Transcript Time independent Schrödinger Equation

Energy is absorbed and emitted in quantum packets
of energy related to the frequency of the radiation:
E  h
h= 6.63  10−34 J·s
Planck constant
Confining a particle to a region of space imposes
conditions that lead to energy quantization.
Copyright (c) Stuart Lindsay 2008
• De Broglie:
The position of freely propagating particles can be
predicted by associating a wave of wavelength
h

mv
“When is a system quantum mechanical and when is it
classical?”
Copyright (c) Stuart Lindsay 2008
m=9.1·10-31 kg;
q=1.6·10-19 C
In un potenziale di 50kV:
1 2
19
15
E  mv  qV  1.6 10  50000  8 10 J
2
2E
8
1
v
 1.3 10 m  s
m
p  mv  9.11031 1.3 108  1.2 1022 m  kg  s 1
h 6.63 1034
12
 
 5.52 10 m  5.52 pm
 22
p 1.2 10
Wave-like behavior
• Waves diffract and waves interfere
Copyright (c) Stuart Lindsay 2008
The key points of QM
• Particle behavior can be predicted only in terms of
probability.
Quantum mechanics provides the tools for making
probabilistic predictions.
• The predicted particle distributions are wave-like.
The De Broglie wavelength associated with
probability distributions for macroscopic particles
is so small that quantum effects are not apparent.
Copyright (c) Stuart Lindsay 2008
The Uncertainty Principle
h
x  p 
4
A particle confined to a tiny
volume must have an enormous
momentum.
Ex. speed of an electron confined to a hydrogen atom (d≈1Å)
34
6.63 10
 25
1
p 


5
.
27

10
kg

m

s
4x 4 10 10
h
p
25
5.27 10
5
1
v 


5
.
8

10
m

s
m
9.11031
E  t ~ h
The uncertainty in energy of a
particle observed for a very short
time can be enormous.
Ex. lifetime of an electronic transition with a band gap of 4eV
h  4.14 10
15
eV  s
h
4.15 1015
t 

 1015 s  1 fs
E
4
Wavefunctions
The values of probability amplitude at all points in
space and time are given by a “wavefunction”
(r, t)
Systems that do not change with time are called
“stationary”:
(r)
Wavefunctions
Since the particle must be somewhere:

*
(r) (r)d r  1
3
r
In the shorthand invented by Dirac this equation is:
  1
Pauli Exclusion Principle
.
• Consider 2 identical particles:
particle 1 in state 1 particle 2 in state  2
• The state could just as well be:
particle 1 in state 2 particle 2 in state  1
• Thus the two particle wavefunction is
total  A 1 (1) 2 (2)  1 (2) 2 (1)
+ for Bosons, - for Fermions
Bosons and fermions
• Fermions are particles with odd spins, where the
quantum of spin is  / 2
Electrons have spin  / 2 and are Fermions
•
3He
3
nuclei have spin
and are Fermions
2
• 4He nuclei have spins 4 and are Bosons
2
Copyright (c) Stuart Lindsay 2008
Pauli Exclusion Principle
Two identical Fermions cannot be found in the same
state.
For fermions the probability amplitudes for exchange
of particles must change sign.
For two fermions:
1
 1( r1 ) 2 ( r2 )  2 ( r1 ) 1( r2 )
 ( r1 , r2 ) 
2
Bosons are not constrained:
an arbitrary number of boson particles can populate
the same state!
For bosons the probability amplitudes for all
combinations of the particles are added.
For two bosons:
1
 1( r1 ) 2 ( r2 )  2 ( r1 ) 1( r2 )
 ( r1 , r2 ) 
2
This increases the probability that two particles will
occupy the same state (Bose condensation).
3He
4He
superfluidity
Bose-Einstein
condensation
Photons are bosons!
E    h 
hc

p  k 
h

Photons have a spin angular momentum (s=1):
spin = ± 
In terms of classical optics the two states correspond to left and
right circularly polarized light.
The Schrödinger Equation
• “Newton’s Law” for probability amplitudes:
 2  2  ( x, t )
 ( x, t )

 U ( x, t ) ( x, t )  i
2
2m x
dt
Time independent Schrödinger Equation
• If the potential does not depend upon time, the
particle is in a ‘stationary state’, and the wavefunction
can be written as the product
 ( x, t )   ( x) (t )
• putting this into the Schrödinger equation gives
 2  2 ( x)
1
 (t )

 U ( x) ( x) 
i
2
2m x
 (t )
dt
Time independent Schrödinger Equation
1  (t )
i
 const .  E
 (t ) dt
 (t )
i
 E (t )
dt
E 

 ( t )  exp  i t 
 

E  
 iE 
 ( x, t )   ( x) exp   t 
  
Note that the probability is NOT a function of time!
Time independent Schrödinger Equation
 2  2 ( x)

 U ( x) ( x)  E ( x)
2
2m x
H ( x)  E ( x)
Solutions of the TISE:
1. Constant potential
   ( x)

 V ( x)  E ( x)
2
2m x
2
2
 2 ( x) 2m( E  V )


 ( x)
2
2
x

 ( x)  A exp ikx
2m( E  V )
k 
2

2
For a free particle (V=0):
2
2
 k
E
2m
Including the time dependence:
( x, t )  A exp ikx  t 
Note the quantum expression for momentum:
p  k
k
2

p
h

2.Tunneling through a barrier
V(x)
V
E
0
X=0
Classically, the electron would just bounce off the barrier but……
But QM requires:


x
(just to the left of a boundary) =

(just to the left of a boundary) =

x
(just to the right of a boundary).
(just to the right of a boundary).
• To the right of the barrier
2m( E  V )
2m(V  E )
k
i
2

