Embed Size (px)
Citation preview
Black Body Radiation
2
3
8( , ) oscT d E d
c
Spectral Density Function
oscEAve. energy of anoscillating dipole
oscE kT
2
3
8( , )T d kT d
c
Energy emitted per unit volume, in over frequency range dv at v, as a function of temperature.
oscE nhEnergy is quantized, and proportional to frequency
1osc h
kT
hE
e
3
3
8 1( , )
1h
kT
hT d d
ce
2 3 08 /
0 00
00
00
TkT c d
T
T
T
Classical Theory predicts that total energy emitted is infinite above 0 K
Photoelectric EffectClassical predictions fail to account for experimental observations
E hE
slope==h
De Broglie Relation
h
p
Why not for particles?
For light
A proton moving at 0.001 C has wavelength?
34
12
27 5
6.626 101.33 10
1.66 10 3.0 10
h h Jsm
mp mv kg s
A 100 g baseball moving at 10 m/s has wavelength?
34
346.626 106.626 10
0.1 10 /
h h Jsm
p mv kg m s
~1000 times its radius
Diffraction
sinn
a
b a
The Double Slit Experiment
A single electron exhibits interference behaviour ???
Emission Spectrum of H
Classical theory predicts that any orbital trajectoryof an electron is unstable as it looses energy through radiation.
13_01fig_PChem.jpg
Energy Levels and The Boltzmann Distribution
System behaves as having a continuous energy spectrum when E≤kT
1i
j
if E kTn
if E kTn
The Schrödinger Equation
2( ( ))( ( )) ( ( )) 2 ( ( ( )))
2n
n n n n n
p x tE V x t p x t m E V x t
m
2 ( )n
h h
p m E V
2 2
2 2 2
1 ( , ) ( , )d x t d x t
v dt dx
22
2
( )( ) 0
d xk x
dx
2 2
2 2
( ) 4( ) 0
d xx
dx
( , ) ( ) ( )x t x t
2k
Consider the space p dependent
Recall
1st Harmonic
The Schrödinger Equation22
2 2
8 ( ( ))( )( ) 0nm E V xd xx
dx h
2 2
2 2
( )( ( )) ( ) 0
8 n
h d xE V x x
m dx
2 2
2
( )( ) ( ) ( )
2 n
d xV x x E x
m dx
( ) ( )n n nH x E x
2 2 2
2
ˆˆ ( ) ( )2 2
d pH V x V x
m dx m
Eigen Relationship
ˆd
p idx
The Time Dependent Schrödinger Equation
22 2
2
( )( ) 0nn
d tv k t
dt
Consider the time dependent part
22
2
( )( ) 0n
n n
d tt
dt
22
2( ) ( ) 0n n n n n
d d dt i i t
dt dt dt
( ) 0 ( ) ( )n n n n n
d di t i t t
dt dt
( ( )) ( )n n n
di i t i t
dt
( ) ( )n n n n
dt i t
dt n nE
( ) ( )n n n
dE t i t
dt
( ) ( ) ( ) ( )n n n n n
dx E t x i t
dt
( ) ( ) ( ) ( )n n n n n
dE x t i x t
dt
ˆ ( ) ( ) ( ) ( )n n n n
dH x t i x t
dt
ˆ ( , ) ( , )n n
dH x t i x t
dt
2
2( ) ( , ) ( , )
2 n n
d dV x x t i x t
m dx dt
The Time Dependent Schrödinger Equation
H Propagates the wave function through time
( ) ( )n n n
dE t i t
dt
The Time Dependent Schrödinger Equation
( ) ( ) 0nn n
Edt t
dt i
( ) ( ) 0
( ) (0) n
n
i tn n
dt i t
dt
t e
( , ) ( ) ( ) (0) ( )ni tn n n n nx t x t e x
nn
E
* 2(0) such that (0) (0) 1 is constant in timen n n or
Im
Re
t
( )t(0)o
ro
Norm is preserved over time
ˆ ( , ) ( , )d
H x t i x tdt
Propagators( )
( ) ( )df t
f t dt f t dtdt
ˆ( , ) ( , ) ( , ) ( , ) ( , )d i
x t dt x t x t dt x t H x t dtdt
ˆ1 ( , )iHdt x t
ˆ( , ) 1 ( , )
n
i Hx t x t
n
ˆ ˆ( , ) lim 1 ( , ) exp ( , )
n
n
i H iHx t x t x t
n
lim 1 exp( )n
n
xx
n
dtn
ˆ ( ) ( , )U x t
ˆˆ ( , ) ( , ) ( , ) ( , )
d d HH x t i x t x t x t
dt dt i
Propagators
(0)( )
!
n n
nn
d f tf t
dt n
2 2
2 2
ˆ ˆ ˆ( , ) ( , ) ( , )
d H H Hx t x t x t
dt i i
ˆ
( , ) ( , )n n
n n n
d Hx t x t
dt i
ˆ( ,0)
( , ) ( ,0)! !
n n n n
n n nn n
d x t H tx t x
dt n i n
ˆ( ,0)
!
n n
n nn
H tx
i n
ˆ ( ) ( ,0) ( , ), such that ( ,0) ( , )U t x x t x x t
ˆˆ 1 ˆ( ,0) ( ,0) ( ) ( ,0)
!
n iHt
n
iHtx e x U t x
n
Unitary Transformation
Summary
( ) ( )n n nH E r r
ˆ ( , ) ( , )d
H t i tdt
r r
( , ) (0) ( )i tt e r r
( , ) ( ) ( )t t r r
ˆ( , ) ( ) ( , )t U t r r
2
ˆ ˆ ˆ ˆ ( )2
H Vm
r
( ) ( )n n nH x E x
( , ) ( ) ( )x t x t
2 2
2ˆ ˆ( )
2
dH V x
m dx
ˆ( , ) ( ) ( , )x t U x t
ˆ ( , ) ( , )d
H x t i x tdt
( , ) (0) ( )i tx t e x
Quantum Mechanics for Many Particles
1 2 3 1 2 3( , , ,..., ) ( , , ,..., )n k n n kH E r r r r r r r r
2
,
ˆ ˆ ˆ ˆ ( , )2 i i ij i j
i i ji
H Vm
r r
1 2 3 1 2 3( , , ,..., , ) ( , ) ( , ) ( , )... ( , )k kt t t t t r r r r r r r r
( , ) ( ) ( )i i it t r r
1r
(0,0,0)
3r
2r4r
m1m3
m2m4
z1z3
z4z2
2
2ˆ ( , )
4
i jij i j i j
o i j
z z eV
r r r r
r r
