View
217
Download
2
Category
Preview:
Citation preview
A description of the program POTHMF for computingmatrix elements of the coupled radial equations for aHydrogen-like atom in a homogeneous magnetic field
O. Chuluunbaatar,A.A. Gusev,V.P. Gerdt,V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich,M.S. Kaschiev, V.V. Serov
Contents:
The problem statementThe Kantorovich methodReduction to algebraic eigenvalue problem
Algorithm for evaluating matrix elements
Algorithm for evaluating asymptoticsof matrix elements
Algorithm for evaluating asymptoticsof radial solutions
Conclusions
The
prob
lem
stat
emen
t The Schrödinger equation for the Hydrogen atom in an axially symmetric magnetic field B=(0,0,B) and in the spherical coordinates (r,θ,φ) in atomic units can be written as the 2D-equation (see Dimova M.G. et al, J.Phys.B 38 (2005) 2337)
with normalization condition
Here m is magnetic quantum number, Z is charge,ωc=B/B0, (B0 ≈ 2.35 · 109 G) is a dimensionless parameter
The wave function satisfies the following boundary conditions in each mσ subspace of the full Hilbert space:
The discrete spectrum wave function is obeyed the asymptotic boundary condition approximated at large r=rmax by a boundary condition of the first type
The continuous spectrum wave functions is obeyed the asymptotic boundary condition approximated at large r=rmax by a boundary condition of the third type
We consider the Kantorovich expansion of the partial solution
using a set of the one-dimensional parametric basis functions :
The
Kan
toro
vich
met
hod
The
Kan
toro
vich
met
hod The unknown coefficients χ are satisfy to eigenvalue problem
for a set of ordinary second-order differential equations
where matrix elements Qij and Hij are given by the relations
The
Kan
toro
vich
met
hod
Stru
ctur
e of
pac
kage
POTH
MF
MATRMevaluation of matrix elements
EIGENFpreparation and solution of algebraic eigenvalue problem
basis functions
matrixelements
asymptotics
boundaryconditions
KANTBPA program for solving the boundary problem for a system of radial equations
MATRA evaluation of asymptotics of matrix elements
ASYMRS evaluation of asymptoticsof radial equations
DIPPOTevaluation of transition matrix elements
solutions
EIG
ENSF
We find eigenfunctions in the form of a series expansion
where P are the unnormalized Legendre polynomials:
The one-dimensional parametric basis functions Φ are eigenfunctions of the eigenvalue problem:
The eigenvalue problem for coefficients at unnormalizedLegendre polynomials takes form
EIG
ENSF
For solution of the algebraic problem one can use the normalized Legendre polynomials
The coefficients of these two expansions are connected by
EIG
ENSF
The eigenvalue problem for coefficients at normalizedLegendre polynomials takes form
ci(0) cj
(0)=δij
EIG
ENSF
cj
Behaviour of normalized coefficients csj (fill circles) and nonnormalized coefficients c̃sj (hollow circles) at r=15 (p=56.25) and ωc=1 for the first (j=1, left panel) and the tenth (j=10, right panel) even solutions.EI
GEN
SF
Some even angular eigenfunctions at m=0 and ωc=1EIG
ENSF
Φ1 Φ2
Φ3 Φ4
Some odd angular eigenfunctions at m=0 and ωc=1EIG
ENSF
Φ1 Φ2
Φ3 Φ4
MA
TRM
for e
valu
atin
g of
mat
rix e
lem
ents
INPUT
OUTPUT
STEP k.M
ATR
Mfo
r eva
luat
ing
of m
atrix
ele
men
ts
STEP k.1.M
ATR
Mfo
r eva
luat
ing
of m
atrix
ele
men
ts
result of STEP k.1.M
ATR
Mfo
r eva
luat
ing
of m
atrix
ele
men
ts
STEP k.2.M
ATR
Mfo
r eva
luat
ing
of m
atrix
ele
men
ts
STEP k.3.M
ATR
Mfo
r eva
luat
ing
of m
atrix
ele
men
ts
λ(k) λ(0)
λ(1)
λ(2)
red, yellow, green are even states cyan, blue, violet are odd states
Remark:Phenomena of avoiding crossings is connected with existence of branching points in a complex plane of parameter p=p(r).See Oguchi T., Radio Sci. 5 (1970) 1207-1214Skorokhodov S.L., Khristoforov D.V.,ZhVM&MF, 46 (2006) 1195-
1210.
