Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Perturbation theory - applications
Quantum mechanics 2 - Lecture 3
Igor Lukačević
UJJS, Dept. of Physics, Osijek
23. listopada 2012.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
Hyperfine structure
Harmonic perturbation
Literature
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Energy level spectra of hydrogen in an electric field near n=15
for m=0.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Johannes Stark (15 April 1874 - 21 June
1957) was a German physicist, and
Energy level spectra of hydrogen in an electric field near n=15
Physics Nobel Prize laureate (1919).
for m=0.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Johannes Stark (15 April 1874 - 21 June
1957) was a German physicist, and
Energy level spectra of hydrogen in an electric field near n=15
Physics Nobel Prize laureate (1919).
for m=0.
Antonino Lo Surdo - independently (same year, 1913)
in Italy: Stark-Lo Surdo effect
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Theoretical explanations
1
W. Voigt (1901) - classical machanical calculations - few orders of
magnitude too low
2
P. Epstein and K. Schwarzschild (1916) - Bohr-Sommerfeld quantum
theory
3
H. Kramers (1920) - spectral transitions line intensities
4
W. Pauli (1926) - Heisenberg’s matrix mechanics
5
E. Schrödinger (1926) - perturbation theory
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Theoretical explanations
1
W. Voigt (1901) - classical machanical calculations - few orders of
magnitude too low
2
P. Epstein and K. Schwarzschild (1916) - Bohr-Sommerfeld quantum
theory
3
H. Kramers (1920) - spectral transitions line intensities
4
W. Pauli (1926) - Heisenberg’s matrix mechanics
5
E. Schrödinger (1926) - perturbation theory
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Theoretical explanations
1
W. Voigt (1901) - classical machanical calculations - few orders of
magnitude too low
2
P. Epstein and K. Schwarzschild (1916) - Bohr-Sommerfeld quantum
theory
3
H. Kramers (1920) - spectral transitions line intensities
4
W. Pauli (1926) - Heisenberg’s matrix mechanics
5
E. Schrödinger (1926) - perturbation theory
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Theoretical explanations
1
W. Voigt (1901) - classical machanical calculations - few orders of
magnitude too low
2
P. Epstein and K. Schwarzschild (1916) - Bohr-Sommerfeld quantum
theory
3
H. Kramers (1920) - spectral transitions line intensities
4
W. Pauli (1926) - Heisenberg’s matrix mechanics
5
E. Schrödinger (1926) - perturbation theory
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Historical overview
Theoretical explanations
1
W. Voigt (1901) - classical machanical calculations - few orders of
magnitude too low
2
P. Epstein and K. Schwarzschild (1916) - Bohr-Sommerfeld quantum
theory
3
H. Kramers (1920) - spectral transitions line intensities
4
W. Pauli (1926) - Heisenberg’s matrix mechanics
5
E. Schrödinger (1926) - perturbation theory
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Consider an one-electron atom in a constant, uniform electric field E in z
direction
=⇒ H
=
=
H
Igor Lukačević
Perturbation theory - applications
0
=
H0 + H 0
L2
Ze 2
pr2
+
−
− eEz ,
2
2m
2mr
r
−eEz = −eEr cos ϑ
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Consider an one-electron atom in a constant, uniform electric field E in z
direction
=⇒ H
=
=
H0
=
H0 + H 0
pr2
L2
Ze 2
+
−
− eEz ,
2
2m
2mr
r
−eEz = −eEr cos ϑ
Can you remember what’s the degeneracy of energy levels in this kind of
system?
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Consider an one-electron atom in a constant, uniform electric field E in z
direction
=⇒ H
=
=
H0
=
H0 + H 0
pr2
L2
Ze 2
+
−
− eEz ,
2
2m
2mr
r
−eEz = −eEr cos ϑ
Can you remember what’s the degeneracy of energy levels in this kind of
system?
