Download A Particle in a 1

Physical Chemistry III (728342)
The Schrödinger Equation
Piti Treesukol
Kasetsart University
Kamphaeng Saen Campus
http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon
Solving Schrödinger Equations

Eigenvalue problem

Simple cases
•
•
•
•
•
•
•
•
 2 d 2

 V ( x)  E
2
2m dx
d 2
2m
E  V 


2
2
dx

The free particle
The particle in a box
The finite potential well
The particle in a ring
The particle in a spherically symmetric potential
The quantum harmonic oscillator
The hydrogen atom or hydrogen-like atom
The particle in a one-dimensional lattice (periodic potential)
Solutions for a Particle in a 1-D Box

Solve for possible r and s in   A sin rx  B cos sx that
satisfy the Schrödinger equation.
r  s  (2mE)1/ 2  1
 II  A sin (2mE)1/ 2  1 x B cos(2mE)1/ 2  1 x for 0  x  a

Not all solutions are acceptable. The wave function
needs to be continuous in all regions
I  0
 III  0
x
 II (0)  0 
 boundary conditions
 II (a)  0
Solutions for a Particle in a 1-D Box
Using the boundary conditions
 II (0)  A sin 0  B cos 0  0
B0
 II (a)  A sin (2mE)1/ 2  1a   0


(2mE)1/ 2
sin (2mE)  a  0 
a   n

n2h2
E
; n  1,2,3...
2
8ma
1/ 2
1
• Only these allowed energy can make the wavefunction
well-behaved.
n=4
E4
E3
Energy

n=3
E2
n=2
E1
n=1
Solutions for a Particle in a 1-D Box

Normalizability of the wavefunction

a

0
1   |  |2 dx   |  II |2 dx | A |2

a
0
sin 2 cxdx 
2  nx 
sin
0  a dx
a
x 1 
  c
2 4 
1/ 2
2
A  i 
a
Normalization constant
1/ 2
2
  
a
sin
nx
for 0  x  a
a

The lowest energy is when n=1

For Macroscopic system,
n is very large
h2
Zero-point energy
E
2
8ma
ground
state
Solutions for a Particle in a 3-D Box

For the 3-D box (axbxc), the potential outside the box is
infinity and is zero inside the box
 2  d 2 d 2 d 2 
 2  2  2   E

2m  dx
dy
dz 
  X ( x)Y ( y ) Z ( z )

 2 X " ( x)  2 Y " ( y )  2 Z " ( z )


E
2m X ( x ) 2m Y ( y ) 2m Z ( z )
1/ 2
2
X ( x)   
a
n x
sin x ;
a
1/ 2
2
Y ( y)   
b
1/ 2
2
Z ( z)   
c
sin
n yy
b
;
n z
sin z ;
c
nx2 h 2
Ex 
8ma 2
Ey 
n y2 h 2
8mb2
nz2 h 2
Ez 
8mc2
Solutions for a Particle in a 3-D Box
• Inside the box:
• Outside the box:
• Total energy
1/ 2
 8 
 

abc


 0
n yy
nxx
n z
sin
sin
sin z
a
b
c
2
h 2  nx2 n y nz2 
E
 2 2
2

8m  a
b
c 
• The ground state is when nx=1 ny=1 nz=1
Degeneracy

A Particle in a 3-D box with a=b=c
2
  
a
3/ 2
sin
n yy
nxx
n z
sin
sin z
a
b
c

h2
E
nx2  n y2  nz2
8ma2
• States n ,n ,n


x
y
z

  211,  121,  112 have the same energy.
An energy level corresponding to more than one states
is said to be degenerate.
The number of different state belonging to the level is
the degree of degeneracy.
Orthogonality and the Bracket Notation

Two wavefunctions are orthogonal if the integral of their
product vanishes
  d  0
Dirac Bracket Notation
*
n

m
n n  n n 1
n m  n m  0
n   n*
bra
m  m
ket
n m   nm
(n  m)
Kronecker Delta
 nm  1 n  m
0 nm
A Finite Depth Potential Box
A potential well with a finite depth
Classically
• Particles with higher energy can get out of the box
• Particles with lower energy is trapped inside the box
Energy


x
A Finite Depth Potential Box
If the potential energy does not rise to infinity at the wall
and E < V, the wavefunction does not decay abruptly to
zero at the wall
• X<0
• X>0
• X>L
V=0
V = constant
V=0
Barrier
Height
Energy

Particle’s
energy
x=0
x=L
The Tunneling Effect
Incident wave
Transmitted wave
Energy
Reflected wave
x=0
x  0   Aeikx  Be ikx
x0
x=L
k  2mE 
1/ 2
 d 2

 V  E
2
2m dx
  Ceix  De ix   2mE  V 1/ 2
x  L   Eeikx  Feikx
k  2mE 
1/ 2
The Tunneling Effect

