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Dilatation transformation and the stark—zeeman effect
CHEMICAL PHYSICS LETTERS
Voiume 58. nilmber 3
DIWTATION
TRANSFORMATION
sbib-I CIIU 1 W- Gibbs LPbomtw,
Depr:nzent
AND THE STARK-ZEEMAN
1 October 1978
EFFECT
of Physics, Yale University, New Haven. Gmnecticut
06520. USA
Received28 June 1978 We discuss the extension of the dilatation transformation to the Zeeman aEd the Stark-Zeemul problems and we apply the method to the study of the level shifts and ionization widths of the H atom in intense crossed eiectric and magnetic fields.
I _ Introduction
2_ Dilatation transformation
The method of coordinate rotation or dilatation transformation [I] (r+ re ia r being an interparticle coordinate) has become a popular analytical tooi for locating resonances in atomic scattering and photoabsorption problems [2] _ More recentiy? this method has been extended by Reinhardt and his co-workers to the dc Stark problem [3] and by Chu and Reinhardt [4] and by Chu [S] to the ac Stark and multiphoton ionization problems where it proves to be a powerful tool for dealing with intense field atomic processesIt is the purpose of this letter to discuss the extension of the method of dilatation transformation to the Zeeman and the Stark-Zeeman problems which are important in solid state and astrophysics. This v.32k has been facilitated by the recent work of Avron et al. f6] who studied the spectral and scattering theory for Sch5dinger operators with a magnetic field. In section 2 we discuss the spectrum of the “rotated” Zeeman hamiltonian and the utility of the dilatation transformation in locating the quasi-Landau resonances_ In section 3 we discuss the applicability of the diIatation transformation to ihe Stark-Zeeman problem. And we ihustmte the utility of the method by studying the resonance states of the hydrogen atom in crossed electric and magnetic fields. This is
The hamiltonian for the hydrogen-like atom in the presence of a uniform magnetic field B, in atomic units, is
folIowed by a discussion on the extension of the
work in section 4_
Mr)
and the Zeeman probIem
= Ho(Y) + v,
(1)
where HO(~) = $(P f eA/c) is the canonical or generalized momentum, V is the Coulomb potential, and 7 = Y&/2& is the (dimensionless) reduced magnetic field strength* (MB = Bohr magneton, R, = Rydberg ener=v). We shall choose the symmetric gauge for the vector potential A(= si3 x I-)_ Then for a magnetic field B in the z-direction, we have the free-electron Zeemau hamiltouian Ho(y) = -
$V’
+ yL,
f $y*?
sin*6,
where Lz is the z-component of the angular momentum operator. Since both Ho(y) and Hz(y) are invariant under rotations about the z-axis and under
space inversion, their eigenvectors have defiiite parity and m (eignvahte of L,, in units of fi). We 1first discuss the spectrum of the free-electron Zeeman hamiltonian Ho(y) and its dilation transformation which has been also studied recently by Avron et al. [6] _ it is well known that Ho(y) has continuous spectrum [PI@, -) and ‘discrete” Landau levels, (lv + &p~e, N= 0, 1,2, . . _ , which are themselves branching points of the continuous spectrum (wc is the cyclotron frequency)_ Upon making diiata*T = 1 Is equivalentto a magneticfieId strengthof 2.35 x IO9 G.
462
(22)
1 October 1978
CHFZWCAL PHYSICS LEITERS
Volume 58, number 3 r-re’=
U
UCHJ /
Cl- [HzCCO]
U&i,+
[H,(Q)]
dyti
~
~
&_
w+y&;k__kt;<
Eow%d *,o*es
Fig. l_ Effects of a dilatation transformation cn ?he spectrum, , of the free-electron Zeeman hanldtonian Ho_ The J_andau thresholds are invariant to the transformation while
-[Ho]
the continua rotate about their respective thresholds into lower half complex energy plane_
tion transformation, Ho(r) =_
f--f reio: (CYreal and positive), +rLz
+ fr2r2e2io
sin’o_
(3)
The spectrum ofHo(r, a)_ u[Ho(y, a)], has an analytic continuation from 01= 0 onto non-physical sheets with cr < a/4. As 0~“turned up” from 0, the continuous spectrum swings out into the lower half plane by angle 2a. As it swings out, it can “uncover” the Landau thresholds which stay fixed (cf. fig. 1). We now turn on the Coulomb potential Vand consider the spectrum of H,(y) and its dilatation transformation_ Since the Coulomb potential is relatively compact with respect to &(Tj, the spectrum o[Hz(y)] differs from o[Ho(y)] only in the point or “discrete” spectrum [6] _ Thus the Landau leveis in Ho(r) are now perturbed and shifted by the Coulomb potential and become “quasi-Landau” levels. In addition, below each quasi-landau level. there is a series of levels (doublets, if electron spinis considered) forming a Coulomb series. All the series but the first (IV = G) have a background of the continuous spectra of all the preceding Landau levels. (They are therefore quasistationary resonance states and possess finite widths.) The quasi-Landau levels themselves are branching points of the continuous spectrum (fig. 2). Applying now the dilatation transformation, r+ie icr, to H,(y), we have H,(y)
-j&(-y.
