![Page 1: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/1.jpg)
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.
Igor Lukacevic Perturbation theory
![Page 2: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/2.jpg)
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
Igor Lukacevic Perturbation theory
![Page 3: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/3.jpg)
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
![Page 4: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/4.jpg)
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
![Page 5: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/5.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Do you remember this?
H0ψ0n = E 0
nψ0n
〈ψ0n|ψ0
m〉 = δnm → complete set
Igor Lukacevic Perturbation theory
![Page 6: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/6.jpg)
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
![Page 7: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/7.jpg)
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 we’d like to solve now is...
Hψn = Enψn
A question
Does anyone have an idea how?
Igor Lukacevic Perturbation theory
![Page 8: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/8.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + λH ′
Igor Lukacevic Perturbation theory
![Page 9: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/9.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + λH ′
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
Igor Lukacevic Perturbation theory
![Page 10: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/10.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + λH′
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
Igor Lukacevic Perturbation theory
![Page 11: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/11.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + λH ′
unperturbed Hamiltonian
perturbation Hamiltonian
small parameter
Igor Lukacevic Perturbation theory
![Page 12: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/12.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Name Description Hamiltonian
L-S coupling Coupling between orbital and H = H0 + f (r)~L · ~Sspin 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′ = eE0zH0 = p2/2m − Ze2/r
Zeeman effect One-electron atom in a constant H = H0 + (e/2mc)~J · ~Buniform magnetic field ~B H′ = (e/2mc)~J · ~B
H0 = p2/2m − Ze2/rAnharmonic Spring with nonlinear restoring H = H0 + K ′x4
oscilator force H′ = K ′x4
H0 = p2/2m + 1/2Kx2
Nearly free Electron in a periodic lattice H = H0 + V (x)electron V (x) =
∑n Vnexp[i(2πnx/a)]
model H0 = p2/2m
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’ 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
Igor Lukacevic Perturbation theory
![Page 14: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/14.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
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
ψn = ψ0n + λψ1
n + λ2ψ2n + . . . ,
En = E 0n + λE 1
n + λ2E 2n + . . .
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’ 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
ψn = ψ0n + λψ1
n + λ2ψ2n + . . . ,
En = E 0n + λE 1
n + λ2E 2n + . . .
and sort
(λ0) . . . H0ψ0n = E 0
nψ0n ,
(λ1) . . . H0ψ1n + H ′ψ0
n = E 0nψ
1n + E 1
nψ0n ,
(λ2) . . . H0ψ2n + H ′ψ1
n = E 0nψ
2n + E 1
nψ1n + E 2
nψ0n ,
...
Igor Lukacevic Perturbation theory
![Page 16: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/16.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Making 〈ψ0n|/
(λ1) and using the normalization property of ψ0n, we get
First-order correction to the energy
E 1n = 〈ψ0
n|H ′|ψ0n〉
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
![Page 17: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/17.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
First-order correction to the energy
E 1n = 〈ψ0
n|H ′|ψ0n〉
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
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
First-order correction to the energy
E 1n = 〈ψ0
n|H ′|ψ0n〉
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
Unperturbed w.f.: ψ0n(x) =
√2
asin(nπ
ax)
Igor Lukacevic Perturbation theory
![Page 19: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/19.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
First-order correction to the energy
E 1n = 〈ψ0
n|H ′|ψ0n〉
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
Unperturbed w.f.: ψ0n(x) =
√2
asin(nπ
ax)
Perturbation Hamiltonian: H ′ = V0
Igor Lukacevic Perturbation theory
![Page 20: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/20.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
First-order correction to the energy
E 1n = 〈ψ0
n|H ′|ψ0n〉
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
Unperturbed w.f.: ψ0n(x) =
√2
asin(nπ
ax)
Perturbation Hamiltonian: H ′ = V0
First-order correction:E 1n = 〈ψ0
n|V0|ψ0n〉 = V0〈ψ0
n|ψ0n〉 = V0
⇒ corrected energy levels: En ≈ E 0n + V0
Igor Lukacevic Perturbation theory
![Page 21: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/21.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
Unperturbed w.f.: ψ0n(x) =
√2
asin(nπ
ax)
Perturbation Hamiltonian: H ′ = V0
First-order correction:E 1n = 〈ψ0
n|V0|ψ0n〉 = V0〈ψ0
n|ψ0n〉 = V0
⇒ corrected energy levels: En ≈ E 0n + V0
Compare this result with an exact solution⇒ for a constant perturbation all thehigher corrections vanish
Igor Lukacevic Perturbation theory
![Page 22: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/22.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 1
Find the first-order corrections to the energy ofa particle in a infinite square well if the “floor”of the well is raised by an constant value V0.
