- Source: Magnetic translation
Magnetic translations are naturally defined operators acting on wave function on a two-dimensional particle in a magnetic field.
The motion of an electron in a magnetic field on a plane is described by the following four variables: guiding center coordinates
(
X
,
Y
)
{\displaystyle (X,Y)}
and the relative coordinates
(
R
x
,
R
y
)
{\displaystyle (R_{x},R_{y})}
.
The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy
[
X
,
Y
]
=
−
i
ℓ
B
2
{\displaystyle [X,Y]=-i\ell _{B}^{2}}
,
where
ℓ
B
=
ℏ
/
e
B
{\displaystyle \ell _{B}={\sqrt {\hbar /eB}}}
, which makes them mathematically similar to the position and momentum operators
Q
=
q
{\displaystyle Q=q}
and
P
=
−
i
ℏ
d
d
q
{\displaystyle P=-i\hbar {\frac {d}{dq}}}
in one-dimensional quantum mechanics.
Much like acting on a wave function
f
(
q
)
{\displaystyle f(q)}
of a one-dimensional quantum particle by the operators
e
i
a
P
{\displaystyle e^{iaP}}
and
e
i
b
Q
{\displaystyle e^{ibQ}}
generate the shift of momentum or position of the particle, for the quantum particle in 2D in magnetic field one considers the magnetic translation operators
e
i
(
p
x
X
+
p
y
Y
)
,
{\displaystyle e^{i(p_{x}X+p_{y}Y)},}
for any pair of numbers
(
p
x
,
p
y
)
{\displaystyle (p_{x},p_{y})}
.
The magnetic translation operators corresponding to two different pairs
(
p
x
,
p
y
)
{\displaystyle (p_{x},p_{y})}
and
(
p
x
′
,
p
y
′
)
{\displaystyle (p'_{x},p'_{y})}
do not commute.
References
Kata Kunci Pencarian:
- Mekanika kuantum
- Albert Einstein
- Sistem Satuan Internasional
- Tektonika lempeng
- IBM
- Metaloid
- Klorin
- Logam berat
- Fenomena 2012
- Pengantar mekanika kuantum
- Magnetic translation
- Magnetic field
- Magnetism
- Compass
- Permeability (electromagnetism)
- Geomagnetic reversal
- Joshua Zak
- Rotating magnetic field
- Ladder operator
- Plate tectonics