cr) =Ho(y_ ff) + ewie l?
Bound shales
Resnnonccs krlPasedl
Fig. 2. Effects of a dilatation transformation on the spectrum,
o[fi,],
Of an
atomic Zeeman hamiltonian Hz_ Tile bound
states (the Coulomb series below the fust (N = 0) quasi-
Landau threshold) are invariant to the transformation while the conthuua rotate about their respective quasi-bndau thresholds, exposing complex “resonance” Coulomb series (above the iv = 0 Landau threshold) in appropriate strips of the comples energy plrtne.
+ Wo(-Y, 01) 1 -2irUv2 Te
!aesonL7nces uudden,
(4)
The spectrum of the rotated hamiltonian, o[&(y, ar)] , is shown in fig- 2. Thus the point spectrum, including bound states (IV= 0 Coulomb series) and resonance
1 Coulomb senes), and the Landau thresholds are invariant, while the continua “rotate” about their thresholds by an angle 2a onto the lower half energy plane, “exposing” complex resonance eigenvalues for appropriate ranges ofo. To test the invariance of the bound states. we made a calculation of the energy of the ground state of the hydrogen atom in a magnetic field (y = 0.1) as a function of the rotational angle Q. The results given in table 1 indicate that it is the case. Perhaps the most useful aspect of applying the dilatation transformation to the Zceman problem, as suggested in fig. 2, is to facilitate the location of the positions and the widths of these “exposed” resonance states via analytic continuation. It should states @r>
Table 1 Test of the invariance of the ground state energy of the H atom in a rnagetic fie!d”) with respect to the dilatatron transformation, r + rer&, o! is the rotational angle. The ener,7 is obtained by the standard matrix linear variational method using 10s 1Od (rn = 0) Lagrerre type basis with exponents (Xs = 1.2, &d = 0.3) (cf. eq. (71), and is in harmony with the e.xact calculation of Cabib et al. [ 181 P (radian)
E&u)
0 0.2 0.4 0.6
-0.49032 -0.49032 -0.49032 -0.49032
a)Magrtetic field strength is -y = 0.1.
463
Volume 58, number 3 be noted that quasi-Landau field strensh and Sr atoms spectroscopy
CHELIICAL PHYSICS LJXLERS
the determination of the posirions of resonances as a function of magnetic has been recently accomplished for Ba thrrugh the use of two-photon laser [T] _
3. Dilatation transformation and the Stark-&man problem
When fast atomic or moiecular particles move in a strong magnetic field with velocity u, the electrons bound to the particles will experience besides the magnetic field B also an electric field E which is perpendicular to both B and U_Under the influence of electric field, the electrons can become detached by tunneling through the barrier. This process is known as the Lorenrz ionizaticn. Its possible technical applications (such as the buildup of piasma) have been pomted out a decade ago [g] _ The Stark-Zeeman effects are also important in solid state physics. Examples are the processes such as the magneto-optical absorption of Mott exciton [9] and the electron tunneling in semiconductors under the influence of crossed electric and magnetic fields :lO] _The behavior of atoms or molecules in intense electric and magnetic fields may be also important in astrophysics, as the e_xistence of superstrong magnetic fields in neutron stars and white dwarf.. has been weil.known [ll]Without loss of generality, we shall now deal with ‘LheprobIem of hydrogen-Iike system under the influence of crossed homogeneous electric and magnetic fields Assuming B 11z and E 11x, the StarkZeeman hamiitonian H,, can be written (in atomic units) as &(Y.
