perturbation theory - quantum mechanics 2 - lecture...
TRANSCRIPT
-
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
-
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
-
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
-
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
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Do you remember this?
H00n = E0n
0n
0n|0m = nm complete set
Igor Lukacevic Perturbation theory
-
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
-
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
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
AssumeH = H0 + H
Igor Lukacevic Perturbation theory
-
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
-
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
-
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
-
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 + eE0zuniform 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 + 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(2nx/a)]
model H0 = p2/2m
Igor Lukacevic Perturbation theory
-
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
-
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
n = 0n +
1n +
22n + . . . ,
En = E0n + E
1n +
2E 2n + . . .
Igor Lukacevic Perturbation theory
-
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
n = 0n +
1n +
22n + . . . ,
En = E0n + E
1n +
2E 2n + . . .
and sort
(0) . . . H00n = E0n
0n ,
(1) . . . H01n + H0n = E
0n
1n + E
1n
0n ,
(2) . . . H02n + H1n = E
0n
2n + E
1n
1n + E
2n
0n ,
...
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Making 0n|/
(1) and using the normalization property of 0n, we get
First-order correction to the energy
E 1n = 0n|H |0n
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 formulationFirst-order theorySecond-order theory
First-order correction to the energy
E 1n = 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
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
First-order correction to the energy
E 1n = 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)
Igor Lukacevic Perturbation theory
-
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 = 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 = V0
Igor Lukacevic Perturbation theory
-
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 = 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 = V0
First-order correction:E 1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E 0n + V0
Igor Lukacevic Perturbation theory
-
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 floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2
asin(n
ax)
Perturbation Hamiltonian: H = V0
First-order correction:E 1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E 0n + V0Compare 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 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 floorof the well is raised by an constant value V0.
Unperturbed w.f.: 0n(x) =
2
asin(n
ax)
Perturbation Hamiltonian: H = V0
First-order correction:E 1n = 0n|V0|0n = V00n|0n = V0
corrected energy levels: En E 0n + V0Compare 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 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
-
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 =2V0a
a/20
sin2(n
ax)dx =
V02
Igor Lukacevic Perturbation theory
-
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 =2V0a
a/20
sin2(n
ax)dx =
V02
HW. Compare this result with an exact one.
Igor Lukacevic Perturbation theory
-
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.(1) and 1n =
m 6=n
cmn0m give
First-order correction to the wavefunction
1n =m 6=n
0m|H |0n(E 0n E 0m)
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 formulationFirst-order theorySecond-order theory
First-order correction to the wavefunction
1n =m 6=n
0m|H |0n(E 0n E 0m)
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 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
-
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
-
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
)~
Igor Lukacevic Perturbation theory
-
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
)~
H0 eigenfunctions:
0n(x) =
1
2nn!
e
x2
2 Hn(x)
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 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
)~
H0 eigenfunctions:
0n(x) =
1
2nn!
e
x2
2 Hn(x)
E 1n = 0n|ax3|0n = 0
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 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
)~
H0 eigenfunctions:
0n(x) =
1
2nn!
e
x2
2 Hn(x)
E 1n = 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 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
)~
H0 eigenfunctions:
0n(x) =
1
2nn!
e
x2
2 Hn(x)
E 1n = 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 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
)~
H0 eigenfunctions:
0n(x) =
1
2nn!
e
x2
2 Hn(x)
E 1n = 0n|ax3|0n = 0For 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 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)
(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
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 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)
(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 3~n+1 ~n-1 ~n-3 3~
Igor Lukacevic Perturbation theory
-
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)
(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 3~n+1 ~n-1 ~n-3 3~
1n =a
2~
[1
3
n(n 1)(n 2)
20n3 + 3n
n
20n1
3(n + 1)
n + 1
20n+1
1
3
(n + 1)(n + 2)(n + 3)
20n+3
]
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationFirst-order theorySecond-order theory
Making 0n|/
(2), using the normalization property of 0n and orthogonality
between 0n and 1n, we get
Second-order correction to theenergy
E 2n =m 6=n
|0m|H |0n|2
E 0n E 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 formulationFirst-order theorySecond-order theory
Now we seek the second-order correction to the wave function.(2) and 1n =
m 6=n
cmn0m give
Second-order correction to the wave function
2n =m 6=n
[
0n|H |0n0m|H |0n
(E 0n E 0m)2
+k 6=n
0m|H |0k0k |H |0n(E 0n E 0m)(E 0n E 0k )
]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
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
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
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
A question
Whats wrong with
E 2n =m 6=n
|0m|H |0n|2
E 0n E 0m, m, n q
if unperturbed eigenstates are degenerate E 01 = E02 = = E 0q ?