Unperturbed w.f.: ψ0n(x) =
√2
asin(nπ
ax)
Perturbation Hamiltonian: H ′ = V0
First-order correction:E 1n = 〈ψ0
n|V0|ψ0n〉 = V0〈ψ0
n|ψ0n〉 = V0
⇒ corrected energy levels: En ≈ E 0n + V0
Compare this result with an exact solution⇒ for a constant perturbation all thehigher corrections vanish
Igor Lukacevic Perturbation theory
![Page 23: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/23.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-wayacross the well
Igor Lukacevic Perturbation theory
![Page 24: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/24.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-wayacross the well
⇒ E 1n =
2V0
a
∫ a/2
0
sin2(nπ
ax)dx =
V0
2
Igor Lukacevic Perturbation theory
![Page 25: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/25.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 1 (cont.)
Now, cut the perturbation to only a half-wayacross the well
⇒ E 1n =
2V0
a
∫ a/2
0
sin2(nπ
ax)dx =
V0
2
HW. Compare this result with an exact one.
Igor Lukacevic Perturbation theory
![Page 26: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/26.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Now we seek the first-order correction to the wave function.(λ1) and ψ1
n =∑m 6=n
cmnψ0m give
First-order correction to the wavefunction
ψ1n =
∑m 6=n
〈ψ0m|H ′|ψ0
n〉(E 0
n − E 0m)
ψ0m
For calculationdetails, see Refs[2], [3] and [4].
Igor Lukacevic Perturbation theory
![Page 27: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/27.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
First-order correction to the wavefunction
ψ1n =
∑m 6=n
〈ψ0m|H ′|ψ0
n〉(E 0
n − E 0m)
ψ0m
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
![Page 28: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/28.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Igor Lukacevic Perturbation theory
![Page 29: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/29.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
Igor Lukacevic Perturbation theory
![Page 30: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/30.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
Igor Lukacevic Perturbation theory
![Page 31: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/31.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
H0 eigenfunctions:
ψ0n(x) =
√1
2nn!
√α
πe−
αx2
2 Hn(x√α)
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
Igor Lukacevic Perturbation theory
![Page 32: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/32.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
H0 eigenfunctions:
ψ0n(x) =
√1
2nn!
√α
πe−
αx2
2 Hn(x√α)
⇒ E 1n = 〈ψ0
n|ax3|ψ0n〉 = 0
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
Igor Lukacevic Perturbation theory
![Page 33: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/33.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
H0 eigenfunctions:
ψ0n(x) =
√1
2nn!
√α
πe−
αx2
2 Hn(x√α)
⇒ E 1n = 〈ψ0
n|ax3|ψ0n〉 = 0
For ψ1n we need expressions 〈ψ0
m|H ′|ψ0n〉
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
Igor Lukacevic Perturbation theory
![Page 34: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/34.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
H0 eigenfunctions:
ψ0n(x) =
√1
2nn!
√α
πe−
αx2
2 Hn(x√α)
⇒ E 1n = 〈ψ0
n|ax3|ψ0n〉 = 0
For ψ1n we need expressions 〈ψ0
m|H ′|ψ0n〉
↔ for m = n ± 2k , k ∈ Z these are zero
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
Igor Lukacevic Perturbation theory
![Page 35: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/35.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for aharmonic oscilator with applied smallperturbation W = ax3.
Hamiltonian: H = − ~2
2m
d2
dx+
1
2kx2 + ax3
H0 eigenenergies: E 0n =
(n +
1
2
)~ω
H0 eigenfunctions:
ψ0n(x) =
√1
2nn!
√α
πe−
αx2
2 Hn(x√α)
⇒ E 1n = 〈ψ0
n|ax3|ψ0n〉 = 0
For ψ1n we need expressions 〈ψ0
m|H ′|ψ0n〉
↔ for m = n ± 2k , k ∈ Z these are zero
⇒ so, we’ll, for example, take only these:m = n + 3, n + 1, n − 1, n − 3
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
Igor Lukacevic Perturbation theory
![Page 36: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/36.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
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)
(2α)3
〈n|ax3|n + 1〉 = 3a ·
√(n + 1)3
(2α)3
〈n|ax3|n − 1〉 = 3a ·
√n3
(2α)3
〈n|ax3|n − 3〉 = a ·
√n(n − 1)(n − 2)
(2α)3
Don’t forget...
n Hn(ξ)0 11 2ξ2 4ξ2 − 23 8ξ3 − 12ξ4 16ξ4 − 48ξ2 + 125 32ξ5 − 160ξ3 + 120ξ
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
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)
(2α)3
〈n|ax3|n + 1〉 = 3a ·
√(n + 1)3
(2α)3
〈n|ax3|n − 1〉 = 3a ·
√n3
(2α)3
〈n|ax3|n − 3〉 = a ·
√n(n − 1)(n − 2)
(2α)3
Energy differences
m En − Em
n+3 −3~ωn+1 −~ωn-1 ~ωn-3 3~ω
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Example 2
Compute the first-order corrections for a harmonic oscilatorwith applied small perturbation W = ax3.