n = %(7-r) +fjc,
(5)
where H,(y) is the atomic Zeeman hamiltonian given in eq_ (I), andf= eEq#R, is the reduced electric field strength. cf= 1 corresponds to an electric field stren,$h of 5.1 X log V/cm.) Unlike the Zeeman potential. the Stark operator fi is non-analytic and not relatively compact with respect to the atomic hamiltonian. Thus the spectrum a[&,], similar to the dc Stark probiem, has no discrete bound states and has a CO~INIOUS spectm running from -to +m on the real axis. The bound 464
1 Ocrober 1978
states (namely, the Coulomb series below the first Landau level) in H,(y) are all Stark shifted and broadened and move down to the lower half complex energy plane and become resonance states [12] _ In the following we shall discuss the determination of these resonance states via the use of dilatation transformation. It has been shown in section 2 that dilatation transformation is applicable to the atomic Zeeman hamiltonian H,(y). IIowever. its applicability to the StarkZeeman problem is not so obvious as the Stark potentialfSr is notoriously not dilatation analytic [13]. Nevertheless, by studying the numetfcul range of the dc Stark hamiltonian, Cerjan et al. f3] showed recently that the dilatation transformation does indeed shift the continuous spectrum in a way as to expose the resonances. Also BrEndas and Froelich [14j have recently justified the use of dilatation transformation in the dc Stark problem by the use of extended virial theorem. The applicability of the dilatation transformation to the Zeeman and to the Stark problems suggests that the method may thus be extended to the combined Stark-Zeeman probIem. To illustrate the utility of the dilatation transformation in the Stark-Zeeman problem, we carry out a calculation of the position and the width of the ground state of the hydrogen atom in crossed electric and magnetic fields. The “rotated” Stark-Zeeman hamiltonian, (6)
will be “discretized” Iaguerre type basis
by use of the orthonormal
@Wn = Gr@~)~*l
exp(-W2)
XL?%W)
%?a@, 9)
0)
C,&) being a normalization constant. A typical structure of the Zeeman-Stark hamiltonian in matrix representation is shown in fig. 3_ It consists of diagonal angular momentum blocks Lm (where L = S, P, D, - . _etc., and m is the magnetic quantum number) and off-diagonal dipole coupling blocks- of types X and Y (type X couples L, to (L f I),,+ 1, whereasrype Y couples L, to (L + llm+l) due to Stark potential and off-diagonal quudrupok coupling blocks of type 2 (which coupies L, to (L f 2)& due
Volume 58, number 3
1 October 1978
CHEMICAL PKYSICS LETTERS -o_so~
(
,
)
)
'I ‘y-0
-0.51
-
T
9 %
w
n” t 5E
f
In
Fig. 3- Structureof the atomic Stark-Zeeman hamiltoni&. See text for notation_
s= "
to the quadratic diamagnetic interaction_ it is interes-
2 fn
-OS2
-
ii 1 z
ting to note that this Stark-Zeeman bloch structure (fig. 3), if the quadratic Zeeman coupling terms 2 are ignored, is identical in form to that of the ytmi-energy operator proposed by Chu in the treatment of multiphoton ionization via circularly polarized radiation [S] . Indeed the similarity of these two processes has been pointed out by Bunkin and Prokhorov [ 15 ] some 14 years ago. The shifted and broadened hydrogenic “Is” resonance state can be determined via an eigenvalue analysis* of the complex non-hermitean matriv shown in fig. 3. Convergence of such calculations as a function of the number of basis functions and the number of angular momentum blocks L, as weIl as the rotational angle or has been performed. Similar to previoils studies [2-S], the a-trajectory appears to have a “kink”, indicating the approaching of near converged results. Fig_ 4 shows the Stark-Zeeman shifts and widths of the “1s” resonance as a function of electric and magnetic field strengths. We choose f = 0.05 and 0-l for the electric field and varied the ratio r/“from 0 to l_ The shifts and widths at r = 0 (i.e. dc Stark effect) are in harmony with earlier calculations 13,161, whereas the results for y + 0 are new results. We note that for a f=ed electric field strength f, the width *If only one root is sought, the method of inverse_itemt.ion may be used instead of diagonabation of the whole matrix. See refk 13-51.
