Quantum mechanical equation of motion of charged particles in magnetic field
In quantum mechanics , the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field . It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light , so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.[1]
Equation [ ]
For a particle of mass
m
{\displaystyle m}
and electric charge
q
{\displaystyle q}
, in an electromagnetic field described by the magnetic vector potential
A
{\displaystyle \mathbf {A} }
and the electric scalar potential
ϕ
{\displaystyle \phi }
, the Pauli equation reads:
Pauli equation (general)
[
1
2
m
(
σ
⋅
(
p
−
q
A
)
)
2
+
q
ϕ
]
|
ψ
⟩
=
i
ℏ
∂
∂
t
|
ψ
⟩
{\displaystyle \left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }
Here
σ
=
(
σ
x
,
σ
y
,
σ
z
)
{\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}
are the Pauli operators collected into a vector for convenience, and
p
=
−
i
ℏ
∇
{\displaystyle \mathbf {p} =-i\hbar \nabla }
is the momentum operator . The state of the system,
|
ψ
⟩
{\displaystyle |\psi \rangle }
(written in Dirac notation ), can be considered as a two-component spinor wavefunction , or a column vector (after choice of basis):
|
ψ
⟩
=
ψ
+
|
↑
⟩
+
ψ
−
|
↓
⟩
=
⋅
[
ψ
+
ψ
−
]
{\displaystyle |\psi \rangle =\psi _{+}|{\mathord {\uparrow }}\rangle +\psi _{-}|{\mathord {\downarrow }}\rangle \,{\stackrel {\cdot }{=}}\,{\begin{bmatrix}\psi _{+}\\\psi _{-}\end{bmatrix}}}
.
The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators .
H
^
=
1
2
m
[
σ
⋅
(
p
−
q
A
)
]
2
+
q
ϕ
{\displaystyle {\hat {H}}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )\right]^{2}+q\phi }
Substitution into the Schrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See Lorentz force for details of this classical case. The kinetic energy term for a free particle in the absence of an electromagnetic field is just
p
2
2
m
{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}}
where
p
{\displaystyle \mathbf {p} }
is the kinetic momentum , while in the presence of an electromagnetic field it involves the minimal coupling
Π
=
p
−
q
A
{\displaystyle \mathbf {\Pi } =\mathbf {p} -q\mathbf {A} }
, where now
Π
{\displaystyle \mathbf {\Pi } }
is the kinetic momentum and
p
{\displaystyle \mathbf {p} }
is the canonical momentum .
The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity :
(
σ
⋅
a
)
(
σ
⋅
b
)
=
a
⋅
b
+
i
σ
⋅
(
a
×
b
)
{\displaystyle ({\boldsymbol {\sigma }}\cdot \mathbf {a} )({\boldsymbol {\sigma }}\cdot \mathbf {b} )=\mathbf {a} \cdot \mathbf {b} +i{\boldsymbol {\sigma }}\cdot \left(\mathbf {a} \times \mathbf {b} \right)}
Note that unlike a vector, the differential operator
p
−
q
A
=
−
i
ℏ
∇
−
q
A
{\displaystyle \mathbf {p} -q\mathbf {A} =-i\hbar \nabla -q\mathbf {A} }
has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function
ψ
{\displaystyle \psi }
:
[
(
p
−
q
A
)
×
(
p
−
q
A
)
]
ψ
=
−
q
[
p
×
(
A
ψ
)
+
A
×
(
p
ψ
)
]
=
i
q
ℏ
[
∇
×
(
A
ψ
)
+
A
×
(
∇
ψ
)
]
=
i
q
ℏ
[
ψ
(
∇
×
A
)
−
A
×
(
∇
ψ
)
+
A
×
(
∇
ψ
)
]
=
i
q
ℏ
B
ψ
{\displaystyle \left[\left(\mathbf {p} -q\mathbf {A} \right)\times \left(\mathbf {p} -q\mathbf {A} \right)\right]\psi =-q\left[\mathbf {p} \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\mathbf {p} \psi \right)\right]=iq\hbar \left[\nabla \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \left[\psi \left(\nabla \times \mathbf {A} \right)-\mathbf {A} \times \left(\nabla \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \mathbf {B} \psi }
where
B
=
∇
×
A
{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
is the magnetic field.
