)=(1,2) have the same energy as
2m
h 2  n12
n22 


 2 
2

8m  L1
L2 

n1  1,2,3,  
E1, 2  E2,1
5h 2

8mL2
• The two states are said to be
degeneracy.
n2  1,2,3,  
If L1=L2 =L, the potential well in 2D is in
a square region of length L.

h2

4mL2

h2
2
2
En1 , n2 
n

n
1
2
8mL2
n1  1,2,3,  , n2  1,2,3,  
• Degeneracy: States of different wave
functions have the same energy.
• If N wave functions have the same
energy, this energy level is said to be
N-fold degeneracy.
Confined motion in an infinite potential well in three dimensions
• Position operator
xˆ  x, yˆ  y, zˆ  z
• Momentum operator
pˆ x 
 
 
 
, pˆ y 
, pˆ z 
i x
i y
i z
̂ 
P 
i
Gradient operator



  iˆ  ˆj  kˆ
x
y
z
if 0  x  L1 ,

 0, 0  y  L ,
2
V ( x, y , z )  
and 0  z  L3 ,

,
otherwise.
L1, L2 and L3 are the lengths of the box.
Time-independent Schrodinger’s equation
2 2
 is a function of

 ψ  Eψ
x, y and z.
2m
Laplacian operator
2    
2 2 2
 2 2 2
x y z
Three-dimensional wave functions
Separation of variables
ψ ( x, y, z )  X ( x)Y ( y) Z ( z )
Solution
 n1x   n2 y   n3z 
8
,
 sin 
 sin 
ψ n1 ,n2 ,n3 ( x, y, z ) 
sin 
L1L2 L3  L1   L2   L3 
n1  1,2,3,  , n2  1,2,3,  , n3  1,2,3,  
A wave function is specified by three
quantum numbers n1,, n2 and n3.
Energy levels and degeneracy
• Energy level
n32 
h 2  n12
n22

En1 , n2 , n3 
 2  2
2

8m  L1
L2
L3 

n1  1,2,3,  ,
n2  1,2,3,  ,
E1,1,1 
3h
8mL2
• The three states of
(n1,n2,n3)=(2,1,1), (1,2,1) and
(1,1,2) have the same energy as
n3  1,2,3,  
If L1=L2 =L3=L, the 3D potential is a box.
En1 , n2 , n3
• For a potential well in a 3D box,
the ground state is
(n1,n2,n3)=(1,1,1) with the
energy
2

h2

n12  n22  n32
2
8mL

E2,1,1  E1, 2,1  E1,1, 2
3h 2

4mL2
• The three states are said to be
degeneracy. So, the next energy
level is 3-fold degeneracy.
Quantum tunneling
•
•
Classical mechanics
It is impossible for a particle to
surmount over a barrier with
potential energy high than its
kinetic energy.
Quantum mechanics
If the barrier is thin and the
barrier energy is not infinite,
particles have the probability to
penetrate into the potential region
forbidden by classical mechanics.
This is called quantum tunneling.
The transmission probably of
quantum tunneling generally
decays exponentially with the
thickness of the barrier and with
the square root of the particle
mass.
V: Potential energy
of a barrier
E: Kinetic energy of
a particle.
Left-hand side
Right-hand side
Transmission probability
Transmission probability T is defined as
the probability of the particle tunneling
through the barrier.
The wave function decays exponentially
inside the forbidden region but is not zero.
| A |2
T 
| A |2
A and A′ are the amplitudes of the incident
and transmitted waves, respectively.
Rectangular potential barrier
with a thickness L

 

e κL  e
T  1 
16ε (1  ε )

