Transcript perturbation theory 2

Time Dependent Perturbation
Theory
• Many possible potentials. Consider one where
V’(x,t)=V(x)+v(x,t)
• V(x) has solutions to the S.E. and so known
eigenvalues and eigenfunctions
 n  eigenfunctions of V ( x )
n ( x, t )  e
 n ( x)
iEn t / 
• let perturbation v(x,t) be small compared to V(x)
• examples:finite square well plus a pulse that starts
at t=0 or atoms in an oscillating electric field
• write wavefunction in terms of known
eigenfunctions but allow the coefficients (cn) vary
with time  fraction with any eigenvalue (say
energy) changes with time
' ( x, t )   cn (t )n  solutionsof V '
n
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Time Dependent
• start from unperturbed with |n> time independent
H0 n  En0 n
(1)
• add on time-dependent perturbation and try to find
new, time-dependent wavefunctions. l keeps track
of the order. Gives Schrodinger Eq.:
d
( H 0  lV (t ))  (t )  H  (t )  i  (t )
dt
( 2)
• write out  in terms of eigenfunctions |n>
   cn (t )e
 iEn0t / 
n
• substitute into (2) and eliminate part that is (1)
i
d
 dcn
iEn0t / 
0  iEn0t / 
c
e
n

i


E
n


n
n cn e

dt
 dt

( H 0  lV ) cn e
iEn0t / 


n   E  lV cn e
0
n
iEn0t / 
n
 dcn
 iEn0t / 
  i
 lVc n e
n 0
dt


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Time Dependent
• take dot product with <m|
1
e
iEm0 t / 
 dcn
 iEn0t / 
m  i
 lVc n e
n 0
 dt

dcm
i ( Em0  En0 ) t / 
 i
 l  cn e
mV n
dt
• this is exact. How you solve depends on the
potential and initial conditions
• an example. start out at t<0 in state k. “turn on”
perturbation at t=0. See what the values of cm are at
a later time
• reminder. cm gives probability to be in state m
Pm (t ) | m  |2 | cm (t ) |2
• for first order, just do transitions from km. ignore
knm and higher (as the assumption that all are
in state k won’t hold for t>0)
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Time Dependent – example
dcm
i ( Em0  En0 ) t / 
i
 l  cn e
mV n
dt
•
try to solve. assume in particular state k at t=0
 (t  0)  k
• and so get diff. eq. for each state m and solution
dcm
i ( Em0  Ek0 ) t / 
i
 le
mV k
dt
cm (t ) 
l
t
e

i
i ( Em0  Ek0 ) t / 
m V (t ) k dt
0
• if there isn’t any time dependence to V (or if it
changes slowly compared to DE/hbar) then can pull
the matrix element out of the integral
vmk
cm (t ) 
e  i ( Ek  Em ) t /   1
Ek  Em
2 sin2 A  (1  cos2 A)


2
| vmk |  2 
2 Ek  Em


| cm | 
sin
2
 2  E k  Em 
• notice DE and matrix element dependence
2
2

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sidenote
cm (t ) 
l
t
e

i
i ( Em0  Ek0 ) t / 
m V (t ) k dt
0
•
if the perturbing potential is sinusoidal (e.g.
photons of a given energy) then have
V (t )  Meit
• then the results are similar (see book for doing the
integral)
2
| vmk |
| cm | 
2
2
2
 2 

 sin 2
 mk   

mk 
2

• which has a large value when the energy difference
is equal to the frequency of the perturbation
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Time Dependent Probability
• The probability for state k to make a transition into
any other state is:
Pk  cn* (t )cn (t )
n k
1  Pk  ck*ck
• evaluate P by assuming closely spaced states and
replacing sum with integral

Pk 
*
2
c
c
dN

|
c
|
 n n n  n  n dEn

dNn
n 
 density final states
dEn
• define density of states (in this case final; after the
perturbation) in the same way as before
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Fermi Golden Rule
Pk  
|vnk |2

2
n 
sin 2 ( En  Ek ) t / 2 
(( En  Ek ) / 2  )
2
dEn
• Assume first 2 terms vary slowly.Pull out of
integral and evaluate the integral at the pole
• this doesn’t always hold--ionization has large dE
offset by larger density
Pk 
|vnk |2
Rk 
dP
dt