MA
TRM
for e
valu
atin
g of
mat
rix e
lem
ents
Qij=-ci(0) cj
(1), Q’ij= ------ =-ci(1) cj
(1) - ci(0) cj
(2),
red, yellow, green are even states; cyan, blue, violet are odd states
Qii+2
Qii+4
Q’ii+2
Q’ii+4
dQijdr
red, yellow, green are even states; cyan, blue, violet are odd states
Hii+4
Hii
H’ii
H’ii+4
Hij=ci(1) cj
(1), H’ij= ------ = ci(2) cj
(1) + ci(1) cj
(2),dHij
dr
Lets us calculate asymptotics of matrix elements at small r using algorithm MATRM for evaluating of matrix elements:
INPUT:
ci(r)cj(r)=δij
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
LOCAL:
r→0
Remark:The finite radius of convergence expansion is connected with existence of branching points in a complex plane of parameter p=p(r).
In OUTPUT we have the asymptotic values of the matrix elements Ej(r), Hjj'(r) and Qjj'(r) at small r, characterized by l=2j-2+|m| for even states and by l=2j-1+|m| for odd states, have the form
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
We describe briefly evaluating matrix elements at large r as series expansions by the inverse power of p without taking into account the exponential small terms. Following Damburg R.J., Propin R.Kh. J. Phys. B, 1 (1968) 681
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents r→∞
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents STEP 1.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents STEP 2.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents STEP 3.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents STEP 4.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents STEP 5.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
Putting the above coefficients to evaluated on step 4 matrix elements we find the series expansion by the inverse power of rwithout the exponential terms
In these formulas asymptotic quantum numbers nl, nr denote transversal quantum numbers that are connected with the unified numbers j, j' by the formulas nl=i1, nr=j-1 for both even and odd parity for threshold energy En(0)= Єth
mσj (ωc).Remark. Evaluating the exponential small corrections can be done using additional series expansion of the solution in the region D2=[0,1-η2], η2<η, η2=o(p-1/2-ε) in accordance with Damburg R.J., Propin R.Kh. J. Phys. B, 1 (1968) 681.
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
Remark:The finite radius of convergence expansion is connected with existence of branching points in a complex plane of parameter p=p(r).
The calculations was performed on a MAPLE till kmax=12, a first terms of series expansions take form
in last formula n=min(nl,nr)
MA
TR
Afo
r ev
alua
ting
the
asym
ptot
icso
f mat
rix
elem
ents
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
ASY
MR
Sfo
r ev
alua
ting
the
asym
ptot
icso
f rad
ial s
olut
ions
0 E=-0.00781242971347E=-0.00617279526323E=-0.00617279808777E=-0.00617279808777
Convergence of the method for energy E(N=9, m=0) (in a.u.) of even wavefunctions γ=1.472 10-5
Nr=nmax=2:nmax=4:nmax=6:nmax=8:
8 E=-0.00499982705326E=-0.00499993540325E=-0.00617243586258E=-0.00617243586258
Nr=nmax=2:nmax=4:nmax=6:nmax=8:
The results are in agreement with so(4,2) algebraic perturbationcalculations Gusev A.A. et al, Programming and computer software v. 27 (2001) p. 27-31.
The wave functions Ψ1 and Ψ2 of first and second open channels of the continuum spectrum states σ = - 1, Z=1, γ =1 and m=0 with energy 2E=3.4 Ry. above second threshold εm2
th=3 Ry.
Ψ1
Ψ2
The
num
eric
al re
sults
Chuluunbaatar et al, 2006, J.Phys. B
Con
clus
ions
A new effective method of calculating the both discrete and continuum spectrum wave functions of a hydrogen atom in a strong magnetic field is developed based on the Kantorovich approach to the parametric eigenvalue problems in spherical coordinates.
The two-dimensional spectral problem for the Schroedinger equation is reduced to a one-dimensional spectral parametric problem for the angular variable and a finite set of ordinary second-order differential equations for the radial variable.
The rate of convergence of the method is examined numerically and is illustrated with a number of typical examples.
The results are in good agreement with calculations of photoionizationcross sections given by other authors.
The developed approach provides a useful tool for calculations of threshold phenomena in the formation and ionization of (anti)hydrogen-like atoms and ions in magnetic traps.
Recommended