-o/53-
Fig- 4. The Stark-Zeeman level si;ifts and the field ionization widthsof rhe groundstate of the hydrogen atom in crossed electric and magnetic tields. fand y are respectively the reduced electric and magnetic field strength in atomic units. The size of basis used is about 15sl5plSdlSf to 20s20p20d20f, depending on the field strengths. The optimum rotation angle is about CY= 0.40 radian. The Laguerre exponents we used are hs = 1.2, Xp = 0.6: Ad = 0.3, and hf = 0.25 _
decreases with increasing magnetic field strength, indicating the stabilizing effect of the masetic fieldz _
4. Discussion We have shown that the dilatation trrtlsformation technique provides us with a viable and convenient tool for the studies of resonance states in external ~Similar results were obtained by Drukarev and hlonozon 1171 who studied the energy and decay probability of a particle in a model (zero-radius-well-zpprotimation)
potential.
Volume 58. number 3
CHEMICAL PHYSICS -RS
magnetic and electric fields. We have determined the hydrogen& level shifts and decay widths in crossed electric and magnetic fields via L’discretization of the continuum and the use of dilatation ana!yticity_ To extend our present work to superstrong (7 %- 1) magnetic field and to iV-body problems. we add two fmal remarks here_ (1) The J._aguerre type basis used in this work is o&y appropriate for moderate strong magnetic tieIds, say y 2 1_ For the case such as neutron stars, -y Z+ 1, one has to take into account explicitly the spherical symmetry of the Coulomb field and the cylindrical symmetry of the magnetic field- l_!seful references can be found in the review [ 1 ! ] _ (2) To extend the method to the IV-body problem involving magnetic fields, an analysis of the separation of the center of mzss may be required [6] Further work in these directions is in progressAcknowledgment The author is grateful to I. Herbst for a useful discussion and to B. Simon for providing the preprints of ref. [6] _This research was supported in part by the National Science Foundation under grant no. PHY 74-09408-A02. Part of this research was carried out at the Center for Astrophysics, Harvard University through the partial support of U-S- Department of Energy under contract no. EY-76-S-02-2887.
References [I] (2 j
466
E. BaJsIev and J-M- Combes, Common. Math- Phys22 (1971) 280; B. Simon, Ann_ Math. 97 (1973) 247. G.D. Doolen, J. NuttaB and R-W- Stagat, Phys- RevA10 (1974) 1612;
1 October 1978
RA. Bain, J-N. BardsJey, B-R. Junker and C-VSukumar; J. Phyr B7 (1974) 2189; A-P- Hickman, A-D- Jsaacson and W-H. MiJler, Chem. Phys. Letters 37 (1976) 63; T-N- Rescigno and V. McKay, Phys Rev_ Al2 (1975) 522; R Yaris and P. Wiier, J. Phys. Bll(l978) 1475, 1481; LT. Choo, M-C- Cracker and J_ Nuttall, J- Phys. Bll (1978) 1313. 131 W-P- Reinhardt, Intern. J- Quantum Chem. SlO (1976) 359; J-J. Wendoloski and W-P- Reinhardt. Phys. Rev- Al7 (1978) 195; C Cerjan. W-P- Reinhardt and J.E. Avron, J. Phys. Bll(l978) L201. [41 S-L Chu and W-P- Reinhardt, Phys- Rev- Letters 39 (1977) 1195. t51 S.I. Chu, Chem. Phys Letters 54 (1978) 367[61 J. Avron, I- Herbst and B. Simon. to be published171 R.J. Fonck, D.H. Tracy, D-C- Wri&t and F-S- Tomkins, Phys Rev_ Letters 40 (1978) 1366. IS1 N-V- Fedorenko, V-A- Ankudinov and R.N. Il’in. Soviet Phys. Tech. Phys. 10 (1965)461. 191 BS. Monozon and A-G- Zhiich, Soviet Phys- Semicond- 2 (1968) 150. 1101 A-G- Aronov and G-E- Pikus, Soviet Phys. JETP 24 (1967) 183,339. t111 R.H. Garstang, Rept- Progr. Phys. 40 (1977) 105, and references therein. EC. Tit&marsh, Proc Roy. Sot A207 (1951) 321. [W I131 J. Avron and I. Herb& Commun. hfath. Phys 52 (1977) 239i141 E BrSmias and P- Froelich, Phys. Rev_ Al6 (1977) 2207. 1151 F.V. Bunkin and AM. Prokhorov, Soviet Phys. JETP 19 (1964) 739. ft61 M.H. Alexander, Phys. Rev- 178 (1969) 34; hf. Hehenberger, H-V. McIntosh and E_ Br%das, Phys Rev- A10 (1974) 1494. G-F. Drukarev and B-S- hfonozon. Soviet Phys JETP 34 (1972) 509. D_ Calib, E_ Fabri and G- Fiorio, Nuovo Cimento 10B (1972) 185.
