Lecture on Perturbation Theory
Transcript of Lecture on Perturbation Theory
-
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
Perturbation theoryQuantum mechanics 2 - Lecture 2
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
17. listopada 2012.
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
1 Time-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
2 Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
3 Time-dependent perturbation theory
4 Literature
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Contents
1 Time-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
2 Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
3 Time-dependent perturbation theory
4 Literature
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Do you remember this?
Igor Lukacevic Perturbation theory
C
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Do you remember this?
H0
0n = E
0n
0n
0n|0m = nm complete set
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Now, let us kick the potential bottom a little...
Igor Lukacevic Perturbation theory
Contents
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Now, let us kick the potential bottom a little...
What wed like to solve now is...
Hn = Enn
A question
Does anyone have an idea how?
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Assume
H = H0
+ H
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + H
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
Igor Lukacevic Perturbation theory
Contents
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + H
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + H
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
Name Description Hamiltonian
L-S coupling Coupling between orbital and H = H0 + f(r)L S
spin angular momentum in a H = f(r)L Sone-electron atom H0 = p2/2m Ze2/r
Stark effect One-electron atom in a constant H = H0 + eE0z
uniform electric field E = ezE0 H = eE0z
H0 = p2/2m
Ze2/rZeeman effect One-electron atom in a constant H = H0 + (e/2mc)J B
uniform magnetic field B H = (e/2mc)J BH0 = p2/2m Ze2/r
Anharmonic Spring with nonlinear restoring H = H0 + Kx4
oscilator force H = Kx4
H0 = p2
/2m + 1/2Kx2
Nearly free Electron in a periodic lattice H = H0 + V(x)electron V(x) =
n Vnexp[i(2nx/a)]
model H0 = p2/2m
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
Assume
H is small compared to H0
eigenstates and eigenvalues of H do not differ much from those of H0
eigenstates and eigenvalues of H0 are known
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ContentsTime inde endent nondegenerate ert rbation theor General form lation
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
Assume
H is small compared to H0
eigenstates and eigenvalues of H do not differ much from those of H0
eigenstates and eigenvalues of H0 are known
expand
n = 0n +
1n +
2
2n + . . . ,
En = E0n + E
1n +
2E
2n + . . .
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
Assume
H is small compared to H0 eigenstates and eigenvalues of H do not differ much from those of H0
eigenstates and eigenvalues of H0 are known
expand
n = 0n +
1n +
22n + . . . ,
En = E0n + E
1n +
2E
2n + . . .
and sort
(0) . . . H00n = E0n 0n ,(1) . . . H01n + H
0n = E0n
1n + E
1n
0n ,
(2) . . . H02n + H1n = E
0n
2n + E
1n
1n + E
2n
0n ,
...
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ContentsTime-independent nondegenerate perturbation theory General formulation
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
Making 0n|
(1) and using the normalization property of 0n, we get
First-order correction to the energy
E1n = 0n|H|0n
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory General formulation
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Time independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
General formulationFirst-order theorySecond-order theory
First-order correction to the energy
E1n = 0n|H|0n
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory General formulation
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p g p yTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
First-order theorySecond-order theory
First-order correction to the energy
E1n = 0n|H|0n
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2a
sin
na
x
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ContentsTime-independent nondegenerate perturbation theory General formulation
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gTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
First-order theorySecond-order theory
First-order correction to the energy
E1n = 0n|H|0n
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0
n(x) =2
a sinn
a x
Perturbation Hamiltonian: H = V0
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ContentsTime-independent nondegenerate perturbation theory General formulation
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Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First-order theorySecond-order theory
First-order correction to the energy
E1n = 0n|H|0n
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2
asin
n
ax
Perturbation Hamiltonian: H = V0First-order correction:E
1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E0n + V0
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Ti i d d d b i hGeneral formulationFi d h
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Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First-order theorySecond-order theory
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2a sin
na x
Perturbation Hamiltonian: H = V0
First-order correction:E
1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E0n + V0
Compare this result with an exact solution for a constant perturbation all thehigher corrections vanish
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Ti i d d t d t t b ti thGeneral formulationFi t d th
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Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First-order theorySecond-order theory
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2a sin
na x
Perturbation Hamiltonian: H = V0
First-order correction:E
1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E0n + V0
Compare this result with an exact solution for a constant perturbation all thehigher corrections vanish
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time independent degenerate perturbation theoryGeneral formulationFirst order theory
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Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First-order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-wayacross the well
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryGeneral formulationFirst-order theory
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Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First-order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-wayacross the well
E1n = 2V0a
a/20
sin2
n
ax
dx =V0
2
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryGeneral formulationFirst-order theory
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Time independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
First order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-way
across the well
E1n = 2V0a
a/20
sin2
n
ax
dx =V0
2
HW. Compare this result with an exact one.