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
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 H1,q+1 . . .
. . . 0H q/2,q/2
0 . . .H qq
H q+1,1...
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
For n = 1,
qm=1
(H pm E npm)anm = 0 appear as
H 11 E 1 H 12 H 13 . . . H 1q
H 21 H22 E 1 H 23 . . . H 2q
......
......
...H q1
a11a12...
a1q
= 0
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
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 detH E nI = 0 are the diagonal elements of
the submatrix of H .
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
Literature
General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}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
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General formulationExample: Two-dimensional harmonic oscilator
H = H0 + H
H00n = E
0n
0n
E 01 q-fold degenerate
m1-
Construct
n =
qi=1
anm0m
which diagonalizes submatrixof H :n|H |k = E nk
m2
Nondegenerateperturbation theorywith basis B
q(H E )a = 0
m3- det
H E nI = 0m8 m6 m5 m46
? XXXXX
XXXXXX
Xy
?
{n}Gives new basis B
m7 {ani} E 1,E 2, . . . ,E q
Igor Lukacevic Perturbation theory
-
ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
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General formulationExample: Two-dimensional harmonic oscilator
n = n +1n +
22n + . . . n qn =
0n +
1n +
22n + . . . n > qEn = E
0n +E
1n +
2E 2n + . . . n q (E 01 = = E 0q )
En = E0n + E
1n +
2E 2n + . . . n > qE n = n|H |n n qE 1n = 0n|H |0n n > q
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
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General formulationExample: Two-dimensional harmonic oscilator
n = n +1n +
22n + . . . n qn =
0n +
1n +
22n + . . . n > qEn = E
0n +E
1n +
2E 2n + . . . n q (E 01 = = E 0q )
En = E0n + E
1n +
2E 2n + . . . n > qE n = n|H |n n qE 1n = 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)
Igor Lukacevic Perturbation theory
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ContentsTime-independent nondegenerate perturbation theory
Time-independent degenerate perturbation theoryTime-dependent perturbation theory
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General formulationExample: Two-dimensional harmonic oscilator
Two-dimensional harmonic oscilator Hamiltonian:
H0 =p2x + p
2y
2m+
K
2(x2 + y 2)
np = n(x)p(y) |np
Igor Lukacevic Perturbation theory
-
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 + p
2y
2m+
K
2(x2 + y 2)
np = n(x)p(y) |np
Enp = ~(n + p + 1) is(n + p + 1)-fold degenerate.Whats the degeneracy of |01state?
Igor Lukacevic Perturbation theory
-
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 + p
2y
2m+
K
2(x2 + y 2)
np = n(x)p(y) |np
Enp = ~(n + p + 1) is(n + p + 1)-fold degenerate.Whats the degeneracy of |01state?E10 = E01 = 2~0.
Igor Lukacevic Perturbation theory
-
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
Igor Lukacevic Perturbation theory
-
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
1 = a10 + b01
2 = a10 + b
01
Igor Lukacevic Perturbation theory
-
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
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 = K
22, 2 =
m0~
Igor Lukacevic Perturbation theory
-
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
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 = K
22, 2 =
m0~
solve the secular equation
E EE E = 0 E = E E10
E+ = E10 + E
E = E10 E
@@@
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
obtain the new w.f. from(E EE E
)(ab
)= 0
=E = +E 1 = 12 (10 + 01)E = E 2 = 12 (10 01)
Igor Lukacevic Perturbation theory
-
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 1 = 12 (10 + 01)E = E 2 = 12 (10 01)
HW
How does the threefold-degenerate energy
E = 3~0
of the two-dimensional harmonic oscilator separate due to the perturbation
H = K xy ?
Igor Lukacevic Perturbation theory
-
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
-
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
-
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) + H(~r , t)
n(~r , t) = n(~r)eit
H0n = E0nn
(~r , t) =n
cn(t)n(~r , t) , t > 0
i~t
= (H0 + H)
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) + H(~r , t)
n(~r , t) = n(~r)eit
H0n = E0nn
(~r , t) =n
cn(t)n(~r , t) , t > 0
i~t
= (H0 + H)
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
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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
e ikl tf (t)dt
2
For calculation details, see Refs [2] and [3].
Igor Lukacevic Perturbation theory
-
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
-
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, Shaums Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.
Igor Lukacevic Perturbation theory
ContentsTime-independent nondegenerate perturbation theoryGeneral formulationFirst-order theorySecond-order theory
Time-independent degenerate perturbation theoryGeneral formulationExample: Two-dimensional harmonic oscilator
Time-dependent perturbation theoryLiterature