Chapter 11 Perturbation Theory (???? p249)

1
Chapter 11 Perturbation Theory (???? p249)
11.1 Stationary Perturbation Theory 11.2
Degeneracy 11.4 Time-Dependent Perturbation 11.5
Time-independent Perturbation
2
11.1 Stationary Perturbation Theory
The Hamiltonian of a real system is described by
H0 is called the Hamiltonian of the unperturbed
system, the perturbation has to be very
small. ? is a real parameter and is also called
the perturbation parameter.
We assume that the eigenfunctions and eigenvalue
of the unperturbed Hamiltonian H0 are known,
3
We are seeking the eigenvalues and eigenfunctions
of the complete Hamiltonian H, i.e.
The exact wave function ? is expanded by means of
the known solution of ?n0 of the unperturbed
system
Insert this into above equation
4
Multiply by , and integrate, we can get
(1)
We have used the fact that the eigenfunctions are
orthonormal,
The matrix element Wmn stands for
Equation (1) can be transformed into
5
When ?0,there is only idealized state, with
If ??0,wave function will change, and other
neighbouring states ?0n with m ? n will be
admixed.
??? ? 0, m ? n, ???????????0m????an????????0n?????
m n ??,????, ?m n??,?????
6
In case of assuming that perturbation is small,
we expand both the desired expansion coefficient
am and the energy eigenvalue Ek in powers of the
perturbation parameter ?
The number in the bracket denotes the degree of
perturbation. Insert those into the following
equation,
We obtain
7
1. 0th Approximation, (?0,no perturbation)
(2)
Consider m k,
8
2. 1st Approximation
0th Approximation
Insert this into equation (2) and only take the
first order in ?, we get
When mk
m?k
9
In case of m k, we can not obtain am k(1)
using the above method. According to the
normalization of wave function, we get 0th wave
function and energy eigenvalue
The complete wave function of system can be
described by
10
For first order, neglect the term of ?2
11
3. 2nd Approximation
For mk, we insert the results of the first
approximation into equation (2), and only
consider the coefficient of ?2, we get
For m?k,
12
Example
The energy of k state in the 2nd approximation is
A particle which is defined in infinite potential
well is effected by the perturbation
For the ground state, solve its energy in the
second approximation and its wave function in the
first approximation.
13
Solution
The Hamiltonian of the complete system is given
by
H0 is the Hamiltonian of the unperturbed system.
we can easily obtain the energy eigenfunction and
eigenvalue of the unperturbed system,
For the ground state
14
Using the formula of the energy of k state in the
2nd approximation,
We get the energy of ground state in the 2nd
approximation
15
Hence
16
Using the formula of the wave function of k state
in the first approximation
We get the wave function of ground state in the
first approximation
17
11.2 Degeneracy perturbation (????)
For a given level of energy En0, a series of
eigenfunction ?n?0, ?1, 2, , fn, might exist.
Such a level is call fn-fold degenerate, the
energy eigenvalue is independent of ?.
We go back to(11.6a, p250)
Set
In the case of fn-fold degenerate, the above
equation becomes
(3)
18
The matrix element is given by
Consider 0th approximation
When mk
Obviously
19
For the kth level, let n k, the double sum over
n and ? reduces to a single sum for the 0th
approximation.
According to
We obtain
Since k is fixed, we neglect the k in the foot
index, and get
20
The index ? runs from 1 to fk. The above
equation represents a system of fk linear
equation for a(0)?. The condition these equations
have nontrivial solutions is that the determinant
of its coefficient is zero. (?????????????????????
?)
This equation is called the secular equation
(????)
21
??????????????,??????????(2l1)???????????????????
?????,?????,????????????????????????????,?????????
????????????
22
Hydrogen atom Stark effect
1913 ? Stark ?????????????????????
????,???????????????,????,??????????????????(????S
tark effect).?????????????????,
??????????????????(Quadratic Stark effect).
???????n2??, ????Stark effect
????????104-105V/cm???????????5?109
V/cm,????,?????,??????????????????,???????????????
?
