The generalized momentum , also generalized , canonical , canonically conjugated , or conjugated momentum , occurs both in Hamiltonian mechanics and in Lagrange mechanics . Together with the location, it characterizes the respective state of the system, which changes over time according to Hamilton's equations of motion .
As a function of location and speed , the generalized momentum is the partial derivative of the Lagrange function with respect to speed:
q
{\ displaystyle q}
q
˙
{\ displaystyle {\ dot {q}}}
L.
{\ displaystyle L}
p
j
=
∂
L.
∂
q
˙
j
,
j
=
1....
n
{\ displaystyle p_ {j} = {\ frac {\ partial L} {\ partial {\ dot {q}} _ {j}}} \ ,, \ j = 1 .... n}
In the transition from classical physics to quantum mechanics , the canonical momentum (in contrast to the kinetic momentum ) is replaced by the momentum operator :
p
^
{\ displaystyle {\ hat {p}}}
p
j
→
p
^
j
=
-
ℏ
i
∂
∂
x
j
{\ displaystyle p_ {j} \ rightarrow {\ hat {p}} _ {j} = - \ hbar i {\ frac {\ partial} {\ partial x_ {j}}}}
Examples
Classic movement
L.
=
1
2
m
x
˙
2
-
V
(
x
,
t
)
{\ displaystyle L = {\ frac {1} {2}} \, m \, {\ dot {\ mathbf {x}}} ^ {2} -V (\ mathbf {x}, t)}
the generalized momentum is equal to the kinetic momentum:
p
=
m
x
˙
{\ displaystyle \ mathbf {p} = m {\ dot {\ mathbf {x}}}}
When a particle of mass moves in a potential in cylindrical coordinates
m
{\ displaystyle m}
V
(
r
,
φ
,
z
,
t
)
{\ displaystyle V (r, \ varphi, z, t)}
L.
=
1
2
m
(
r
˙
2
+
r
2
φ
˙
2
+
z
˙
2
)
-
V
(
r
,
φ
,
z
,
t
)
{\ displaystyle L = {\ frac {1} {2}} \, m {\ bigl (} {\ dot {r}} ^ {2} + r ^ {2} {\ dot {\ varphi}} ^ { 2} + {\ dot {z}} ^ {2} {\ bigr)} - V (r, \ varphi, z, t)}
the generalized momentum conjugated to the angle is the component of the angular momentum in the direction of the cylinder axis:
p
φ
˙
=
∂
L.
∂
φ
˙
=
m
r
2
φ
˙
{\ displaystyle p _ {\ dot {\ varphi}} = {\ frac {\ partial L} {\ partial {\ dot {\ varphi}}}} = m \, r ^ {2} {\ dot {\ varphi} }}
L.
=
1
2
m
x
˙
2
-
q
ϕ
(
t
,
x
)
+
q
x
˙
⋅
A.
(
t
,
x
)
{\ displaystyle L = {\ frac {1} {2}} \, m \, {\ dot {\ mathbf {x}}} ^ {2} -q \, \ phi (t, \ mathbf {x}) + q \, {\ dot {\ mathbf {x}}} \ cdot \ mathbf {A} (t, \ mathbf {x})}
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum :
A.
{\ displaystyle \ mathbf {A}}
p
=
m
x
˙
+
q
A.
(
t
,
x
)
{\ displaystyle \ mathbf {p} = m \, {\ dot {\ mathbf {x}}} + q \, \ mathbf {A} (t, \ mathbf {x})}
Relativistic movement
With the relativistic movement of a particle of mass in a potential without constraints in Cartesian coordinates
m
0
{\ displaystyle m_ {0}}
V
(
x
,
t
)
{\ displaystyle V (\ mathbf {x}, t)}
L.
=
-
m
0
c
2
1
-
x
˙
2
c
2
-
V
(
x
,
t
)
{\ displaystyle L = -m_ {0} \, c ^ {2} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x}}} ^ {2}} {c ^ {2}} }}} - V (\ mathbf {x}, t)}
the generalized momentum is equal to the kinetic momentum:
p
=
m
0
x
˙
1
-
x
˙
2
c
2
{\ displaystyle \ mathbf {p} = {\ frac {m_ {0} \, {\ dot {\ mathbf {x}}}} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x} }} ^ {2}} {c ^ {2}}}}}}}
With the relativistic movement of a point charge with the mass in the electromagnetic field
q
{\ displaystyle q}
m
0
{\ displaystyle m_ {0}}
L.
=
-
m
0
c
2
1
-
x
˙
2
c
2
-
q
ϕ
(
t
,
x
)
+
q
x
˙
⋅
A.
(
t
,
x
)
{\ displaystyle L = -m_ {0} \, c ^ {2} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x}}} ^ {2}} {c ^ {2}} }}} - q \, \ phi (t, \ mathbf {x}) + q \, {\ dot {\ mathbf {x}}} \ cdot \ mathbf {A} (t, \ mathbf {x})}
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum:
p
=
m
0
x
˙
1
-
x
˙
2
c
2
+
q
A.
(
x
,
t
)
{\ displaystyle \ mathbf {p} = {\ frac {m_ {0} \, {\ dot {\ mathbf {x}}}} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x} }} ^ {2}} {c ^ {2}}}}}} + q \, \ mathbf {A} (\ mathbf {x}, t)}
literature
Wolfgang Nolting: Basic Course Theoretical Physics 2 Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .
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