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ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Relativistic quantum mechanicsQuantum mechanics 2 - Lecture 11
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
January 15, 2013
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Contents
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Non-relativistic physics
E =p2
2m99K
E 7→ E = i~ ∂∂t
~p 7→ p = −i~∇
=⇒ i~∂ψ∂t
= − ~2
2m∆ψ
↓free-particle S.E.
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Non-relativistic physics
E =p2
2m99K
E 7→ E = i~ ∂∂t
~p 7→ p = −i~∇
=⇒ i~∂ψ∂t
= − ~2
2m∆ψ
↓free-particle S.E.
Statistical interpretation of ψ(r, t):
ρ(r, t) = |ψ(r, t)|2
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Relativistic physics
E 2 = m2c4 + c2p2 99K
E 7→ E = i~ ∂∂t
~p 7→ p = −i~∇
=⇒
(i~ ∂∂t
)2
ψ =[m2c4 + c2 (−i~∇)2
]ψ
↓relativistic S.E. (Fock’s equation)
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Relativistic physics
E 2 = m2c4 + c2p2 99K
E 7→ E = i~ ∂∂t
~p 7→ p = −i~∇
=⇒
(i~ ∂∂t
)2
ψ =[m2c4 + c2 (−i~∇)2
]ψ
=⇒(�− κ2
)ψ = 0 Klein-Gordon equation
� = ∆− 1
c2∂2
∂t2
κ =mc
~,
1
κ=
~mc reduced Compton wavelength [3]
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
A question
What is Compton wavelength for an electron?
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Statistical interpretation of ψ in Schrodinger theory:
S.E. ⇒ equation of continuity
∂ρ
∂t+ divj = 0
ρ = ψ∗ψ probability density
j = − ~2im
[ψ∗∇ψ − (∇ψ∗)ψ]
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Statistical interpretation of ψ in Klein-Gordon theory:
K-G equation ⇒ equation of continuity
∂ρ
∂t+ divj = 0
ρ = ψ∗ψ − ψ∗ψj = −c2 [ψ∗∇ψ − (∇ψ∗)ψ]
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Statistical interpretation of ψ in Klein-Gordon theory:
K-G equation ⇒ equation of continuity
∂ρ
∂t+ divj = 0
ρ = ψ∗ψ − ψ∗ψ R 0 problem!
j = −c2 [ψ∗∇ψ − (∇ψ∗)ψ]
Problem
ρ depends on the initial conditions: ψ(0) and ψ(0)
ρ cannot be interpreted as the probability density
Good side
K-G equation describes well the spinless bosons, like π-mesons.
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Contents
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Requirements for the relativistic wave equation
1 keep the statistical interpretation of ψ
2 must be relativistically invariant
3 must be of the 1st order in time variable
4 agrees with the K-G equation in the limit of large quantum numbers
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Requirements for the relativistic wave equation
1 keep the statistical interpretation of ψ
2 must be relativistically invariant
3 must be of the 1st order in time variable
4 agrees with the K-G equation in the limit of large quantum numbers
(2) ⇒ symmetrical in spatialand time derivatives
(3) in analogy with S.E.
⇒ must be linear in spatial derivatives:
H = cα · p + βmc2
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Factorisation of K-G equation gives [5](E − cα · p− βmc2
)(E + cα · p + βmc2
)ψ = 0
Comparisson with K-G equation imposes
β2 = 1 , αkβ + βαk = 0 , (1)α2x = 1 , αxαy + αyαx = 0 , (2)α2y = 1 , αyαz + αzαy = 0 , (3)α2z = 1 , αzαx + αxαz = 0 (4)
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
A task
1 Write down conditions (1)-(4) using anticommutators. (Hint: consult Ref.[1].)
2 Pauli matrices
σx =
[0 11 0
], σy =
[0 −ii 0
], σx =
[1 00 −1
],
satisfy conditions (2)-(4). Please, verify if they satisfy condition (1).
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Dirac’s matrices
αi =
[0 σi
σi 0
], β =
[I 00 −I
]
αx =
0 0 0 10 0 1 00 1 0 01 0 0 0
, αy =
0 0 0 −i0 0 i 00 −i 0 0i 0 0 0
,
αz =
0 0 1 00 0 0 −11 0 0 00 −1 0 0
, β =
1 0 0 00 1 0 00 0 −1 00 0 0 −1
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
A task
Please, verify if Dirac’s matrices satisfy condition (1).