〈n|ax3|n + 3〉 = a ·
√(n + 1)(n + 2)(n + 3)
(2α)3
〈n|ax3|n + 1〉 = 3a ·
√(n + 1)3
(2α)3
〈n|ax3|n − 1〉 = 3a ·
√n3
(2α)3
〈n|ax3|n − 3〉 = a ·
√n(n − 1)(n − 2)
(2α)3
Energy differences
m En − Em
n+3 −3~ωn+1 −~ωn-1 ~ωn-3 3~ω
⇒ ψ1n =
a
2~ωα
[1
3
√n(n − 1)(n − 2)
2αψ0
n−3 + 3n
√n
2αψ0
n−1
−3(n + 1)
√n + 1
2αψ0
n+1 −1
3
√(n + 1)(n + 2)(n + 3)
2αψ0
n+3
]
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Making 〈ψ0n|/
(λ2), using the normalization property of ψ0n and orthogonality
between ψ0n and ψ1
n, we get
Second-order correction to theenergy
E 2n =
∑m 6=n
|〈ψ0m|H ′|ψ0
n〉|2
E 0n − E 0
m
For calculationdetails, see Refs[2], [3] and [4].
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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 we seek the second-order correction to the wave function.(λ2) and ψ1
n =∑m 6=n
cmnψ0m give
Second-order correction to the wave function
ψ2n =
∑m 6=n
[− 〈ψ
0n|H ′|ψ0
n〉〈ψ0m|H ′|ψ0
n〉(E 0
n − E 0m)2
+∑k 6=n
〈ψ0m|H ′|ψ0
k〉〈ψ0k |H ′|ψ0
n〉(E 0
n − E 0m)(E 0
n − E 0k )
]ψ0
m
For calculationdetails, see Refs[2], [3] and [4].
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
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 formulationExample: Two-dimensional harmonic oscilator
Symmetry Degeneracy Perturbation
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Symmetry Degeneracy Perturbation
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
A question
What’s wrong with
E 2n =
∑m 6=n
|〈ψ0m|H ′|ψ0
n〉|2
E 0n − E 0
m, m, n ≤ q
if unperturbed eigenstates are degenerate E 01 = E 0
2 = · · · = E 0q ?
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Matrix H ′ in basis B
H ′ =
H ′11 H ′1,q+1 . . .. . . 0
H ′q/2,q/2
0 . . .
H ′qqH ′q+1,1
...
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
For n = 1,
q∑m=1
(H ′pm − E ′nδpm)anm = 0 appear as
H ′11 − E ′1 H ′12 H ′13 . . . H ′1q
H ′21 H ′22 − E ′1 H ′23 . . . H ′2q...
......
......
H ′q1
a11
a12
...a1q
= 0
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
The q roots of secular equation det∣∣H ′ − E ′nI
∣∣ = 0 are the diagonal elements ofthe submatrix of H ′.
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
![Page 53: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/53.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H ′
H0ψ0n = E 0
nψ0n
E 01 q-fold degenerate
m1-
Construct
ϕn =
q∑i=1
anmψ0m
which diagonalizes submatrixof H ′:〈ϕn|H ′|ϕk〉 = E ′δnk
m2 ����
���Nondegenerateperturbation theorywith basis B
q∑(H ′ − E ′δ)a = 0
m3- det
∣∣H ′ − E ′nI∣∣ = 0
m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{ϕn}Gives new basis B
m7� {ani} E ′1,E
′2, . . . ,E
′q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
ψn = ϕn +λϕ1n +λ2ϕ2
n + . . . n ≤ qψn = ψ0
n +λψ1n +λ2ψ2
n + . . . n > qEn = E 0
n +λE ′1n +λ2E ′2n + . . . n ≤ q (E 01 = · · · = E 0
q )
En = E 0n + λE 1
n + λ2E 2n + . . . n > q
E ′n = 〈ϕn|H ′|ϕn〉 n ≤ qE 1n = 〈ψ0
n|H ′|ψ0n〉 n > q
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
ψn = ϕn +λϕ1n +λ2ϕ2
n + . . . n ≤ qψn = ψ0
n +λψ1n +λ2ψ2
n + . . . n > qEn = E 0
n +λE ′1n +λ2E ′2n + . . . n ≤ q (E 01 = · · · = E 0
q )
En = E 0n + λE 1
n + λ2E 2n + . . . n > q
E ′n = 〈ϕn|H ′|ϕn〉 n ≤ qE 1n = 〈ψ0
n|H ′|ψ0n〉 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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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Two-dimensional harmonic oscilator Hamiltonian:
H0 =p2x + p2
y
2m+
K
2(x2 + y 2)
ψnp = ϕn(x)ϕp(y)� |np〉
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Two-dimensional harmonicoscilator Hamiltonian:
H0 =p2x + p2
y
2m+
K
2(x2 + y 2)
ψnp = ϕn(x)ϕp(y)� |np〉
Enp = ~ω(n + p + 1) is(n + p + 1)-fold degenerate.What’s the degeneracy of |01〉state?
