Transcript Document

Chapter 5: Quantum Mechanics
Limitations of the Bohr atom necessitate a more general approach
de Broglie waves –> a “new” wave equation
“probability” waves
classical mechanics as an approximation
Wave Function Y
probability amplitude
2
P(x)  Y  Y  Y
Y is generally complex
Y  A  iB  Re i
Y  Y  A2  i 2 B 2  A2  B 2
P(x), Y
2
Y   A  iB  Re i
Y  Y  R 2 e i i  R 2
real, non- negative quantities
cphys351 c5:1
Mathematical properties of the wave function
P(x)  Y
2
Y
2
dV finite, non - zero
all space
convention : choose P(x)  Y
 PdV   Y
all space
2
2
dV  1
all space
"normalize" Y  Y'
N
Y
all space
2
dV ,Y' 
1
Y
N
cphys351 c5:2
More mathematical properties of the wave function
Y must be continuous and single valued
Y Y Y
,
,
must be continuous and single valued
x y z
Y must be normalizable
Y  0 as x, y, z  
cphys351 c5:3
The classical wave equation as an example of a wave equation:
2 y
x
2

1 2 y
v 2 t 2

solutions of the form : y  F t

y  Ae i (t
x v)
x 
 * verify
v 
(plane wave solution)
Re[y]physically relevant for classical waves
Re[y]  Re[A]cos (t
 A'cos{ (t
x v)  Im[A]sin  (t
x v)
x v)  }
cphys351 c5:4
alternative form for plane waves
2
y  Ae i ( kxt ) | k | ,   2

y
y
 ik y
 i y verify solution to CWE
x
t
for matter wav es :
h
p    k E  h   

for (non relativistic particle) particle
p2
 U ( x, t )  E
2m
 2k 2
 U ( x, t )  
2m
2
iff 2  v 2
k
2
vs CWE 2  v 2
k
cphys351 c5:5
plane wave type solutions
Y
 ik Y
x
with
Y
 i Y
t
2
k2
 U (x,t)  
2m
2
2Y
Y

 UY  i
1 d
2
2m x
t
2
 2 Y  2 Y  2 Y 
Y
  2  2  2  UY  i
2m x
t
y
z 

2
2m
 2 Y  UY  i
Y
t


3  d


Time dependent Schrödinger Equation
linear (in Y) partial differential equation
cphys351 c5:6
Expectation values (average values)
P(x) | Y(x) | 2

2
|
Y(x)
|
dx  1

Y normalized

x 




 xP(x)dx 
2
x
|
Y(x)
|
dx





G(x)   G(x)P(x)dx 
2
G(x)
|
Y(x)
|
dx

but, statistics and averages for momentum? will look at

G(x) 
*
Y
 (x)G(x)Y(x)dx

cphys351 c5:7
If the potential energy U is time independent,
Schrödinger equation can be simplified by “factoring”
separation of variables
Total energy can have a constant (and well defined) value
Consider plane wave:
Y  Aei(kxt)  Aei ( pxEt)  Aei (px) e i (Et)

  (x)e i (Et) so i
 (x)e i (Et)  E  (x)e i (Et)
t
in the S.E.

 

2

2
i
Y
Y  UY 
2
t
2m x
2
2
2
2






i (Et)
i (Et)
Ee
e

  U  or 
 U   E
2
2
 2m x
  2m x

An eigenvector, eigenvalue problem!
cphys351 c5:8
The time independent Schrödinger equation
 2   2

2
 2 
 


 U   E
 2mx 2 y 2 z 2 



 2 2

   U   E
 2m



Allowed values for (some) physical quantities such as energy
are related to the eigenvalues/eigenvectors of differential
operators
eigenvalues will depend on the details of the wave
equation (especially in U) and on the boundary conditions
cphys351 c5:9
U
Particle in a box: (infinite) potential well
L
 2 d 2

 2m dx 2  U    E


inside, U  0, outside, U      0
V0
x
 2 d 2

 E
2
2m dx
 2k 2
  A sin kx  B cos kx,
E
2m
BC :  (0)  A sin 0  B cos 0  0  B  0
n
 ( L)  A sin kL  0  kL  n(n  1,2, )  k 
L
n 2 2 2 n 2 h 2
nx
En 

; n ( x)  A sin
2mL
8mL
L
cphys351 c5:10
Wavefunction normalization
 A sin nx ,0  x  L
n  
L
 0,otherwise


 P( x)dx  1   

2
n
dx

2nx 
2 1 
2  nx 
 A  sin 
dx  A  1  cos
dx
20
L 
 L 
0
L
L
2
1
2
2
L A 
2
L
2
chooseA 
L
2
nx
n 
sin
n  1,2,
L
L
 A
2
cphys351 c5:11
Example 5.3 Find the probability that a particle trapped in a box L wide can be found
between .45L and .55L for the ground state and for the first excited state.
Example 5.4 Find <x> for a particle trapped in a box of length L
cphys351 c5:12
U
Particle in a box: finite potential well
L
 2 d 2



U
 2m dx 2
   E


inside, U  0,
 2 d 2

 E
2
2m dx
V0
E
x
I
II
III
 2k 2
  II  A sin kx  B cos kx,
E
2m
outside, U  V0  E
 2 d 2
2m
2

(
V

E
)

with
a

(V0  E )
0
2
2
2m dx

 I  C I e  ax  DI e ax
 III  C III e  ax  DIII e ax
cphys351 c5:13
Boundary Conditions
 I  C I e  ax  DI e ax