2
 ( x)  A exp  x
 ( x) 2  exp( ikx)  exp( ikx)
Real part of
Is constant here
 ( x) 2  A2 exp  2x 
Decays exponentially here
Decay length for electrons that “leak” out of a metal is ca. 0.04 nm
 ( x) 2  A2 exp  2x 
2
(0)  A
2
2
1 2 A
(
) 
2
e
The distance over which the probability falls to 1/e of its
value at the boundary is 1/2k.
Per V-E=5eV (Au ionization energy):
E 511keV
m 2 
c
c2
  6.6 10 16 eV  s
o 1
2m( E  V )
2  511103  5
k

 1.14 A
2
18 2
16 2

( 3 10 ) ( 6.6 10 )
o
1
 0.44 A
2k
3. Particle in a box
• Infinite walls so  must go to zero at edges
• This requirement is satisfied with
  B sin kx
• The energy is
and kL=n i.e. k=n/L n=1,2,3....
n 2  2 2
En 
2mL2
• And the normalized wave function is
n 
2
 nx 
sin 

L  L 
En ,n 1

2n  1 

2
2
2
2mL
The energy gap of semiconductor crystals that are just
a few nm in diameter (quantum dots) is controlled
primarily by their size!
4. Density of states for a free particle
Density of states = number of quantum states available
per unit energy o per unit wave vector.
• The energy spacing of states may be infinitesimal, but
the system is still quantized.
• Periodic boundary conditions: wavefunction repeats after
a distance L (we can let L → )
exp( ik x L)  exp( ik y L)  exp( ik z L)  exp( ik x , y , z 0)  1
2nx
kx 
L
ky 
2n y
L
2n z 
kz 
L
Copyright (c) Stuart Lindsay 2008
• For a free particle:

 2k 2  2 2
E

k x  k y2  k z2
2m
2m
2nx
kx 
L
ky 
2n y
L

2n z 
kz 
L
k-space: the allowed states are points in a space with
coordinates kx, ky and kz.
The “volume” of k-space occupied by each allowed point
is
3
 2 


 L 
k-space is filled with an uniform grid of points each
separated in units of 2π/L along any axis.
r-space:
4 r dr
V
2
k-space:
4 k 2 dk 4 L3 k 2 dk

Vk
8 3
• Number of states in shell dk (V=L3):
4Vk 2 dk Vk 2 dk
dn 

3
8
2 2
dn Vk 2

dk 2 2
The number of states per unit wave vector increases as the
square of the wave vector.
5. A tunnel junction
The gap
Electrode 2
Electrode 1
Real part of
 ( x) L  exp ikx  r exp  ikx
 ( x) M  A exp x  B exp  x
 ( x) R  t exp ikx
Electrode 1
The gap
Electrode 2
• Imposing the two boundary conditions on  (x )
and continuous  ( x) :
x
T
1
2
0
V
1
sinh 2 L
4 E (V0  E )
Transmission
coefficient
Or with 2L>>1:
L in Å,  in eV
i( L)  i0 exp  1.02  L
 = V0-E workfunction
[Φ(gold)=5eV]
Copyright (c) Stuart Lindsay 2008
The scanning tunneling microscope
i( L)  i0 exp  1.02  L
The current decays a
factor 10 for each Å
of gap.
L=5Å V=1 Volt
i ≈ 1nA
Approximate Methods for solving the
Schrödinger equation
• Perturbation theory works when a small perturbing term
can be added to a known Hamiltonian to set up the
unknown problem:
Hˆ  Hˆ 0  Hˆ 
• Then the eigenfunctions and eigenvalues
approximated by a power series in :
can
be
E n  E n( 0)  E n(1)  2 E n( 2)  
 n   n( 0)   n(1)  2 n( 2)  
Copyright (c) Stuart Lindsay 2008
• Plugging these expansions into the SE and equating each
term in each order in  so the SE to first order becomes:
E 
(1)
n
( 0)
n
( 0)
( 0)
(1)
ˆ
ˆ
 H  n  ( H 0  En ) n
• The (infinite series) of eigenstates for the Schrödinger
equation form a complete basis set for expanding any other
function:

(1)
n
  a nm
(0)
m
m
Copyright (c) Stuart Lindsay 2008
Substitute this into the first order SE and multiply from the
left by  n( 0) * and integrate, gives, after using
 n  m   nm
E
(1)
n
 
(0)
n
(0)
ˆ

H n
The new energy is corrected by the perturbation Hamiltonian
evaluated between the unperturbed wavefunctions.
Copyright (c) Stuart Lindsay 2008
 n   n( 0)
  m( 0)
H mn
  (0)
( 0)
E

E
m n
n
m
   m( 0 ) Hˆ   n( 0 )
H mn
• The new wave functions are mixed in using the degree to
which the overlap with the perturbation Hamiltonian is
significant and by the closeness in energy of the states.
• If some states are very close in energy, a perturbation
generally results in a new state that is a linear combination
of the originally degenerate unperturbed states.
Copyright (c) Stuart Lindsay 2008
Time Dependent Perturbation Theory
Turning on a perturbing potential at t=0 and applying the previous
procedure to the time dependent Schrödinger equation:
1
 Em  Ek  
ˆ
P(k , t )     m H   k exp i
t  dt 


 
 i 0
t
2
For a cosinusoidal perturbation:
H (t )  cos(t )
P peaks at
Copyright (c) Stuart Lindsay 2008
Em  Ek


Conservation of energy
in the transition
leading to Fermi’s Golden Rule, that the probability
per unit time, dP/dt is
dP(m, k ) 2

 m Hˆ   k
dt

2
 ( Em  Ek   )
For a system with many levels that satisfy energy conservation
dP(m, k ) 2

 m Hˆ   k
dt

2
 ( Ek )
Density of States