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryGeneral formulationFirst-order theory
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p g p yTime-dependent perturbation theory
Literature
ySecond-order theory
Now we seek the first-order correction to the wave function.(1) and 1n =
m=n
cmn0m give
First-order correction to the wavefunction
1n =
m=n
0m|H|0n(E0n E0m)
0m
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryGeneral formulationFirst-order theory
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p g p yTime-dependent perturbation theory
Literature
ySecond-order theory
First-order correction to the wavefunction
1n = m=n
0m|H|0n(E0n
E0m)
0m
For calculationdetails, see Refs[2], [3] and [4].
In conclusion
First-order perturbation theory gives:
often accurate energies
poor wave functions
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryGeneral formulationFirst-order theory
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Time-dependent perturbation theoryLiterature
Second-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTi d d b i h
General formulationFirst-order theoryS d d h
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Time-dependent perturbation theoryLiterature
Second-order theory
Example 2
Compute the first-order corrections for a
harmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = 2
2m
d2
dx+
1
2kx
2 + ax3
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTi d d t t b ti th
General formulationFirst-order theoryS d d th
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Time-dependent perturbation theoryLiterature
Second-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied small
perturbation W = ax3
.
Hamiltonian: H = 2
2m
d2
dx+
1
2kx
2 + ax3
H0 eigenenergies: E0n =
n +
1
2
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time dependent perturbation theory
General formulationFirst-order theorySecond order theory
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Time-dependent perturbation theoryLiterature
Second-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H =
2
2m
d2
dx +1
2 kx2
+ ax3
H0 eigenenergies: E0n =
n +
1
2
H0 eigenfunctions:
0n(x) =
12nn!
e
x2
2 Hn(x)
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
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ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theory
General formulationFirst-order theorySecond-order theory
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Time-dependent perturbation theoryLiterature
Second-order theory
Example 2Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H =
2
2m
d2
dx
+1
2
kx2 + ax3
H0 eigenenergies: E0n =
n +
1
2
H0 eigenfunctions:
0n(x) = 1
2n
n!
ex2
2 Hn(x
)
E1n = 0n|ax3|0n = 0
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
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ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theory
General formulationFirst-order theorySecond-order theory
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Time dependent perturbation theoryLiterature
Second order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = 2
2m
d2
dx+
1
2kx
2 + ax3
H0 eigenenergies: E0n =
n +
1
2
H0 eigenfunctions:
0n(x) =
1
2nn!
ex2
2 Hn(x
)
E1n = 0n|ax3|0n = 0For 1n we need expressions 0m|H|0n
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theory
General formulationFirst-order theorySecond-order theory
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p p yLiterature
y
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = 2
2m
d2
dx+
1
2kx
2 + ax3
H0 eigenenergies: E0n =
n + 12
H0 eigenfunctions:
0n(x) =
1
2nn!
ex2
2 Hn(x
)
E1n = 0n|ax3|0n = 0For 1n we need expressions 0m|H|0n for m = n 2k, k Z these are zero
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theory
General formulationFirst-order theorySecond-order theory
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p p yLiterature
y
Example 2
Compute the first-order corrections for a
harmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = 2
2m
d2
dx+
1
2kx
2 + ax3
H0 eigenenergies: E0n = n + 12
H0 eigenfunctions:
0n(x) =
1
2nn!
ex2
2 Hn(x
)
E1n =
0n
|ax3
|0n
= 0
For 1n we need expressions 0m|H|0n for m = n 2k, k Z these are zero so, well, for example, take only these:m = n + 3, n + 1, n
1, n
3
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theory
General formulationFirst-order theorySecond-order theory
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Literature
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
n|ax3
|n + 3 = a (n + 1)(n + 2)(n + 3)(2)3n|ax3|n + 1 = 3a
(n + 1)3
(2)3
n|ax3
|n 1 = 3a n3
(2)3
n|ax3|n 3 = a
n(n 1)(n 2)(2)3
Dont forget...
