Download Document

Modern Physics (II)
Chapter 9: Atomic Structure
Chapter 10: Statistical Physics
Chapter 11: Molecular Structure
Chapter 12-1: The Solid State
Chapter 12-2: Superconductivity
Serway, Moses, Moyer: Modern Physics
Tipler, Llewellyn: Modern Physics
Modern Physics I
Chap 3: The Quantum Theory of Light
Blackbody radiation, photoelectric effect, Compton effect
Chap 4: The Particle Nature of Matter
Rutherford’s model of the nucleus, the Bohr atom
Chap 5: Matter Waves
de Broglie’s matter waves, Heisenberg uncertainty principle
Chap 6: Quantum Mechanics in One Dimension
The Born interpretation, the Schrodinger equation, potential wells
Chap 7: Tunneling Phenomena (potential barriers)
Chap 8: Quantum Mechanics in Three Dimensions
Hydrogen atoms, quantization of angular momentums
Chapter 5: Matter Waves
de Broglie’s intriguing idea of “matter wave” (1924)
Extend notation of “wave-particle duality” from light to matter
For photons, P  E  hf  h
c
Suggests
for matter,
c

The wavelength is detectable
only for microscopic objects
h

de Broglie wavelength
P
P: relativistic
momentum
E
f 
de Broglie frequency
h
E: total relativistic
energy
(x,t ) contains within it all the information that can be
known about the particle
(x, t ) is an infinite set of numbers
corresponding to the wavefunction
value at every point x at time t
Properties of wavefunction
  x, t 
Finite, single-valued, and continuous on x and t
Normalization:
  x, t 
x



  x, t  dx  1
2
The particle can be
found somewhere with
certainty
must be “smooth” and continuous where U(x) has a
finite value
The one-dimensional Schrödinger wave equation
  x, t 
2

  x, t   U  x, t    x, t   i
2
2m x
t
2
Time-independent Schrödinger equation
U(x,t ) = U(x), independent of time
d 2 ( x )

 U  x   x   E  x 
2
2m dx
2
Solution:
  x, t     x  e
E: total energy of
the particle
E
i t
Probability density at any given position x (independent of time)
P  x     x,t    x,t      x   x 
stationary states
Time-independent Schrödinger equation
Separation of variables : (x,t ) = (x)·(t )
d 2 ( x )
d  t 

  t   U  x   x    t   i   x 
2
2m dx
dt
2
1 d 2 ( x )
1 d  t 

 U  x  i
= E = constant
2
2m   x  dx
  t  dt
2
Independent of t
Independent of x
2
2

d
 ( x)
spatial
  2m dx 2  U  x   x   E  x 

E
i
t


d

t
temporal i
 E  t 
 t  e

dt
E 
A particle in a finite square well
0,
U(x)  
U o ,

0xL
Region II
elsewhere
Region I, III
d 2  ( x ) 2m
 2 [U ( x )  E ]  ( x )  0
2
dx
Region I:
Region II:
 I ( x )  C1e
U0
I
II
0
Need to solve Schrodinger wave
equation in regions I, II, and III
x
U(x)
>0
 II ( x)  B1 sin kx  B2 cos kx
 x
Region III:  III ( x )  D2e
Consider:
III
L
x
E < U0
The wavefunctions look
very similar to those for
the infinite square well,
except the particle has a
finite probability of
“leaking out” of the well
Finite square Well
(x)
U0
I
n=1
0
U(x)
U0
II
U0
III
L
I
II
x
x
0
U(x)
Penetration depth

I
L
III
x
0
n=2
U0
II
2m U o  E 
No classical analogy !!
III
n=3
Example: A particle in an infinite square well of width L
n
n 2
Ln 
2 2 kn
n
 kn 
L
Pn  kn
Pn 2 n 2 2 2
En 

2m
2mL2
Momentum is quantized. Energy is quantized !
The notion of quantum number: n
The Square Barrier Potential
Uo where 0  x  L
U ( x)  
elsewhere
 0
Resonance transmission
at certain energies E > U0
Ux
Uo
I
0
n 2 2 2
E  Uo 
2mL2
A finite transmission through the barrier at E < U0
if the barrier is made sufficiently thin
III
II
L
x
Expectation values
For a given wavefunction (x,t ), there are two types of
measurable quantities: eigenvalues, expectation values
Observables
(可觀測量)
and Operators
(算符)
An “observable” is any particle property that can be measured
Expectation value Q predicts the average value for Q

Q     x,t  Q    x,t  dx

Q   x,t   q   x,t 
 (x,t ) is the “eigenfunction”
and q is the “eigenvalue”
q = constant
The Schrödinger wave equation:
 H Ψ=[E]Ψ
Examples of eigenvalues and eigenfunctions
U = central forces
 Lz Yl
ml
 ,   ml Yl  , 
ml
 L  Yl  ,   l (l  1)
2
ml
U = 0, a free particle
2
Yl  , 
ml
  Aeikx t 

i kx  t 
[ P] 
  A  ik  e 
 ( k )
i x
i

i kx t
[ E ]  i
  i A  i  e    (  )
t
Three-dimensional Schrödinger equation
h2  2
2
2 









x
,
y
,
z
,
t

U
(
x
,
y
,
z
,
t
)

x
,
y
,
z
,
t

ih
  x , y , z ,t 
 2
2
2 
2m  x y z 
t
h2 2


   x, y , z ,t   U ( x, y , z, t )   x, y , z,t   ih   x, y , z,t 
2m
t
Time-independent Schrödinger equation:

2
   r   U  r   r   E  r 
2
2m
Particle in a system with central forces
ẑ
electron

nucleus

U  r   U (r ) 
r
ŷ
(  Ze)( e)
4 o r
a central force !!
x̂
Require use of spherical coordinates
 ( r , t )   ( x , y , z , t )  ( r ,  ,  , t )
Time-independent Schrödinger equation

2
2m
2  r , ,   U  r   r , ,   E  r , , 
principal quantum number
orbital quantum number
magnetic quantum number
d 2
2

m
0
2
d
1 d 
d    m2

 sin 
   2 

sin   d 
d   sin 
 1 d  2 dR   2m
 r 2 dr  r dr    2  E  U  r   

 



 
ml

 1    0

 
l
 1 
R  0
2
r

R r
n
  r     r, ,   R  r       
For any central force U(r ), angular momentum is quantized
by the rules
L  l (l  1)
 = 1, 2, 3, … (n-1)
and
Lz  ml
ml = 0, 1, 2, … 
Since |L| and Lz are quantized
differently, L cannot orient in the
z-axis direction. |L| > Lz
Z 
En  
 
8 o ao  n 2 
e
2
Degeneracy for a given n
2
n = 1, 2, 3,…
n 1
2


2


1

n

 0
Probability of finding electron of a hydrogen-like atom in
the spherical shell between r and r + dr from the nucleus
Pn,  r  dr  r 2 Rn,  r  Rn,  r  dr
0.52 Å
l=0
Excited States of Hydrogen-like Atoms
The first excited state: n = 2
 200
fourfold degenerate
2s state, is spherically symmetric
 210 ,  211,  211
2p states, is not spherically symmetric
 210
2
 211
2
Rn,  r   e Zr / na0
(2/23/2009, 2h)