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Two-dimensional harmonicoscilator Hamiltonian:
H0 =p2x + p2
y
2m+
K
2(x2 + y 2)
ψnp = ϕn(x)ϕp(y)� |np〉
Enp = ~ω(n + p + 1) is(n + p + 1)-fold degenerate.What’s the degeneracy of |01〉state?E10 = E01 = 2~ω0.
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Now, turn on the perturbation: H ′ = K ′xy
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Now, turn on the perturbation: H ′ = K ′xy
find w.f. which diagonalize H ′
ϕ1 = aψ10 + bψ01
ϕ2 = a′ψ10 + b′ψ01
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Now, turn on the perturbation: H ′ = K ′xy
find w.f. which diagonalize H ′
ϕ1 = aψ10 + bψ01
ϕ2 = a′ψ10 + b′ψ01
calculate the elements of submatrix of H ′ in the basis {ψ10, ψ01}
H ′ = K ′(〈10|xy |10〉 〈10|xy |01〉〈01|xy |10〉 〈01|xy |01〉
)= E
(0 11 0
)
E =K ′
2β2, β2 =
mω0
~
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
Now, turn on the perturbation: H ′ = K ′xy
find w.f. which diagonalize H ′
ϕ1 = aψ10 + bψ01
ϕ2 = a′ψ10 + b′ψ01
calculate the elements of submatrix of H ′ in the basis {ψ10, ψ01}
H ′ = K ′(〈10|xy |10〉 〈10|xy |01〉〈01|xy |10〉 〈01|xy |01〉
)= E
(0 11 0
)
E =K ′
2β2, β2 =
mω0
~solve the secular equation
∣∣∣∣ −E ′ EE −E ′
∣∣∣∣ = 0⇒ E ′ = ±EE10
E+ = E10 + E
E− = E10 − E
���
@@@
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
obtain the new w.f. from(−E ′ EE −E ′
)(ab
)= 0
=⇒E ′ = +E ϕ1 = 1√
2(ψ10 + ψ01)
E ′ = −E ϕ2 = 1√2(ψ10 − ψ01)
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
obtain the new w.f. from(−E ′ EE −E ′
)(ab
)= 0
=⇒E ′ = +E ϕ1 = 1√
2(ψ10 + ψ01)
E ′ = −E ϕ2 = 1√2(ψ10 − ψ01)
HW
How does the threefold-degenerate energy
E = 3~ω0
of the two-dimensional harmonic oscilator separate due to the perturbation
H ′ = K ′xy ?
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
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
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
Problem
If the system is initially in H0, what is the probability that, after time t,transition to another state (of H0) occurs?
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
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) + λH ′(~r , t)
ψn(~r , t) = ϕn(~r)e−iωt
H0ϕn = E 0nϕn
Ψ(~r , t) =∑n
cn(t)ψn(~r , t) , t > 0
i~∂Ψ
∂t= (H0 + λH ′)Ψ
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
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) + λH(~r , t)
ψn(~r , t) = ϕn(~r)e−iωt
H0ϕn = E 0nϕn
Ψ(~r , t) =∑n
cn(t)ψn(~r , t) , t > 0
i~∂Ψ
∂t= (H0 + λH ′)Ψ
Can you remember the meaning of these coefficients?
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
Inserting Ψ(~r , t) and cn(t) = c0n + λc1
n (t) + λ2c2n (t) + . . . into time-dependent
S.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 ψl to stateψk at time t
Pl→k = Plk = |cn|2 =
∣∣∣∣H′kl~∣∣∣∣2∣∣∣∣ ∫ t
−∞e iωkl t
′f (t′)dt′
∣∣∣∣2
For calculation details, see Refs [2] and [3].
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
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
![Page 73: Perturbation theory - Quantum mechanics 2 - Lecture 2](https://reader035.vdocument.in/reader035/viewer/2022071602/613d5d53736caf36b75c7510/html5/thumbnails/73.jpg)
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
Literature
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, Shaum’s Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.
Igor Lukacevic Perturbation theory