 κL 2

1
  E / V  1
2m(V  E )
1
 ,

L
T  16ε (1  ε )e  2 κL

E<V
E>V
Application of quantum tunneling: Scanning tunneling microscopy
Nobel prize in physics 1986
Ca atoms on a
GaAs surface
Visualization of the reaction SiH3 →SiH +H2
on a Si surface
http://www.almaden.ibm.com/vis/stm/stm.html
Vibrational motion in a parabolic potential
In classical mechanics, a particle attaching
to a spring that obeys the Hook’s law
vibrates back and forth. The vibrational
motion is called harmonic oscillations. The
stretching and bending motions of atoms
in a molecule is described as harmonic
oscillations.
Schrodinger’s Eq.
 2 d 2ψ(x) 1 2

 kx ψ ( x)  Eψ ( x)
2
2m dx
2
Boundary conditions
ψ ( x) and ψ ( x)  0, as x  .
Wave function of the ground state
1/ 4
V ( x) 
1
kx2
2
d 2x
m
 kx  0
dt 2
x(t )  x0 cos( t )

k
m
k: Force constant, : Angular
frequency
 1 
ψ 0 ( x)   2 
  
 1  x 2 
exp    
 2    
Wave function of the first excited state
1/ 4
 4 
ψ1 ( x)   2 
  
 1  x 2 
x
exp    

 2    
1/ 4
 2 

  
mk


Wave functions of a harmonic oscillator
• The wave functions are specified by the vibrational qunatum number v = 0, 1,
2, 3, …, which is a non-negative integer.
• The wave function of the v-th state has v nodes.
• The mean displacement <x> in any state is zero, but the mean square
displacement <x2> = (v +1/2)2.
• The wave functions extend beyond the turning points of the parabolic potential
in classical mechanics. So, the particle has a probability to penetrate out of the
potential with an exponential decay in distance.
• In quantum mechanics, a particle in a double-well potential may tunnel
through the forbidden region in classical mechanics.
Energy levels of a harmonic oscillator
1

E     
2

  0 ,1, 2 , 3,   
E  E 1  E
 
• The lowest energy level (v = 0) is E0 = ћ/2, which is
the zero-point energy of the oscillator.
• The energy levels are uniform spacing of ћ.
• The energy levels follow the Planck’s hypothesis.
• The higher quantum number a state of the oscillator,
the more number of nodes in its wave function and the
higher the particle energy is.
Rotation in two dimensions
Kinetic energy and angular momentum
E
J2

J2
2 I 2mr 2
J  pr  mvr
I  mr 2
The rotational motion of a particle on a plane is
described by angular momentum J.
I is the moment of inertia of the particle about the
rotational axis.
Quantization for two-dimensional rotation
Quantization of angular momentum (Bohr’s rule) for a fixed radius
2r  n, n  0,1,2,3,  
h
nh
p 
 J  n
 2r
Wave functions
J z  ml 
ml  0,  1,  2,...
ψ ml () 
1
eiml 
2
Energy levels
ml2  2
Eml 
2I
E  Eml 1  Eml
( 2ml  1) 2

2I
The energy levels are quantized
with double degeneracy for the
excited states (ml ≠ 0) and nondegeneracy for the ground state
(ml = 0).
Rotation in three dimensions
Spherical polar coordinates
In quantum mechanics, the wave function of the
particle r, q,  is a solution of the Schrodinger’s
equation and subject to two boundary conditions.
(r , q,   2)  (r , q, )
(r ,   q,   )  (r , q, )
In classical mechanics, the rotation of
a particle in 3D is on the surface of a
sphere, with its position specified by
the polar coordinates (q, ), where
0 ≤ q ≤  and 0 ≤   2.
Quantization in angular momentum
L2  l (l  1) 2 , l  0,1, 2, 3,  
l: Orbital angular momentum quantum number
Orbital angular momentum and
rotational kinetic energy

 
L  mr  v
L2
E
2mr 2
Energy levels
l (l  1) 2
El 
2mr 2
Angular momentum in three dimensions
• The angular momentum about
the z-axis is also quantized.
Representation of wave functions for ml=0
Lz  ml 
ml  0,  1, .... ,l
• Each level is specified by two
quantum number l and ml.
• l : Orbital angular momentum
quantum number
• ml : Magnetic quantum number
• Both quantum numbers are values of
integer.
• l has non-negative integral values, 0,
1, 2,….
• ml is limited to 2l+1 values from –l,
-l+1, …, l-1, 1.
Vector model for angular momentum l =2
The energy levels El,ml of each orbiatl quantum
number l are (2l+1)-fold degeneracy.
Exercises
• 8A.5(a), 8A.12(b), 8A.13(b)
• 8B.3(a), 8B.4(a), 8B.6(a), 8B.9(a)
• 8C.4(a), 8C.5(a), 8C.6(a)