Boundary conditions (x=0, x=L)
 i (bc)   j (bc)
 i (bc)  j (bc)

x
x
Function
Function
First Derivative
x0
A B  C  D
xL
CeL  De L  EeikL  Feikl
First Derivative
x0
ikA  ikB  C  D
xL
CeL  De L  ikEeikL  ikFeikl
• F = 0 because there is no particle traveling to the left on
the right of the barrier
Prob( incident )  A
2
Prob( transmitted )  E
2
The Tunneling Effect

Transmission Probability
T
 transmitted
 incident
2
2


L
L 2 

E
e e

 2  1 

16 1    
A

2
1
when   E / V

The leakage by penetration through
classically forbidden zones is called
tunneling.
Transmission Coefficient
• For high, wide barriers L  1, the
probability is simplified to T  16 1   e2L
• The transmission probability decrease
exponentially with the thickness of the
barrier and with m1/2.
1.0
0.5
0.0
0.0
1.0
2.0
E/V0
3.0
4.0
The Harmonic Oscillator (Vibration)
A Harmonic Oscillator and Hooke’s Law
• Harmonic motion: Force is proportional to its displacement: F= –kx
when k is the force constant.
l
d 2l
m 2   k (l  l0 )
dt
d 2x
m 2  kx  0
dt
l0
l
f ( x)  
V ( x) 
f   k (l  l0 )   kx
V

dV
dx
0
Equilibrium
x(t )  c1 sin t  c2 cos t
1 2
kx Harmonic Potential
2
k
  
m
1
2
x
The 1-D Harmonic Oscillator
Symmetrical well V ( x)  x  a 
Schrödinger equation with harmonic potential:
2
V ( x) 
6
5
4
3
2
Solving the equation by using boundary
conditions that     0 .
 The permitted energy levels are

1/ 2
1
0
Displacement, x
1 2
kx
2
 2 d 2 1 2

 kx   E
2
2m dx
2
7
Potential Energy, V



k
Ev  v  12     
m
v  0, 1, 2, 3 ...
The zero-point energy of a harmonic oscillator
is E  
0
1
2
Solutions for 1-D Harmonic Oscillator

Wavefunctions for a harmonic oscillator
 ( x)  N   polynomial in x  Gaussian fn.
x
  ( x)  N H  e ( x /  )
 
2
/2

  
 mk 
• Hermite polynomials; H  y 

H0 y  1
H1  y   2 y
H2 y  4 y2  2
H 3  y   8 y 3  12 y
H 4 ( y )  16 y 4  48 y 2  12
H 5 ( y )  32 y 5  160 y 3  120 y
H 6 ( y )  64 y 6  480 y 4  720 y 2  120

1/ 4
 
2
Wavefunctions
 3
 8x3
x  ( x /  ) 2 / 2
 3 ( x)  N3   3  12 e


2
 4x2

 2 ( x)  N 2   2  2 e ( x /  ) / 2


 2 x  ( x /  ) 2 / 2
 1 ( x )  N1    e
 
 0 ( x)  N 0 e
 2
( x / )2 / 2
 1
Normalizing Factors
1/ 2
1/ 2
1


N   1/ 2  
  2  ! 
k
    
m
 0
At high quantum number (>>) harmonic
oscillator has their highest amplitudes near
the turning points of the classical motion
(V=E)
 The properties
of oscillators
• Observables
ˆ  dx   
ˆ
   
• Mean displacement



*

x     x  dx  N
2

  2 N2  ( H e  y
2
/2


N 
  N
2
2
0
2
2




7

*

n
H yH e




( H e
) y ( H e  y
x2
2 2
2
/2
6

) x( H e
x2
2 2
5
)dx
4
3
)dy
2
 y2
dy
1
H H 1  12 H 1 e  y dy
0
2



H ' H e  y dy 
2
if    '
 1/ 2 2  ! if    '
0
• Mean square displacement
x 2    12 
• Mean potential energy
V 
• Mean kinetic energy
1
2

(mk )1/ 2
kx2   12 k   12 
 12   12   12 E

(mk )1/ 2
EK  E  V  12 E


The tunneling probability decreases
quickly with increasing .
Macroscopic oscillators are in states
with very high quantum number.
 8%
Rigid Rotor (Rotation)

2-D Rotation
• A particle of mass m constrained to move in a circular path
of radius r in the xy plane.
z
 E = EK+V
V = 0
 EK = p2/2m
• Angular momentum Jz= pr
• Moment of inertia I = mr2
r
x
p2
E
; J z   pr ; I  mr 2
2m
J z2
Not all the values of the angular
E
momentum are permitted!
2I
y
• Using de Broglie relation, the angular momentum about the z-axis
is
J z   pr ; p  h
Jz  
hr

