Name: Anno 2011-02-16 11:29
Please disprove this idea wrong , Its doing my head in... FacePalmfailz....
Below link for full head fuk
Time Optimal Quantum State Control:
By P. G. Morrison BSc. MSc.
Synopsis:
There exists a Hamiltonian operator, for a given constraint, that drives the system from an ini-
tial state to a target nal state in least time. This Hamiltonian operator is a time-dependent
matrix with symmetry properties that represent the internal spectroscopy of the atomic system.
It is possible to nd the time optimal trajectory for the quantum state once the Hamiltonian is
de ned, and hence de ne a minimum time required for the evolution in question.
Results indicate that the time-dependence contained within the Hamiltonian matrix for partic-
ular choices of groups on SU(N) is of a periodic nature. This enables great simpli cation in
analysis as many results from the theory of periodic di erential equations may be directly
applied to physical problems.
In the practical area of semiconductor design and quantum dots on substrates, this requires that
the coupling between the dots be of an AC-type. Quantum dot scenarios are ideal for the appli-
cation of this method as the positions in space which the particle may occupy are limited to a
nite set of locations.
1
Mathematical apparatus:
Matrices:
~
A = [Aij] 1 6 i; j 6 n
~
Tr(A) = Ak k
k
j > = [c1(t); c2(t); c3(t); ; cn(t)]T
H11(t) H1n(t)
; Hjk(t) = Hk j(t) ;
k
^ ^ ^
Tr(F ) = 0 ; F = F
^ ^ ^ ^ ^
i = H (t)A(t) ? A(t)H (t) for any Hermitean A
^ ~ ^ ~ ~
Take the ansatz A = H (t) + F (t), and using [H (t); H (t)] = 0 we may rewrite the Heisenberg
equation of motion as:
i (H (t) + F (t)) = H (t)F (t) ? F (t)H (t)
with our quantum state obeying the Schrodinger equation:
~
i = H (t)j (t) >
The time dependent Hamiltonian has the property that it may not commute at di erent times,
~ ~
i.e. [H (t1); H (t2)] 0 for t1 t2.
2
Some simple time optimal operators and boundary conditions:
SU(2):
0 exp( ? 2i t)
exp( + 2i t) 0
; j (tf) > =
j"(0)j tf =
0
e?i sin(kt) ; F = 0 !2 0 = const:
e+i sin(kt)
(0) > = [1; 0; 0]T
j"(0)j
H = j"(0)j( ^
j (0) > = [1; 0; 0; 0]T ; j
j"(0)j
In general it seems that, given the existence of a Hamiltonian operator with su cient smooth-
ness, the Heisenberg Uncertainty Principle for energy and time is obeyed in the form:
(Energy strength of Hamiltonian) (time of evolution for state) = constant
Also, the number of possible choices of a Hamiltonian and constraint law for SU(N) is equal to:
#(N) = 2N ?1
which diverges rather rapidly as N ! 1. It is worse than the proverbial needle in a haystack!
3
http://www.sendspace.com/file/1ir58k
full doc , if you can dissprove 1k!!!
Below link for full head fuk
Time Optimal Quantum State Control:
By P. G. Morrison BSc. MSc.
Synopsis:
There exists a Hamiltonian operator, for a given constraint, that drives the system from an ini-
tial state to a target nal state in least time. This Hamiltonian operator is a time-dependent
matrix with symmetry properties that represent the internal spectroscopy of the atomic system.
It is possible to nd the time optimal trajectory for the quantum state once the Hamiltonian is
de ned, and hence de ne a minimum time required for the evolution in question.
Results indicate that the time-dependence contained within the Hamiltonian matrix for partic-
ular choices of groups on SU(N) is of a periodic nature. This enables great simpli cation in
analysis as many results from the theory of periodic di erential equations may be directly
applied to physical problems.
In the practical area of semiconductor design and quantum dots on substrates, this requires that
the coupling between the dots be of an AC-type. Quantum dot scenarios are ideal for the appli-
cation of this method as the positions in space which the particle may occupy are limited to a
nite set of locations.
1
Mathematical apparatus:
Matrices:
~
A = [Aij] 1 6 i; j 6 n
~
Tr(A) = Ak k
k
j > = [c1(t); c2(t); c3(t); ; cn(t)]T
H11(t) H1n(t)
; Hjk(t) = Hk j(t) ;
k
^ ^ ^
Tr(F ) = 0 ; F = F
^ ^ ^ ^ ^
i = H (t)A(t) ? A(t)H (t) for any Hermitean A
^ ~ ^ ~ ~
Take the ansatz A = H (t) + F (t), and using [H (t); H (t)] = 0 we may rewrite the Heisenberg
equation of motion as:
i (H (t) + F (t)) = H (t)F (t) ? F (t)H (t)
with our quantum state obeying the Schrodinger equation:
~
i = H (t)j (t) >
The time dependent Hamiltonian has the property that it may not commute at di erent times,
~ ~
i.e. [H (t1); H (t2)] 0 for t1 t2.
2
Some simple time optimal operators and boundary conditions:
SU(2):
0 exp( ? 2i t)
exp( + 2i t) 0
; j (tf) > =
j"(0)j tf =
0
e?i sin(kt) ; F = 0 !2 0 = const:
e+i sin(kt)
(0) > = [1; 0; 0]T
j"(0)j
H = j"(0)j( ^
j (0) > = [1; 0; 0; 0]T ; j
j"(0)j
In general it seems that, given the existence of a Hamiltonian operator with su cient smooth-
ness, the Heisenberg Uncertainty Principle for energy and time is obeyed in the form:
(Energy strength of Hamiltonian) (time of evolution for state) = constant
Also, the number of possible choices of a Hamiltonian and constraint law for SU(N) is equal to:
#(N) = 2N ?1
which diverges rather rapidly as N ! 1. It is worse than the proverbial needle in a haystack!
3
http://www.sendspace.com/file/1ir58k
full doc , if you can dissprove 1k!!!