For the full Pauli equation, one then obtains[2]
Pauli equation (standard form)
H
^
|
ψ
⟩
=
[
1
2
m
[
(
p
−
q
A
)
2
−
q
ℏ
σ
⋅
B
]
+
q
ϕ
]
|
ψ
⟩
=
i
ℏ
∂
∂
t
|
ψ
⟩
{\displaystyle {\hat {H}}|\psi \rangle =\left[{\frac {1}{2m}}\left[\left(\mathbf {p} -q\mathbf {A} \right)^{2}-q\hbar {\boldsymbol {\sigma }}\cdot \mathbf {B} \right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }
Weak magnetic fields [ ]
For the case of where the magnetic field is constant and homogenous, one may expand
(
p
−
q
A
)
2
{\textstyle (\mathbf {p} -q\mathbf {A} )^{2}}
using the symmetric gauge
A
=
1
2
B
×
r
{\textstyle \mathbf {A} ={\frac {1}{2}}\mathbf {B} \times \mathbf {r} }
, where
r
{\textstyle \mathbf {r} }
is the position operator . We obtain
(
p
−
q
A
)
2
=
|
p
|
2
−
q
(
r
×
p
)
⋅
B
+
1
4
q
2
(
|
B
|
2
|
r
|
2
−
|
B
⋅
r
|
2
)
≈
p
2
−
q
L
⋅
B
,
{\displaystyle (\mathbf {p} -q\mathbf {A} )^{2}=|\mathbf {p} |^{2}-q(\mathbf {r} \times \mathbf {p} )\cdot \mathbf {B} +{\frac {1}{4}}q^{2}\left(|\mathbf {B} |^{2}|\mathbf {r} |^{2}-|\mathbf {B} \cdot \mathbf {r} |^{2}\right)\approx \mathbf {p} ^{2}-q\mathbf {L} \cdot \mathbf {B} \,,}
where
L
{\textstyle \mathbf {L} }
is the particle angular momentum and we neglected terms in the magnetic field squared
B
2
{\textstyle B^{2}}
. Therefore we obtain
Pauli equation (weak magnetic fields)
[
1
2
m
[
(
|
p
|
2
−
q
(
L
+
2
S
)
⋅
B
)
]
+
q
ϕ
]
|
ψ
⟩
=
i
ℏ
∂
∂
t
|
ψ
⟩
{\displaystyle \left[{\frac {1}{2m}}\left[\left(|\mathbf {p} |^{2}-q(\mathbf {L} +2\mathbf {S} )\cdot \mathbf {B} \right)\right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }
where
S
=
ℏ
σ
/
2
{\textstyle \mathbf {S} =\hbar {\boldsymbol {\sigma }}/2}
is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g -factor . The term in
B
{\textstyle \mathbf {B} }
, is of the form
−
μ
⋅
B
{\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} }
which is the usual interaction between a magnetic moment
μ
{\textstyle {\boldsymbol {\mu }}}
and a magnetic field, like in the Zeeman effect .
For an electron of charge
−
e
{\textstyle -e}
in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum
J
=
L
+
S
{\textstyle \mathbf {J} =\mathbf {L} +\mathbf {S} }
and Wigner-Eckart theorem . Thus we find
[
|
p
|
2
2
m
+
μ
B
g
J
m
j
|
B
|
−
e
ϕ
]
|
ψ
⟩
=
i
ℏ
∂
∂
t
|
ψ
⟩
{\displaystyle \left[{\frac {|\mathbf {p} |^{2}}{2m}}+\mu _{\rm {B}}g_{J}m_{j}|\mathbf {B} |-e\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }
where
μ
B
=
e
ℏ
2
m
{\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2m}}}
is the Bohr magneton and
m
j
{\textstyle m_{j}}
is the magnetic quantum number related to
J
{\textstyle \mathbf {J} }
. The term
g
J
{\textstyle g_{J}}
is known as the Landé g-factor , and is given here by
g
J
=
3
2
+
3
4
−
ℓ
(
ℓ
+
1
)
2
j
(
j
+
1
)
,
{\displaystyle g_{J}={\frac {3}{2}}+{\frac {{\frac {3}{4}}-\ell (\ell +1)}{2j(j+1)}},}
[a]
where
ℓ
{\displaystyle \ell }
is the orbital quantum number related to
L
2
{\displaystyle L^{2}}
and
j
{\displaystyle j}
is the total orbital quantum number related to
J
2
{\displaystyle J^{2}}
.