23
If we arrange the electric field in the z
direction, the potential energy is given by
Due to
Spherical harmonic function
Hence
Hamiltonian of the system
24
The degeneracy of hydrogen atom is n2. For n 2,
four eigenfunction belong to the energy E0n E02
of the unperturbed system, i.e.
Energy level of 0th approximation
Wave functions of 0th approximation
25
The matrix elements of the perturbation V?? is
Spherical harmonic function
26
(No Transcript)
27
For the other matrix V?? , such as
We only calculate the angular part dependent of
?,
Therefore V13, V14, V23, V24, V34 0
28
Because of the degeneracy of the system, the
general solution of the Hamilton to the energy
eigenvalue E20 is given by a linear combination
of the function ??.
Except for V12 and V21,all matrix elements are
zero, the system reduces to four equations, namely
29
The solutions of the determinat are
30
Obviously, the degeneracy does not vanish owing
to the symmetry of system does not be broken.
31
???n2????????????
??2S?2P?????,???????????????,????????P,?????E
(Ex0, Ey0, EzE ), ???????
??????????z???????
??????V???,????P3a0e, ?1???0 ?2????, ??3
??4????? ?/2? ?????????,?????????????????????n2
????????????????????
32
Example
a particle is defined in
one-dimension infinite potential with the length
L, and the wave function satisfies the period
condition
The wave function is described by
Set the particle is effected by a trap,
Problem solve the first approximation of energy
level using degenerate perturbation.
33
Solution
The degeneracy of energy level is two, the
corresponding eigenfunctions are
Insert the expression of H?, we get
34
Due to a ltlt L, the range of integrate
approximately expands to infinite, namely
According to the integrate formula
35
We obtain
Using the same method, we get
Since the integrand is odd function,
36
The secular equation is given by
Due to
We get
or
Therefore the first approximation of energy level
is
37
Exercise
1. A particle which is defined in one-dimension
infinite potential (0ltxlta) is perturbed by a
potential
Problem solve the first approximation of energy
level of ground state
2. A particle which is defined in
two-dimension infinite potential V (x, y
) is perturbed by a potential
and
Problem solve the first approximation of the
lowest two energy level (?????????????).
38
11.4 Time-Dependent Perturbation (p269)
?????????????????????????????.
?????????????,??????. ?V(r, t)?????t 0?t T
????,????????,?????????????
V(r, t)???,???????????,??????????????????
?V(r, t)???,?????????????
39
After considering perturbation, Schordinger can
be written
(1)
H0 is the whole Hamiltonian of the system without
perturbation, the index 0 indicates the time
independent. V(r, t) is the perturbation
potential.
First we consider eigenvalue of the unperturbed
system.
If the stationary part of the normalized wave
function satisfies
40
Then time-dependent function
are solutions of the unperturbed system.
The solutions of the perturbed system are the
linear superposition of those of the unperturbed
system, namely
Insert this into equation (1), get
41
Due to
Therefore
42
Set
Considering the normalization of wave function
?k, and integrate over dV, we get
For ?mm0,
The frequency ?mk are sometimes called Bohr
frequency for the transition Em?Ek.
43
At the beginning t 0, we assume that the system
is in the state En, namely
with an(0)1, ak(0) 0 for k?n
When t gt0, in general, am?0 (for all m)
denotes the probability of finding the system at
time t in the state ?k with the energy Ek,
44
The next we will calculate am(t).
In general, the solution of the following
equation can only obtained approximately.
Considering V(r, t) is a small perturbation, in
the absence of perturbation, the system remains
unchanged in its initial state. So we consider
1st order approximation.
45
For first order approximation
For t lt0 and t gtT, we assume that V(r, t) 0, so
When t gtT, am1(t) is constant. The perturbation
has ceased and the system settles into a new
state.
46
11.5 Time-independent Perturbation
If perturbation is independent of time,
The probability for the transition from the state
fn to fm is
47
?t??,W????m??,??????Vmn ?f(t, ?)???? ?mn0, f(t,
?)????2?/t, ?????,??????E 2? ? /t
????????t???????,????????????????,?????????,??????
??,