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Dirac’s equation (i~ ∂∂t− cα · p− βmc2
)ψ = 0
Solution is a four-component column matrix (spinor)
ψ(r, t) =
ψ1(r, t)ψ2(r, t)ψ3(r, t)ψ4(r, t)
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of ψ in Dirac equation
Dirac equation ⇒ equation of continuity
∂ρ
∂t+ divj = 0
ρ = ψ†ψ = |ψ1|2 + |ψ2|2 + |ψ3|2 + |ψ4|2 ≥ 0 probability density
j = −cψ†αψ probability density
current
requirement (1) is satisfied
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Contents
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Dirac equation (E − cα · p− βmc2
)ψ = 0
Suppose a plane wave solution
ψ(r, t) = uei~ (p·r−Et) , E =
p2
2m
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Dirac equation (E − cα · p− βmc2
)ψ = 0
Suppose a plane wave solution
ψ(r, t) = uei~ (p·r−Et) , E =
p2
2m
=⇒(Eu − cα · pu − βmc2u
)= 0
E −mc2 0 0 00 E −mc2 0 00 0 E +mc2 00 0 0 E +mc2
u1
u2u3u4
−c
0 0 pz px − ipy0 0 px + ipy −pzpz px − ipy 0 0
px + ipy −pz 0 0
u1
u2u3u4
= 0
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
∣∣∣∣∣∣∣∣E −mc2 0 −c(pz) −c(px − ipy )
0 E −mc2 −c(px + ipy ) c(pz)−c(pz) −c(px − ipy ) E + mc2 0
−c(px + ipy ) c(pz) 0 E + mc2
∣∣∣∣∣∣∣∣ = 0
E 2 = c2p2 + m2c4
E = ±√
c2p2 + m2c4
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
1 E+ = +√
c2p2 + m2c4
u(+)↑ = N
10cpz
E+ + mc2c(px + ipy )
E+ + mc2
, u(+)↓ = N
01
c(px − ipy )
E+ + mc2−cpz
E+ + mc2
2 E− = −
√c2p2 + m2c4
u(−)↑ = N
cpz
E− −mc2c(px + ipy )
E− −mc2
10
, u(−)↓ = N
c(px − ipy )
E− −mc2−cpz
E− −mc2
01
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
HW
Calculate the normalization constant N. (Solution can be found, for example, inRef. [1])
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Why are there ↑ and ↓ arrows in the subscripts?
u(+) relate to nonrelativistic limit
cpzE+ + mc2
c(px + ipy )
E+ + mc2
∼v
cv�c−−−→ 0
u(+)↑ (r, t) ∼
[10
]e
i~ (p·r−Et) ,
u(+)↓ (r, t) ∼
[01
]e
i~ (p·r−Et)
free spin 1/2 particles
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Interpretation of E+ and E−
Positron - experimentaldiscovery:
D. Skobelstyn (1929).
C.-Y. Chao (1929).
C. D. Anderson (1932). -Nobel Prize (1936).
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Contents
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Consider an electron in an emg field:
H = cα ·(
p− q
cA)
+ βmc2 + qφ(r)
D.E. ⇒ [cα ·
(p− q
cA)
+ βmc2 + qφ(r)]ψ = Eψ
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Consider an electron in an emg field:
H = cα ·(
p− q
cA)
+ βmc2 + qφ(r)
D.E. ⇒ [cα ·
(p− q
cA)
+ βmc2 + qφ(r)]ψ = Eψ
α symmetry ⇒[c(
p− q
cA)· σ]
+(mc2 + qφ(r)
)W = EW[
c(
p− q
cA)· σ]−(mc2 − qφ(r)
)V = EV
where
W =
[ψ(1)
ψ(2)
], V =
[ψ(3)
ψ(4)
]
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
V =
[c (p− (q/c)A) · σE − qφ+ mc2
]W
c2[(
p− q
cA)· σ(E − qφ+ mc2
)−1
(p− (q/c)A) · σ]W
+(mc2 + qφ(r)
)W = EW
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
V =
[c (p− (q/c)A) · σE − qφ+ mc2
]W
c2[(
p− q
cA)· σ(E − qφ+ mc2
)−1
(p− (q/c)A) · σ]W
+(mc2 + qφ(r)
)W = EW
v/c → 0⇒
‖V ‖‖W ‖ → 0
E ′ = E −mc2 , ‖qφ‖ � mc2
⇒(E ′ − qφ+ 2mc2
)−1
=1
2mc2
(1− E ′ − qφ
2mc2+ · · ·
)
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
D.E. in nonrelativistic limit[1
2m
(p− q
cA)2− q
mcS · B + qφ
]W = E ′W
where
S =~2σ
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
D.E. in nonrelativistic limit[1
2m
(p− q
cA)2− q
mcS · B︸ ︷︷ ︸
µ · B
+qφ
]W = E ′W
where
S =~2σ
⇒ µ =q
mcS magnetic moment of an electron
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Contents
1 Klein-Gordon equation
2 Dirac equation
3 Free-electron solution of Dirac equation
4 Electron magnetic moment
5 Literature
Igor Lukacevic Relativistic quantum mechanics
ContentsKlein-Gordon equation
Dirac equationFree-electron solution of Dirac equation
Electron magnetic momentLiterature
Literature
1 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, SanFrancisco, 2003.
2 I. Supek, Teorijska fizika i struktura materije, II. dio, Skolska knjiga,Zagreb, 1989.
3 Compton wavelength
4 P.A.M. Dirac - life & interesting facts
5 P. A. M. Dirac, ”The Quantum Theory of the Electron”, Proceedings ofthe Royal Society A: Mathematical, Physical and Engineering Sciences 117(778): 610 (1928).
6 C. D. Anderson - Nobel lecture about the dicovery of positron
Igor Lukacevic Relativistic quantum mechanics
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