n Hn()0 11 22 42 23 83 124 164 482 + 125 325 1603 + 120
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLi
General formulationFirst-order theorySecond-order theory
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Literature
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
n|ax3
|n + 3 = a (n + 1)(n + 2)(n + 3)(2)3n|ax3|n + 1 = 3a
(n + 1)3
(2)3
n|ax3
|n 1 = 3a n3
(2)3
n|ax3|n 3 = a
n(n 1)(n 2)(2)3
Energy differences
m En Emn+3 3n+1 n-1 n-3 3
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Time-dependent perturbation theoryLit t
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Literature
Example 2
Compute the first-order corrections for a harmonic oscilator
with applied small perturbation W = ax
3
.
n|ax3|n + 3 = a
(n + 1)(n + 2)(n + 3)
(2)3
n|ax3|n + 1 = 3a
(n + 1)3
(2)3
n|ax3|n 1 = 3a
n3
(2)3
n|ax3|n 3 = a
n(n 1)(n 2)
(2)3
Energy differences
m En Emn+3 3n+1
n-1 n-3 3
1n = a2
1
3
n(n 1)(n 2)
2
0n3 + 3n
n
2
0n1
3(n + 1)
n + 1
2
0n+1 13
(n + 1)(n + 2)(n + 3)
2
0n+3
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Literature
Making 0n|
(2), using the normalization property of 0n and orthogonality
between 0n and 1n, we get
Second-order correction to theenergy
E2n =
m=n
|0m|H|0n|2E0n E0m
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
Time-dependent perturbation theoryLiterature
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Literature
Now we seek the second-order correction to the wave function.(2) and 1n =
m=n
cmn0m give
Second-order correction to the wave function
2n =
m=n
0n|H|0n0m|H|0n
(E0n E0m)2
+k=n
0m|H|0k0k|H|0n(E0n
E0m)(E0n
E0k)
0m
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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Literature
Contents
1 Time-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
2 Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
3 Time-dependent perturbation theory
4 Literature
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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Symmetry Degeneracy Perturbation
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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Symmetry Degeneracy Perturbation
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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A question
Whats wrong with
E2n =m=n
|0m|H|
0n|
2
E0n E0m , m, n q
if unperturbed eigenstates are degenerate E01 = E02 = = E0q?
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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H = H0 + HH0
0n = E
0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
qi=1
anm
0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
y
c
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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Matrix H in basis B
H =
H11 H1,q+1 . . .
. . . 0
Hq/2,q/2
0. . .
HqqHq+1,1
...
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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H = H0 + HH0
0n = E
0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
qi=1 a
nm
0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
y
c
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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For n = 1,
qm=1
(Hpm Enpm)anm = 0 appear as
H11
E1 H
12 H
13 . . . H
1q
H21 H22 E1 H23 . . . H2q...
......
......
Hq1
a11
a12...
a1q
= 0
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H = H0 + HH0
0n = E
0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
q
i=1 a
nm
0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
y
c
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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The q roots of secular equation detH EnI = 0 are the diagonal elements of
the submatrix of H.
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H = H0 + HH00n = E0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
q
i=1
anm
0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
y
c
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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H = H0 + HH00n = E0n
0n
E01 q-fold degenerate
m1 E
Construct
n=
q
i=1
anm0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
yc
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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H = H0 + HH00n = E0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
q
i=1
anm0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
yc
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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H = H0 + HH00n = E0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
q
i=1
anm0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
yc
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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H = H0 + HH00n = E0n
0n
E01 q-fold degenerate
m1 E
Construct
n =
q
i=1
anm0
m
which diagonalizes submatrixof H:n|H|k = Enk
m2
%Nondegenerateperturbation theorywith basis B
q(H E)a = 0
m3E detH EnI
= 0
m8 m6 m5 m4T
c
yc
{n}Gives new basis B
m7' {ani} E1, E2, . . . , Eq
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n = n +1n +
22n + . . . n qn =
0n +
1n +
22n + . . . n > qEn = E
0n +E
1n +
2E2n + . . . n
q (E01 =
= E0q)
En = E0n + E
1n +
2E2n + . . . n > qEn = n|H|n n qE1n = 0n|H|0n n > q
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n = n +1n +22n + . . . n qn =
0n +
1n +
22n + . . . n > qEn = E
0n +E
1n +
2E2n + . . . n q (E01 = = E0q)En = E
0n + E
1n +
2E2n + . . . n > qEn =
n
|H
|n
n
q
E1n = 0n|H|0n n > q
So, what do we get from degenerate perturbation theory:
1st-order energy corrections
corrected w.f. (with nondegenerate states they serve as a basis forhigher-order calculations)
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Two-dimensional harmonic oscilator Hamiltonian:
H0 =
p2x + p2y
2m +
K
2 (x
2
+ y
2
)
np = n(x)p(y) |np
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Two-dimensional harmonic
oscilator Hamiltonian:
H0 =p2x + p
2y
2m+
K
2(x2 + y2)
np = n(x)p(y) |np
Enp = (n + p+ 1) is(n + p+ 1)-fold degenerate.Whats the degeneracy of
|01
state?