2
n 

2

sin 2 Qt
Q
2
dEn 
2

| v |  nt
2
|v| 
2
• n are states near k. Conserve energy. Rate depends
on both the matrix element (which includes
“physics”) and the density of states. Examples later
on in 461
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Transition Rates and Selection Rules
• Electrons in a atom can make transitions from one
energy level to another
• not all transitions have the same probability (and
some are “forbidden”). Can use time dependent
perturbation theory to estimate rates.
X  X    E  EX  EX 
• The atom interacts with the electric field of the
emitted photon. Use time reversal (same matrix
element but different phase space): have
“incoming” radiation field. It “perturbs” the
electron in the higher energy state causing a
transition to the lower energy state
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EM Dipole Transitions
• Simplest is electric dipole moment. It dominates
and works well if photon wavelength is much
larger than atomic size. The field is then essentially
constant across the atom. Can be in any direction.


F  eE

V  er  rˆb  e( x  y  z )b
 b includes EM terms. Other electric and magnetic
moments can enter in, usually at smaller rates, but
with different selection rules
• Use Fermi Golden Rule from Time Dep. Pert.
Theory to get transition rate
2
Rk 
| vnk |2   n

• phase space for the photon gives factor:
n  E3
• and need to look at matrix elements between initial
and final states:
*
vnk  eb  n r  rˆ k dVolume
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Parity
• Parity is the operator which gives a mirror image
x , y , z   x,  y ,  z
r , ,   r ,    ,   
• Parity has eigenfunctions and is conserved in EM
interactions The eigenfunctions for Hydrogen are
also eigenfunctions of parity
P  l
P ( P )  l2    l  1
P  ( 1) l for Ylm
• A photon has intrinsic parity with P=-1. See by
looking at the EDM term
• So must have a parity change in EM transitions and
the final-initial wave functions must be even-odd
combinations (one even and one odd). Must have
different parities for the matrix element to be nonzero
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Spherical Harmonics-reminder
Ylm  lm m
 spherical harm onics
• The product of the theta and phi terms are called
Spherical Harmonics. Also occur in E&M.
• They hold whenever V is function of only r. Seen
related to angular momentum
iml
( )  e
 00  1
ml  0,1,2  l
10  z
1, 1  (1  z )
2
1
2
z  cos
 20  1  3 z 2
(1  z 2 )1/ 2  sin 
1
2
 2, 1  (1  z ) z
2
 2, 2  (1  z 2 )
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Matrix Elements: Phi terms
• Calculate matrix elements. Have 3 terms (x,y,z)

r  r sin  cos xˆ  r sin  sin  yˆ  r cos zˆ
| vnk | x   Rnl  rRkl r 2 dr  Yl m sin  cosYlm d
   l m sin   lm d cos  e im cos e im d
• look at phi integral (will be same for y component)
cos 
e i  e  i
2i
 e
i ( m  m  ) e i   e  i
2i
d  0
unless m  m  1
• no phi dependence in the z component of the matrix
element and so m=m’ for non-zero terms
• gives selection rules
Dm  0,1
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Matrix Elements: Theta terms
• The (x,y,z) terms of the matix elements give
integrals proportional to:
 lm sin  lmd cos

l m
cos lmd cos ,
• these integrals are = 0 unless l-l’=+-1. The EM
term of the photon is (essentially) a L=1 term (sin
and cos terms).
• Legendre polynomials are power series in
cos(theta). The extra sin/cos “adds” one term to the
power series. Orthogonality gives selection rules.
x(1  x  x 2 )  ........x3
• Relative rates: need to calculate theta integrals. And
calculate r integral. They will depend upon how
much overlap there is between the different states’
wave functions
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First Order Transitions
• Photon has intrinsic odd parity (1 unit of angular
momentum, S=1). For:
 nlm  nlm  
n \ l
0
1
2
3
4
4S
4P
4D
4F
3
3S
3P
3D
2
2S
2P
1
1S
• Selection Rues
Dm  0,1 Dl  1
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