⇒ n2 -fold
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Let us consider n = 2 states:
|200i, |211i, |210i, |21 − 1i
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Let us consider n = 2 states:
|200i, |211i, |210i, |21 − 1i
Secular equation:
h200|H 0 |200i − E 0
h211|H 0 |200i
h210|H 0 |200i
h21 − 1|H 0 |200i
Igor Lukačević
Perturbation theory - applications
h200|H 0 |211i
h211|H 0 |211i − E 0
h210|H 0 |211i
h21 − 1|H 0 |211i
h200|H 0 |210i
h211|H 0 |210i
h210|H 0 |210i − E 0
h21 − 1|H 0 |210i
h200|H 0 |21 − 1i
h211|H 0 |21 − 1i
h210|H 0 |21 − 1i
h21 − 1|H 0 |21 − 1i − E 0
=0
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Let us consider n = 2 states:
|200i, |211i, |210i, |21 − 1i
Secular equation:
h200|H 0 |200i − E 0
h211|H 0 |200i
h210|H 0 |200i
h21 − 1|H 0 |200i
h200|H 0 |211i
h211|H 0 |211i − E 0
h210|H 0 |211i
h21 − 1|H 0 |211i
h200|H 0 |210i
h211|H 0 |210i
h210|H 0 |210i − E 0
h21 − 1|H 0 |210i
h200|H 0 |21 − 1i
h211|H 0 |21 − 1i
h210|H 0 |21 − 1i
h21 − 1|H 0 |21 − 1i − E 0
=0
A question
Can you “guess” which of these matrix elements are non-zero?
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
Let us consider n = 2 states:
|200i, |211i, |210i, |21 − 1i
Secular equation:
h200|H 0 |200i − E 0
h211|H 0 |200i
h210|H0 |200i
h21 − 1|H 0 |200i
h200|H 0 |211i
h211|H 0 |211i − E 0
h210|H 0 |211i
h21 − 1|H 0 |211i
h200|H0 |210i
h211|H 0 |210i
h210|H 0 |210i − E 0
h21 − 1|H 0 |210i
h200|H 0 |21 − 1i
h211|H 0 |21 − 1i
h210|H 0 |21 − 1i
h21 − 1|H 0 |21 − 1i − E 0
=0
A question
Can you “guess” which of these matrix elements are non-zero?
h210|H 0 |200i = h200|H 0 |210i = −3|e|Ea0 = −E
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
So, secular equation has 4 roots:
E 0 = 0, 0, +E, −E
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
So, secular equation has 4 roots:
E 0 = 0, 0, +E, −E
E20 + E
E20
E2
n = 2 ⇒ n2 = 4 ???
@
@
@
Igor Lukačević
Perturbation theory - applications
E20 − E
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Theory
From matrix equations for expansion coefficients we get
1
E2+ = E20 + E 99K ϕ+ = √ (|200i − |210i)
2
1
E2− = E20 − E 99K ϕ− = √ (|200i + |210i)
2
ϕ = |211i
(0)
E20 = E2
Igor Lukačević
Perturbation theory - applications
*
H
HH
j ϕ = |21 − 1i
H
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
General features
Hamiltonian:
H=−
Igor Lukačević
Perturbation theory - applications
~2
e2
∆−k
2m
r
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
General features
Hamiltonian:
H=−
~2
e2
∆−k
2m
r
Hierarchy of corrections to the Bohr energies of H-atom
Correction
Order
Bohr energies
Fine structure
Lamb shift
Hyperfine splitting
α2 mc 2
α4 mc 2
α5 mc 2
(m/mp )α4 mc 2
Igor Lukačević
Perturbation theory - applications
α=
e2
4π0 ~c
≈
1
137.036
fine structure factor
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
General features
Fine structure
relativistic correction
spin-orbit coupling
Hierarchy of corrections to the Bohr energies of H-atom
Correction
Order
Bohr energies
Fine structure
Lamb shift
Hyperfine splitting
α2 mc 2
α4 mc 2
α5 mc 2
(m/mp )α4 mc 2
Igor Lukačević
Perturbation theory - applications
α=
e2
4π0 ~c
≈
1
137.036
fine structure factor
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
General features
comes from the quantization of
electric field
Willis Lamb - Nobel Prize
(1955)
H. Bethe - explained the Lamb
shift in hydrogen (1947)
Hierarchy of corrections to the Bohr energies of H-atom
Correction
Order
Bohr energies
Fine structure
Lamb shift
Hyperfine splitting
α2 mc 2
α4 mc 2
α5 mc 2
(m/mp )α4 mc 2
Igor Lukačević
Perturbation theory - applications
α=
e2
4π0 ~c
≈
1
137.036
fine structure factor
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
The relativistic correction
H=−
~2
e2
∆−k
2m
r
This, actually, comes from nonrelativistic kinetic energy.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
The relativistic correction
H=−
~2
e2
∆−k
2m
r
This, actually, comes from nonrelativistic kinetic energy.
