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
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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
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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
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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 ( kxt ) | 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
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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(kxt) Aei ( pxEt) 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)
Ee
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
2mx 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
nx
En
; n ( x) A sin
2mL
8mL
L
cphys351 c5:10
Wavefunction normalization
A sin nx ,0 x L
n
L
0,otherwise
P( x)dx 1
2
n
dx
2nx
2 1
2 nx
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
nx
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
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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
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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
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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 Deax
2
with a
2
2m
2
(V 0 E)
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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
m0
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 h0
E0 12 0 0(zero point energy )
cphys351 c5:23
Quantum Harmonic Oscillator Solutions
m0
mk
2E 2E
y
x
x
2
0
m0
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(kxt) ) 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
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