 II  Asin kx  Bcoskx
  C e  ax  D e ax
III
III
 III
  0 as x    C I  0
U
L
V0
  0 as x    D III  0
 continuous
 I (0)   II (0)  DI  B
E
x
I
II
III
 II (L)   III (L)  Asin kL BcoskL  C III e aL
 ' continuous
 I '(0)   II '(0)  aDI  kA
 II '(L)   III '(L)  kAcoskL kBsin kL  aC III e  aL
cphys351 c5:14
k
DI  A  B
a
k
Asin kL AcoskL  C III e  aL
a
k
AkcoskL k Asin kL  aC III e aL
a

'
2
k
k
k 
sin kL coskL   coskL   sin kL
a
a
a
2
k 
k
   1  2 cotkL
a
a
2


aL kL
kLcotkL     1
2 aL


cphys351 c5:15
2


aL kL
kLcotkL     1
2 aL

2mV0
2
a  2  k 2  aL    (kL) 2 ,
kLcotkL 
2(kL) 2  
2   (kL)
2
 ucotu 

 = 4
2
(2mL2 )
2u 2  
2   u2
6
4
2
-2
-4
-6
V0
20
15
10
5
2
4
6
 = 100
8
10
-5
-10
-15
-20
10
20
30
40
 =
1600
cphys351 c5:16
U
Tunneling
L


d
 2m dx 2  U   E


outside, U  0,
2
2
d 2

 E
2
2m dx
 I  Aeikx  Be ikx

,
ikx
 III  Fe
V0
E
x
2
I
II
III
2
k2
E
2m
inside, U  V 0  E
d 2
 (V0  E)
2
2m dx
 II  Ceax  Deax
2
with a 
2
2m
2
(V 0  E)
cphys351 c5:17
Boundary Conditions
 I  Aeikx  Be ikx   I    I 

 II  Ceax  De ax
  Feikx
 III
 continuous
 I (0)   II (0)  A  B  C  D
U
L
V0
E
x
I
II
III
 II (L)   III (L)  CeaL  De aL  FeikL
 ' continuous
 I '(0)   II '(0)  ik(A  B)  a(C  D)
 II '(L)   III '(L)  a(CeaL  De aL )  ikFeikL
transmission/reflection related to group velocities
T
|  III | 2 v III
|  I  | vI
2
R
|  I  |2 v I
|  I  |2 v I
cphys351 c5:18
A B  C  D
ik ( A  B )  a (C  D)
a
a
 2 A  (1  )C  (1  ) D
ik
ik
ik ikL
(
1

)e F  2e aL C
ikL
aL
 aL
e F e C e D
a

ik
ikeikL F  a (e aL C  e  aL D)
(1  )eikL F  2e  aL D
a

F
1 a k
2
T
 cosh aL 
A
4 aik

2
2
2 2

sinh aL

1
2
1
 1 a  k   2 aL
largeaL : T  1 
 e  2 aL
 e
 4 ak

E  V0 : cosh, sinh  cos, sin  resonant t ransmissio n
2
2 2
n II  L
cphys351 c5:19
Example 5.5: Electrons with 1.0 eV and 2.0 eV are incident on a barrier 10.0 eV high and
0.50 nm wide.
(a) Find their respective transmission probabilities.
(b) How are these affected if the barrier is doubled in width?
cphys351 c5:20
Harmonic Oscillator: classical treatment
F  kx
U  12 kx 2
x  Acos( 0 t   )
d 2x
F  ma  m 2  kx  
d t
  0  k m
as an approximation for any system with equilibrium
equilibrium position x 0
U (x)has a minimum at x 0
dU (x)

 U '(x 0 )  0
dx x x0
Taylor Series expansion/approximation
U (x)  U (x 0 )  U '(x 0 ) (x  x 0 )  12 U "(x 0 ) (x  x 0 ) 2 
 U 0  kxr
1
2
2
cphys351 c5:21
Quantum Oscillator
 2 d 2ψ 1 2
KE  PE  E  
 kx ψ  Eψ
2
2m dx
2
d 2ψ  mk 2 2mE 
  2 x  2 ψ
2
dx
 

Assymptoti c behavior for E  12 kx2 (large x)
d 2ψ mk 2
 2 xψ
2
dx

a 2
2 2 x
d e
dx 2
  ae
try ψ  e
 a2 x 2
a 2
2 2 x
a x e
2
 a2 x 2
a 2
2 2 x
a x e
2
mk
a  2

2
cphys351 c5:22
convenient dimensionl ess parameteri zation
m0
mk
2E 2E
y
x
x 

2


 0
d 2
2

(


y
)  0
2
dy
Eigenvalue s :
  2n  1,n  0,1,2 
0
 En  (2n  1)
 (n  12 )0  (n  12 )h
2
 E  n  h0
 E0  12 0  0(zero point energy )
cphys351 c5:23
Quantum Harmonic Oscillator Solutions
m0
mk
2E 2E
y
x
x 

2


 0
 m0 
n
1 2  y 2 2
n  
H n ( y)
 (2 n!) e
  
H n  Hermite Polynomials
14
n
0
Hn
1
2
2y
3 4 y2  2


cphys351 c5:24
Operators
Momentum in a plane wave ( Y = e i(kxt) ) is "well defined"

ipx

 ikY 
Y  pˆ x 
x
i x
For general wave functions, general operators

p x   Y *pˆ x Y dx   Y *
Y dx
i
i x
G   Y *Gˆ Y dx
Energy  Hamiltonian operator
pˆ 2
E  KE PE  Hˆ 
 U (x)
2m
Time independent Schrödinger equation as ev problem
Hˆ   E
cphys351 c5:25
Example 5.6: An eigenfunction of the operator d 2 /dx 2 is y = e2x. Find the corresponding
eigenvalue.
cphys351 c5:26
Chapter 5 exercises: 4, 5, 6, 11, 23
cphys351 c5:27