On contrary, if we take the relativistic kinetic energy and expand in
T =
p
mc
p2
p4
− 3 2 +···
2m | 8m
{z c }
Hr0
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
The relativistic correction
H=−
~2
e2
∆−k
2m
r
This, actually, comes from nonrelativistic kinetic energy.
On contrary, if we take the relativistic kinetic energy and expand in
T =
p
mc
p2
p4
− 3 2 +···
2m | 8m
{z c }
Hr0
A question
p
Is
appropriate for this expansion?
mc
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
The relativistic correction
H=−
~2
e2
∆−k
2m
r
This, actually, comes from nonrelativistic kinetic energy.
On contrary, if we take the relativistic kinetic energy and expand in
T =
p
mc
p2
p4
− 3 2 +···
2m | 8m
{z c }
Hr0
A question
p
Is
appropriate for this expansion?
mc
Yes...Telec in H ∼ 10 eV, Eelec rest ∼ 511000 eV.
7−→ Hr0 lowest-order relativistic correction to the Hamiltonian.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
The relativistic correction
First-order corrections to energy En
Er1
=
=
En
mc 2
Igor Lukačević
Perturbation theory - applications
=
1
hHr0 i = − 3 2 hψ|p 4 |ψi
8m
c
"
#
2
(En )
4n
−
−3
2mc 2 l + 21
2 · 10−3
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory
~
H = −~
µ·B
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
The magnetic field of the proton
B = µ2r0 I
L = rmv
Igor Lukačević
Perturbation theory - applications
~ =
B
I
e ~
L
4π0 mc 2 r 3
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
The magnetic dipole moment of the electron
µ = IA =
L = Iω =
qr 2 π
T
2πmr 2
T
)
µ
~e = −
e~
S
m
Calculation details can be found in
Ref. [5].
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Now let’s get back to the Hamiltonian:
2 e
1 ~ ~
H=
S ·L
4π0 m2 c 2 r 3
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Now let’s get back to the Hamiltonian:
2 X
X
e XX1~~
X
S
·
L
H=
X
3
m2 c 2 rX
X
0
4π
X
Spin-orbit interaction
0
Hso
Igor Lukačević
Perturbation theory - applications
=
e2
8π0
1 ~ ~
S ·L
m2 c 2 r 3
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Now let’s get back to the Hamiltonian:
2 X
X
e XX1~~
X
H=
S
·
L
X
2
2
3
0 m c rXX
4π
X
Spin-orbit interaction
0
Hso
=
e2
8π0
A problem
H does not commute with ~L and ~
S
anymore.
Igor Lukačević
Perturbation theory - applications
1 ~ ~
S ·L
m2 c 2 r 3
A solution
But it does with L2 , S 2 , J 2 and Jz .
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Having this in mind, we get
First-order energy corrections due to SO
(En )2 n[ j(j + 1) − l(l + 1) − 3/4]
1
ESO
=
mc 2
l(l + 1/2)(l + 1)
Calculation details can be found in Ref [2].
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Having this in mind, we get
First-order energy corrections due to SO
1
ESO
=
(En )2
mc 2
n[ j(j + 1) − l(l + 1) − 3/4]
l(l + 1/2)(l + 1)
Remember
First-order corrections due to relativistic effects
"
#
(En )2
4n
Er1 = −
−
3
2mc 2 l + 12
Igor Lukačević
Perturbation theory - applications
Notice that they have
the same order of
magnitude:
(En )2
mc 2
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Together they give
First-order energy correction due to fine structure
"
#
(En )2
4n
1
Efs =
3−
2mc 2
j + 12
Which in turn gives
Energy levels of hydrogen with fine structure
13.6 eV
α2
n
3
Enj = −
1
+
−
n2
n2 j + 1/2
4
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
Together they give
First-order energy correction due to fine structure
"
#
(En )2
4n
Efs1 =
3
−
2mc 2
j + 12
Which in turn gives
Energy levels of hydrogen with fine structure
α2
n
13.6 eV
3
Enj = −
1
+
−
n2
n2 j + 1/2
4
HW
Solve Problem 6.18 from Ref. [2].
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Spin-orbit coupling
Theory (cont.)
∆λfs = 0.016 nm
⇒ ∆Efs = 4.5 · 10−5 eV.