From Dirac equation [ ]
The Pauli equation is the non-relativistic limit of Dirac equation , the relativistic quantum equation of motion for particles spin-½.[3]
Derivation [ ]
Dirac equation can be written as:
i
ℏ
∂
t
(
ψ
1
ψ
2
)
=
c
(
σ
⋅
Π
ψ
2
σ
⋅
Π
ψ
1
)
+
q
ϕ
(
ψ
1
ψ
2
)
+
m
c
2
(
ψ
1
−
ψ
2
)
{\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)=c\,\left({\begin{array}{c}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{2}\\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{1}\end{array}}\right)+q\,\phi \,\left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)+mc^{2}\,\left({\begin{array}{c}\psi _{1}\\-\psi _{2}\end{array}}\right)}
,
where
∂
t
=
∂
∂
t
{\textstyle \partial _{t}={\frac {\partial }{\partial t}}}
and
ψ
1
,
ψ
2
{\displaystyle \psi _{1},\psi _{2}}
are two-component spinor , forming a bispinor.
Using the following ansatz:
(
ψ
1
ψ
2
)
=
e
−
i
m
c
2
t
ℏ
(
ψ
χ
)
{\displaystyle \left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)=\mathrm {e} ^{-\displaystyle i{\frac {mc^{2}t}{\hbar }}}\left({\begin{array}{c}\psi \\\chi \end{array}}\right)}
,
with two new spinors
ψ
,
χ
{\displaystyle \psi ,\chi }
,the equation becomes
i
ℏ
∂
t
(
ψ
χ
)
=
c
(
σ
⋅
Π
χ
σ
⋅
Π
ψ
)
+
q
ϕ
(
ψ
χ
)
+
(
0
−
2
m
c
2
χ
)
{\displaystyle i\hbar \partial _{t}\left({\begin{array}{c}\psi \\\chi \end{array}}\right)=c\,\left({\begin{array}{c}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\chi \\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi \end{array}}\right)+q\,\phi \,\left({\begin{array}{c}\psi \\\chi \end{array}}\right)+\left({\begin{array}{c}0\\-2\,mc^{2}\,\chi \end{array}}\right)}
.
In the non-relativistic limit,
∂
t
χ
{\displaystyle \partial _{t}\chi }
and the kinetic and electrostatic energies are small with respect to the rest energy
m
c
2
{\displaystyle mc^{2}}
.
Thus
χ
≈
σ
⋅
Π
ψ
2
m
c
.
{\displaystyle \chi \approx {\frac {{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi }{2\,mc}}\,.}
Inserted in the upper component of Dirac equation, we find Pauli equation (general form):
i
ℏ
∂
t
ψ
=
[
(
σ
⋅
Π
)
2
2
m
+
q
ϕ
]
ψ
.
{\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .}
From a Foldy-Wouthuysen transformation [ ]
One can also rigorously derive Pauli equation, starting from Dirac equation in an external field and performing a Foldy-Wouthuysen transformation .[3]
Pauli coupling [ ]
Pauli's equation is derived by requiring minimal coupling , which provides a g -factor g =2. Most elementary particles have anomalous g -factors, different from 2. In the domain of relativistic quantum field theory , one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor
p
μ
→
p
μ
−
q
A
μ
+
a
σ
μ
ν
F
μ
ν
{\displaystyle p_{\mu }\to p_{\mu }-qA_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }}
where
p
μ
{\displaystyle p_{\mu }}
is the four-momentum operator,
A
μ
{\displaystyle A_{\mu }}
if the electromagnetic four-potential ,
a
{\displaystyle a}
is the anomalous magnetic dipole moment ,
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}
is electromagnetic tensor , and
σ
μ
ν
=
i
2
[
γ
μ
,
γ
ν
]
{\textstyle \sigma _{\mu \nu }={\frac {i}{2}}[\gamma _{\mu },\gamma _{\nu }]}
are the Lorentzian spin matrices and the commutator of the gamma matrices
γ
μ
{\displaystyle \gamma ^{\mu }}
.[4] [5] In the context, of non-relativistic quantum mechanics, instead of working with Schrödinger equation, Pauli coupling is equivalent to use Pauli equation (or to postulate Zeeman energy ) for an arbitrary g -factor.
See also [ ]
[ ]
^ The formula used here is for a particle with spin ½, with a g -factor
g
S
=
2
{\textstyle g_{S}=2}
and orbital g -factor
g
L
=
1
{\textstyle g_{L}=1}
.
References [ ]
Books [ ]
Schwabl, Franz (2004). Quantenmechanik I . Springer. ISBN 978-3540431060 .
Schwabl, Franz (2005). Quantenmechanik für Fortgeschrittene . Springer. ISBN 978-3540259046 .
Claude Cohen-Tannoudji; Bernard Diu; Frank Laloe (2006). Quantum Mechanics 2 . Wiley, J. ISBN 978-0471569527 .
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