Tarun Kanti Ghosh, Physics Department, IIT Kanpur
Department of Physics, IIT-Kanpur
Instructor: Tarun Kanti Ghosh
Quantum Mechancs-II (PHY432)
AY 2024-25, SEM-II
Homework-6 (Relativistic quantum mechanics)
1. Klein-Gordon equation: The Klein-Gordon (KG) equation is given by
m c 2
1 ∂ 2 ψ(r, t)
e
2
=
∇
ψ(r,
t)
−
ψ(r, t),
c2 ∂t2
ℏ
where ψ(r, t) is a single-component wave function.
(a) Obtain the following continuity equation from the KG equation:
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∂P
+ ∇ · J = 0,
∂t
where the probability density is given by
iℏ
∂ψ ∗ (r, t)
∂ψ(r, t)
∗
P (r, t) =
− ψ(r, t)
ψ (r, t)
2me c2
∂t
∂t
and the probability current density is given by
J(r, t) =
ℏ
[ψ ∗ (r, t)∇ψ(r, t) − ψ(r, t)∇ψ ∗ (r, t)] .
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(b) Show that the probability density is negative for negative energy branch.
What will be the probability density if ψ(r, t) is real?
(c) Can we obtain/define a velocity operator from the KG equation?
Ho
m
2. Properties of the Dirac matrices: The Dirac matrices are given as
0 σj
I 0
αj =
,β =
,
σj 0
0 −I
where j = x, y, z, Pauli matrices σj , 2 × 2 null matrix 0 and 2 × 2 dentity matrix I.
(a) Check that αj2 = I4×4 and β 2 = I4×4 .
(b) Show that {αi , αj } = 2I4×4 δij and {αi , β} = 0.
(c) Show that
σl 0
αj αk = iϵjkl
,
0 σl
where j, k, l : x, y, z.
3. The Dirac equation for a free particle is given by
∂ψ(r, t)
= Hψ(r, t),
∂t
where the Dirac Hamiltonian H is given by
iℏ
H = cα · p + βme c2
and the Dirac spinor is represented as ψ † = (ψ1∗ , ψ2∗ , ψ3∗ , ψ4∗ , ).
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(1)
Tarun Kanti Ghosh, Physics Department, IIT Kanpur
(a) Write down the Dirac equation explictly in 4 × 4 matrix form.
(b) Show that the velocity operator for a Dirac particle is v̂ = cα.
Note that the velocity operator in Dirac theory is a purley matrix operator α, as opposed
to the differential operator (in real space) in non-relativistic case.
(c) Obtain the following continuity equation from the Dirac equation:
∂P
+ ∇ · J = 0,
∂t
where P is the probability density and J is the probability current density. Show that
the expressions of P and J are P (r, t) = ψ † (r, t)ψ(r, t) and J(r, t) = cψ † (r, t)αψ(r, t).
Here ψ(r, t) is the four-component Dirac spinor.
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4. Total angular momentum is a constant of motion:
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(a) Show that the orbital angular momentum L = r×p is not a conserved quantity: [L, H] =
iℏc(α × p).
Orbital angular momentum is not a good quantum number for the Dirac
equation.
(b) Defining S = (ℏ/2)Σ, where components of the operator Σ are Σx = −iαy αz , Σy =
−iαz αx and Σz = −iαx αy . Show that
σ 0
Σ=
.
0 σ
2
(c) Find the eigenvalues of S. Show that S2 = 3ℏ4 I4×4 .
So it proves that S = ℏ/2 and the Dirac equation describes particles with spin 1/2.
Thus S is the spin angular momentum operator for spin 1/2 particle.
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(d) Show that the total angular momentum is J = L + S is a constant of motion: [J, H] = 0.
Total angular momentum is a good quantum number for the Dirac equation.
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(e) Show that the operator S · p is a constant of motion: [H, S · p] = 0.
(f) Helicity is defined as the projection of the spin along the motion of the particle. Mathematically, the helicity operator is defined as
ĥ =
Σ · p̂
.
|p|
Find the eigenvalues and the eigenfunctions of the helicity operator ĥ.
Helicity is also a good quantum number.
5. Non-relativistic approximation to the Dirac equation:
(a) The Dirac equation for an electron in a hydrogen atom is given by
[cα · p + βme c2 + V (r)]Ψ(r) = EΨ(r),
where E is the total energy, V (r) is the Coulomb interaction between nucleus and the
electron and Ψ(r) = (ψ(r), χ(r)) with ψ(r), and χ(r) are being two-compnenet spinors.
Show that in the non-relativistic limit (p ≪ me c), it is reduced to the Schördinger
equation for the hydrogen atom along wit the three relativistic correction terms.
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Tarun Kanti Ghosh, Physics Department, IIT Kanpur
(b) Consider an electron described by the Dirac equation is subjected to the magnetic field
B. The Dirac equation is given by
[cα · (p + eA) + βme c2 ]Ψ(r) = EΨ(r),
where A is a vector potential corresponding to the magnetic field B and E is the total
energy. Here Ψ(r) = (ψ(r), χ(r)) with ψ(r), and χ(r) are being two-componenet spinors.
Show that in the non-relativistic limit, it reduces to
(p + eA)2
e
+g
S · B ψ(r) = ϵψ(r),
2me
2me
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where g = 2, S = (ℏ/2)σ and ϵ = E − me c2 is the energy measured with respect to the
rest mass energy.
It clearly demonstrate how the Zeeman interaction energy appears along with the correct
g-factor and the spin angular momentum operator S = (ℏ/2)σ.
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