Energies are not in scale.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
discovers the splitting of
spectral lines in magnetic field
Noble prize (1902)
with Lorentz predicted the
existence of “electron”
P. Zeeman, The Effect of
Magnetisation on the Nature of
Light Emitted by a Substance,
Nature 55, 347, 1897.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Applications
1
magnetic field measurements (astronomy)
2
NMR spectroscopy
3
ESR spectroscopy
4
MRI
5
Mösbauer spectroscopy
6
AAS
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
HZ0
Harmonic perturbation
Literature
~ ext
−(~
µL + µ
~S) · B
e~
µ
~S = − S
m
e ~
µ
~L = −
L
2m
e ~
~ ext
L + 2~
S ·B
⇒ HZ0 =
2m
Igor Lukačević
Perturbation theory - applications
=
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Critical battle Bext vs. Bint
Field strength
Dominates
Small pert.
Bext Bint
Bext ≈ Bint
Bint Bext
FS
Zeeman
Zeeman
FS
Bint ≈ 14 T
BEarth ≈ 30 µT
BNMR ≈ 12 T
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
Bext Bint
First-order energy Zeeman corrections
EZ1 = hnljmj |HZ0 |nljmj i =
Igor Lukačević
Perturbation theory - applications
D
E
e ~
S
Bext · ~L + 2~
2m
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
Bext Bint
First-order energy Zeeman corrections
EZ1 = hnljmj |HZ0 |nljmj i =
D
E
e ~
Bext · ~L + 2~
S
2m
It can be prooved that
D
E j(j + 1) − l(l + 1) + 3/4 D~ E
~L + 2~
S = 1+
J
2j(j + 1)
|
{z
}
gJ → Landé g-factor
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
Bext Bint
First-order energy Zeeman corrections
EZ1 = hnljmj |HZ0 |nljmj i =
D
E
e ~
Bext · ~L + 2~
S
2m
It can be prooved that
D
E j(j + 1) − l(l + 1) + 3/4 D~ E
~L + 2~
S = 1+
J
2j(j + 1)
|
{z
}
gJ → Landé g-factor
A question
D E
~ ext to be aligned with z-axis?
Can you remember the value of ~J if we choose B
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
Bext Bint
First-order energy Zeeman corrections
EZ1 = hnljmj |HZ0 |nljmj i =
D
E
e ~
Bext · ~L + 2~
S
2m
It can be prooved that
D
E j(j + 1) − l(l + 1) + 3/4 D~ E
~L + 2~
S = 1+
J
2j(j + 1)
|
{z
}
gJ → Landé g-factor
A question
D E
~ ext to be aligned with z-axis
Can you remember the value of ~J if we choose B
D E
? ~J = mj ~
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
So,
First-order energy Zeeman corrections
EZ1 = µB gJ Bext mj ,
Igor Lukačević
Perturbation theory - applications
µB =
e~
= 5.788 · 10−5 eV/T
2m
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
So,
First-order energy Zeeman corrections
EZ1 = µB gJ Bext mj ,
µB =
e~
= 5.788 · 10−5 eV/T
2m
Total energy due to SO and Zeeman
α2
E (n = 1, l = 0, j = 1/2) = −13.6 eV 1 +
± µB Bext
4
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Weak-field Zeeman effect
Total energy due to SO and Zeeman
α2
E (n = 1, l = 0, j = 1/2) = −13.6 eV 1 +
± µB Bext
4
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Strong-field Zeeman effect
~
B=Bẑ
Bext Bint −−−→ HZ0 =
e
Bext (Lz + 2Sz )
2m
First-order energy Zeeman corrections
EZ1 = µB Bext (ml + 2ms )
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Strong-field Zeeman effect
~
B=Bẑ
Bext Bint −−−→ HZ0 =
e
Bext (Lz + 2Sz )
2m
First-order energy Zeeman corrections
EZ1 = µB Bext (ml + 2ms )
First-order energy FS correction
Efs1 =
13.6 eV 2
α
n3
l(l + 1) − ml ms
3
−
4n
l(l + 1/2)(l + 1)
Calculation details can be found in Ref. [2].