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Two-dimensional harmonic
oscilator Hamiltonian:H0 =
p2x + p2y
2m+
K
2(x2 + y2)
np = n(x)p(y) |np
Enp = (n + p+ 1) is(n + p+ 1)-fold degenerate.Whats the degeneracy of |01state?E10 = E01 = 20.
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Now, turn on the perturbation: H = Kxy
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Now, turn on the perturbation: H = Kxy
find w.f. which diagonalize H
1 =a
10 +b
012 = a
10 + b
01
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Now, turn on the perturbation: H = Kxy
find w.f. which diagonalize H
1 = a10 + b01
2 = a10 + b
01
calculate the elements of submatrix of H in the basis {10, 01}
H = K
10|xy|10 10|xy|0101|xy|10 01|xy|01
= E
0 11 0
E = K22 , 2 = m0
Igor Lukacevic Perturbation theory
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theory
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General formulationExample: Two-dimensional harmonic oscilator
Now, turn on the perturbation: H = K xy
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Now, turn on the perturbation: H K xy
find w.f. which diagonalize H
1 = a10 + b012 = a
10 + b
01
calculate the elements of submatrix of H in the basis {10, 01}
H = K
10|xy|10 10|xy|0101|xy|10 01|xy|01
= E
0 11 0
E =K
22, 2 =
m0
solve the secular equation
E EE E = 0 E = E E10
E+ = E10 + E
E = E10 E
ddd
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obtain the new w.f. from
E EE
E
a
b = 0= E
= +E 1 = 12 (10 + 01)E = E 2 = 12 (10 01)
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obtain the new w.f. from E EE E
a
b
= 0
= E = +E 1 = 12 (10 + 01)
E = E 2 = 12 (10 01)
HW
How does the threefold-degenerate energy
E = 30
of the two-dimensional harmonic oscilator separate due to the perturbation
H = Kxy?
Igor Lukacevic Perturbation theory
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theoryTime-dependent perturbation theory
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1 Time-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
2 Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
3 Time-dependent perturbation theory
4 Literature
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Problem
If the system is initially in H0, what is the probability that, after time t,transition to another state (of H0) occurs?
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ProblemIf the system is initially in H0, what is the probability that, after time t,transition to another state (of H0) occurs?
Let us assume:
H(r, t) = H0(r) + H(r, t)
n(r, t) = n(r)eit
H0n = E0n n
(r, t) =
ncn(t)n(r, t) , t > 0
it
= (H0 + H)
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Problem
If the system is initially in H0, what is the probability that, after time t,transition to another state (of H0) occurs?
Let us assume:
H(r, t) = H0(r) + H(r, t)n(r, t) = n(r)e
it
H0n = E0n n
(r, t) =n
cn(t)n(r, t) , t > 0
i
t = (H0 + H)
Can you remember the meaning of these coefficients?
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Inserting (r, t) and cn(t) = c0n + c
1n(t) +
2c2n (t) + . . . into time-dependentS.E. and factorizing the perturbation Hamiltonian as H(r, t) = H(r)f(t) gives
Probability that the system has undergone a transition from state l to statek at time t
Plk = Plk = |cn|2 =Hkl
2
t
eiklt
f(t)dt2
For calculation details, see Refs [2] and [3].
Igor Lukacevic Perturbation theory
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Time-independent nondegenerate perturbation theoryTime-independent degenerate perturbation theoryTime-dependent perturbation theory
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1 Time-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
2 Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
3 Time-dependent perturbation theory
4 Literature
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1 I. Supek, Teorijska fizika i struktura materije, II. dio, Skolska knjiga,Zagreb, 1989.
2 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., PearsonEducation, Inc., Upper Saddle River, NJ, 2005.
3 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, SanFrancisco, 2003.
4 A. Szabo, N. Ostlund, Modern Quantum Chemistry, Introduction toAdvanced Electronic Structure theory, Dover Publications, New York,1996.
5
Y. Peleg, R. Pnini, E. Zaarur, Shaums Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.
Igor Lukacevic Perturbation theory
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