• A particle is restricted to the circular path thus  cannot take
arbitrary value, otherwise it would violate the requirements for
satisfied wavefunction.
Allowed wavelengths
2r

ml
• The angular momentum is limited to the values
ml hr ml h
Jz   

 ml 

2r
2
ml  0,  1,  2,
hr
• The possible energy levels are
J z2 ml2 2
E

2I
2I
ml < 0
ml > 0
Solutions for 2-D rotation

Hamiltonian of 2-D rotation
2  2
2 
 2  2 
H  x, y   
2m  x
y 
2  2 1  1 2 
 2 

H r ,    
 2
2 
2m  r
r r r  
• The radius of the path is fixed then

2 d 2
2 d 2
H 

2
2
2mr d
2 I d 2
The Schrödinger equation is
d 2
2IE
 2 
2
d

eiml
 ml ( ) 
(2 )1/ 2
ml
1/ 2

2 IE 


• Cyclic boundary condition
 ( )   (  2 )
eiml   2  eiml eiml 2
 ml (  2 ) 

1/ 2
(2 )
(2 )1/ 2
  ml ( )ei 2ml
e i   1
 m (  2 )   12 m  m ( )
l
l
l
 12m
l
must be positive
ml  0,  1,  2, 
• The probability density is independent of 
 m*  m
l
l
 e iml  eiml  1


 
1/ 2 
1/ 2 
 2   2   2
Spherical Coordinates
 Coordinates defined by r, , 
Azimuthal angle
0r 
0   2
Polar angle
0  
Radius
*
ds  dr r̂  rd φ̂  r sin  d θ̂
da  r 2 sin  d  r̂
dV  r sin  d d dr
 1 
1

  r̂  φ̂

θ̂
r r  r sin  
2
2 2  1  1 2
1 
 

  2
 2  2

sin

2
r
r r r  sin  
sin  
 
2
r  x2  y2  z 2
 y
  tan 1  
x
z
  cos 1  
r
x  r cos  sin 
y  r sin  sin 
z  r cos 
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3-D Rotation

A particle of mass m that free to move
anywhere on the surface of a sphere

radius r.

• The Schrödinger equation
2 2
H 
 V
2m
V  0 whenever it is free to travel
r is fixed
2 2

   E
2m
 (r , ,  )   ( ,  )  ( )( )
r

Using spherical coordinate
2 1  1 2
  2
 2
r
r r r
1 2
1 

2
 

sin 
2
2
sin  
sin  

2
Legendrian
• Discard terms that involve differentiation wrt. r
1 2 1  1 2
1 
 


  2  2 2

sin 
2
r
r  sin  
sin  
 
1 2
2mE




2
r

2r 2 mE
2
 
  

2 IE

; I  mr 2

2

Plug the separable wavefunction into the Schrödinger
equation
2  
1  2 
1 
 

sin

 
2
2
sin  
sin  

 d 2  
 d
d  

sin

 
2
2
sin  d
sin  d
d
1 d 2   sin  d
d  
2

sin



sin

2
 d
 d
d

The equation can be separated into two equations
1 d 2  
sin  d
d 
2
2


m

sin



sin

l
 d 2
 d
d
Solutions for 3-D Rotation

The normalized wavefunctions are denoted Y ( , ), which
depend on two quantum numbers, l and ml, and are called
the spherical harmonics.
l , ml
l
ml
Yl ,ml ( ,  )
1/ 2
0
0
 1 


 4 
1/ 2
0
1
 3 


 4 
cos 
1/ 2
1
 3 
 
 8 
0
 5 


 16 
sin  e i
1/ 2
2
1
2
3 cos
2
  1
1/ 2
 5 
 
 8 
cos  sin  e i
1/ 2
 5 


 32 
sin 2  e  2i
l  0, 1, 2 ml  l , l  1, l  2,  ,l
• For a given l, the most probable location of the particle
migrates towards the xy-plane as the value of |ml|
increases
• The energy of the particle is restricted to the values
E  l (l  1)

2I
l  0, 1, 2, 
 Energy is quantized.
 Energy is independent of ml values.
 A level with quantum number l is (2l+1) degenerate.
Angular Momentum & Space Quantization

Magnitude of angular momentum
l l  11/ 2 

l  0, 1, 2, ...
z-component of angular momentum
ml  ml  l , l  1,...,l

The orientation of a rotating body is quantized
ml = +1
z
ml = +2
+2
+1
0
–1
ml = 0
ml = -2
ml = -1
–2
Spin

The intrinsic angular momentum is called
“spin”
ms = +½
• Spin quantum number; s = ½
• Spin magnetic quantum number; ms = s,
s–1, … –s

Element particles may have different s
values
• half-integral spin: fermions
(electron, proton)
• integral spin: boson (photon)
ms = –½
Key Ideas

Wavefunction

Modes of motion (Functions to explain the motion)

Tunneling effect
• Acceptable
• Corresponding to boundary conditions
• Translation
• Vibration
• Rotation
 Particle in a box
 Harmonic oscillator
 Rigid rotor