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The fine structure of hydrogen
The Stark effect
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Strong-field Zeeman effect
The total energy
E
=
13.6 eV 13.6 eV 2
−
+
α
n2 }
n3
| {z
|
Bohr energy
+
l(l + 1) − ml ms
3
−
4n
l(l + 1/2)(l + 1)
{z
}
FS correction
µB Bext (ml + 2ms )
|
{z
}
strong-field Zeeman correction
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Intermediate-field Zeeman effect
Bext ≈ Bint
H 0 = HZ0 + Hfs0
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The fine structure of hydrogen
The Stark effect
The Zeeman effect
Harmonic perturbation
Hyperfine structure
Literature
Intermediate-field Zeeman effect
H 0 = HZ0 + Hfs0
Bext ≈ Bint







0
Hsub = 






5γ − β
0
0
0
0
5γ + β
0
0
0
0
γ − 2β
0
0
0
0
γ + 2β
0
0
0
0
0
0
0
0
γ − 23 β
0
0
0
0
2
0
0
0
0
0
0
0
0
where
γ≡
Igor Lukačević
Perturbation theory - applications
α 2
8
· 13.6 eV ,
√
β
3
0
0
0
0
0
0
√
2
3
0
0
0
0
β
5γ − 13 β
0
0
0
0
γ + 32 β
√
2
β
3
0
0
0
0

0
0
√
3
2
β
5γ + 13 β













β ≡ µB Bext .
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Intermediate-field Zeeman effect
Energy levels for n = 2 states of hydrogen, with FS
and Zeeman splitting
1
=
E2 − 5γ + β
2
=
E2 − 5γ − β
3
=
E2 − γ + 2β
4
=
E2 − γ − 2β
5
=
E2 − 3γ + β/2
q
+ 4γ 2 + (2/3)γβ + β 2 /4
6
=
E2 − 3γ + β/2
q
4γ 2 + (2/3)γβ + β 2 /4
−
7
=
E2 − 3γ − β/2
q
+ 4γ 2 − (2/3)γβ + β 2 /4
8
=
E2 − 3γ − β/2
q
− 4γ 2 − (2/3)γβ + β 2 /4
Zeeman splitting of the n = 2 states of
hydrogen, in all regimes.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Anomalous Zeeman effect
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
South Africa’s KAT-7 telescope.
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
NGC 3109 (a small spiral galaxy), about
4.3 million light-years away from Earth,
located in the constellation of Hydra.
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
µ
~p =
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
gp e ~
Sp ,
2mp
µ
~e = −
Hyperfine structure
Harmonic perturbation
Literature
e ~
Se
me
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
µ
~p =
The Zeeman effect
gp e ~
Sp ,
2mp
µ
~e = −
Hyperfine structure
Harmonic perturbation
Literature
e ~
Se
me
A question
Whose dipole moment is larger?
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
First-order energy HF correction
µ0 gp e 2 D~ ~ E
Sp · Se
Ehf1 =
3πmp me a3
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Calculation details can be
found in Ref. [2].
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
First-order energy HF correction
Ehf1 =
E
µ0 g p e 2 D
S̃p · S̃e
3
3πmp me a
spin-spin coupling
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
First-order energy HF correction
Ehf1 =
µ0 gp e 2 D~ ~ E
Sp · Se
3πmp me a3
A problem
~
Sp and ~
Se do not commute with the
Hamiltonian.
Igor Lukačević
Perturbation theory - applications
Solution
But ~
S =~
Sp + ~
Se does.
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
First-order energy HF correction
Ehf1 =
µ0 gp e 2 D~ ~ E
Sp · Se
3πmp me a3
A problem
~
Sp and ~
Se do not commute with the
Hamiltonian.
Solution
But ~
S =~
Sp + ~
Se does.
Don’t forget...
Hydrogen ground state has two possibilities
Igor Lukačević
Perturbation theory - applications
triplet
singlet
S =1
S =0
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Thus,
First-order energy HF correction
Ehf1 =
Igor Lukačević
Perturbation theory - applications
4gp ~4
3mp me2 c 2 a4
+1/4 ,
−3/4 ,
triplet
singlet
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Thus,
First-order energy HF correction
Ehf1 =
4gp ~4
3mp me2 c 2 a4
+1/4 ,
−3/4 ,
triplet
singlet
Energy gap:
∆E = 5.88 · 10−6 eV
(Egs = −13.6 eV)
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
Igor Lukačević
Perturbation theory - applications
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Example: an atom in a (weak) electromagnetic field.
0,
t<0
H 0 (~r , t) =
2H 0 (~r ) cos ωt , t ≥ 0
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Example: an atom in a
(weak) electromagnetic field.
0,
t<0
H 0 (~r , t) =
2H 0 (~r ) cos ωt , t ≥ 0
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
ck (t) = −
2iH0kl
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
Calculation details can be found in Ref. [1].
