bradley j. nartowt Saturday, July 06, 2013, 12:22:10
PHYS 6246 – classical mechanics Dr. Whiting
*
Show that if and are taken as two independent field variables, the following Lagrangian density leads to the Schrödinger equation and its complex conjugate, L
EL-equation is
d dx
h
2
2
8 m
(
,
*
h
??????
4 i
*
V
*
*
h
2
ih d
2
2
8 m
V
2 dt
[I.1]
L ) L . Rho index is *,** (complex conjugate and no complex conjugate). So, Euler
Lagrange equation both equation both with and and without expanding expanding out the “comma “comma notation” is,
d L
L d L d L * * * dx , dt dx ( ddx
*
d L d ) dy ( dy
*
d L d ) dz ( dz
*
d L d L i * * d t ) t dx ,i
[I.2]
Putting the lagranginan density [I.1] into [I.2], [I.2] , and computing each of the three “ingredients” separately, d L
h d d h2 ( * ) 2 (1) i * 2 i i * 2 dx ,i 8 m dx dx ( ) 8 m
Putting these three ingredients [I.3] into [I.2], which is * L
V
h 4 i
h 4 i
h
d dt
d L
( * L ) dxd i ( * L ) , and we get, ,i
h 2
2
2
2
8 m
h L [I.3] V 4 i *
h d
dt * 4 i dt
2
8 m
2 V
ih
d
2 dt
[I.4]
Just for fun, let’s try varying wrt ; in this case, [I.3] appears as,
d L
dx ,i i
h
d d ( )
8 m dx dx ( ) 2
i
i
*
2 h 2 * 8 2m (1)
Putting these three ingredients [I.5] into [I.2], which is L
h2 2
8 m
d dt
d L
h d
4 i dt
*
dt
* L h * V [I.5] 4 i
( L ) dxd i ( ,i L ) , and we get,
2 * V * i
h 2
*
[I.6]
So: the lagrangian density [I.1] “knows” what complex conjugation is. What are the canonical momenta?
, * , *
L 4 h i , *
[I.7]
Obtain the Hamiltonian density corresponding to this lagrangian density [I.1]. we start with legendre transform and wind up with, H
ii L
h
h
4 i
*
, L
,
2
2
8 m
V *
*
*
h(
*
* )
4 i
h2 2 2 2 V ei 8 m 1
h
2
2
8 m
* V *
h
4 i *
*
[I.8]