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
ck (t) = −
2iH0kl
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
A question
For which ω’s we have dominant terms in ck ?
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
ck (t) = −
2iH0kl
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
A question
For which ω’s we have dominant terms in ck ?
ω ≈ ±ωkl
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
ck (t) = −
2iH0kl
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
Resonant frequencies correspond to:
Igor Lukačević
Perturbation theory - applications
ω ≈ +ωkl
ω ≈ −ωkl
99K
99K
Ek > El
Ek < El
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
ck (t) = −
2iH0kl
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
Resonant frequencies correspond to:
ω ≈ +ωkl
ω ≈ −ωkl
99K
99K
Ek > El
Ek < El
Dominant transition process:
absorption (Ek > El ):
Ek = El + ~ω
(stimulated) emission
(Ek < El ): Ek = El − ~ω
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Expansion coefficients
2iH0kl
ck (t) = −
~
e i(ωkl −ω)t/2 sin(ωkl − ω)t/2
e i(ωkl +ω)t/2 sin(ωkl + ω)t/2
+
ωkl − ω
ωkl + ω
Resonant frequencies correspond to:
ω ≈ +ωkl
ω ≈ −ωkl
99K
99K
Ek > El
Ek < El
absorption (Ek > El ):
Ek = El + ~ω
(stimulated) emission
(Ek < El ): Ek = El − ~ω
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Assume ωkl + ω |ωkl − ω|
A question
Is this a valid assumption?
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Assume ωkl + ω |ωkl − ω|
The transition probability l→k
Plk = |ck |2 =
Igor Lukačević
Perturbation theory - applications
4|H0kl |2
2 1
sin
(ω
−
ω)t
kl
~2 (ωkl − ω)2
2
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Assume ωkl + ω |ωkl − ω|
The transition probability l→k
Plk = |ck |2 =
Igor Lukačević
Perturbation theory - applications
4|H0kl |2
2 1
sin
(ω
−
ω)t
kl
~2 (ωkl − ω)2
2
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Assume ωkl + ω |ωkl − ω|
The transition probability l→k
Plk = |ck |2 =
Igor Lukačević
Perturbation theory - applications
4|H0kl |2
2 1
sin
(ω
−
ω)t
kl
~2 (ωkl − ω)2
2
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Transition probability
Assume ωkl + ω |ωkl − ω|
The transition probability l→k
Plk = |ck |2 =
4|H0kl |2
2 1
sin
(ω
−
ω)t
kl
~2 (ωkl − ω)2
2
Transition probability rate
wlk =
Igor Lukačević
Perturbation theory - applications
2π 0 2
|Hkl | δ(ωkl ∓ ω)
~2
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Emission and absorption od radiation
Einstein’s approach
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Emission and absorption od radiation
Wkl = [Akl + Bkl u(ν)] Nk = wkl Nk
Akl
spontaneous emission
Bkl
stimulated emission
Wlk = Blk u(ν)Nl = wlk Nl
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Emission and absorption od radiation
In equilibrium: Bkl = Blk
(proof in Ref. [1])
Planck’s radiation law:
u(ν) =
8πν 2
1
c 3 e khν
BT − 1
A
8πν 2
=
B
c3
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Laser
hν = Ek − El
Boltzmann’s distribution:
Nk
− hν
= e kB T
Nl
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Contents
1
The Stark effect
Historical overview
Theory
2
The fine structure of hydrogen
General features
The relativistic correction
Spin-orbit coupling
3
The Zeeman effect
4
Hyperfine structure
5
Harmonic perturbation
Transition probability
Emission and absorption od radiation
Laser
6
Literature
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek
Contents
The Stark effect
The fine structure of hydrogen
The Zeeman effect
Hyperfine structure
Harmonic perturbation
Literature
Literature
1
R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, San
Francisco, 2003.
2
D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson
Education, Inc., Upper Saddle River, NJ, 2005.
3
I. Supek, Teorijska fizika i struktura materije, II. dio, Školska knjiga,
Zagreb, 1989.
4
Y. Peleg, R. Pnini, E. Zaarur, Shaum’s Outline of Theory and Problems of
Quantum Mechanics, McGraw-Hill, 1998.
5
V. Knapp, P. Colić, Uvod u električna i magnetna svojstva materijala,
Školska knjiga, Zagreb, 1997.
Igor Lukačević
Perturbation theory - applications
UJJS, Dept. of Physics, Osijek