Voiume 58. nilmber 3
DIWTATION
TRANSFORMATION
sbib-I CIIU 1 W- Gibbs LPbomtw,
Depr:nzent
AND THE STARK-ZEEMAN
1 October 1978
EFFECT
of Physics, Yale University, New Haven. Gmnecticut
06520. USA
Received28 June 1978 We discuss the extension of the dilatation transformation to the Zeeman aEd the Stark-Zeemul problems and we apply the method to the study of the level shifts and ionization widths of the H atom in intense crossed eiectric and magnetic fields.
I _ Introduction
2_ Dilatation transformation
The method of coordinate rotation or dilatation transformation [I] (r+ re ia r being an interparticle coordinate) has become a popular analytical tooi for locating resonances in atomic scattering and photoabsorption problems [2] _ More recentiy? this method has been extended by Reinhardt and his co-workers to the dc Stark problem [3] and by Chu and Reinhardt [4] and by Chu [S] to the ac Stark and multiphoton ionization problems where it proves to be a powerful tool for dealing with intense field atomic processesIt is the purpose of this letter to discuss the extension of the method of dilatation transformation to the Zeeman and the Stark-Zeeman problems which are important in solid state and astrophysics. This v.32k has been facilitated by the recent work of Avron et al. f6] who studied the spectral and scattering theory for Sch5dinger operators with a magnetic field. In section 2 we discuss the spectrum of the “rotated” Zeeman hamiltonian and the utility of the dilatation transformation in locating the quasi-Landau resonances_ In section 3 we discuss the applicability of the diIatation transformation to ihe Stark-Zeeman problem. And we ihustmte the utility of the method by studying the resonance states of the hydrogen atom in crossed electric and magnetic fields. This is
The hamiltonian for the hydrogen-like atom in the presence of a uniform magnetic field B, in atomic units, is
folIowed by a discussion on the extension of the
work in section 4_
Mr)
and the Zeeman probIem
= Ho(Y) + v,
(1)
where HO(~) = $(P f eA/c) is the canonical or generalized momentum, V is the Coulomb potential, and 7 = Y&/2& is the (dimensionless) reduced magnetic field strength* (MB = Bohr magneton, R, = Rydberg ener=v). We shall choose the symmetric gauge for the vector potential A(= si3 x I-)_ Then for a magnetic field B in the z-direction, we have the free-electron Zeemau hamiltouian Ho(y) = -
$V’
+ yL,
f $y*?
sin*6,
where Lz is the z-component of the angular momentum operator. Since both Ho(y) and Hz(y) are invariant under rotations about the z-axis and under
space inversion, their eigenvectors have defiiite parity and m (eignvahte of L,, in units of fi). We 1first discuss the spectrum of the free-electron Zeeman hamiltonian Ho(y) and its dilation transformation which has been also studied recently by Avron et al. [6] _ it is well known that Ho(y) has continuous spectrum [PI@, -) and ‘discrete” Landau levels, (lv + &p~e, N= 0, 1,2, . . _ , which are themselves branching points of the continuous spectrum (wc is the cyclotron frequency)_ Upon making diiata*T = 1 Is equivalentto a magneticfieId strengthof 2.35 x IO9 G.
462
(22)
1 October 1978
CHFZWCAL PHYSICS LEITERS
Volume 58, number 3 r-re’=
U
UCHJ /
Cl- [HzCCO]
U&i,+
[H,(Q)]
dyti
~
~
&_
w+y&;k__kt;<
Eow%d *,o*es
Fig. l_ Effects of a dilatation transformation cn ?he spectrum, , of the free-electron Zeeman hanldtonian Ho_ The J_andau thresholds are invariant to the transformation while
-[Ho]
the continua rotate about their respective thresholds into lower half complex energy plane_
tion transformation, Ho(r) =_
f--f reio: (CYreal and positive), +rLz
+ fr2r2e2io
sin’o_
(3)
The spectrum ofHo(r, a)_ u[Ho(y, a)], has an analytic continuation from 01= 0 onto non-physical sheets with cr < a/4. As 0~“turned up” from 0, the continuous spectrum swings out into the lower half plane by angle 2a. As it swings out, it can “uncover” the Landau thresholds which stay fixed (cf. fig. 1). We now turn on the Coulomb potential Vand consider the spectrum of H,(y) and its dilatation transformation_ Since the Coulomb potential is relatively compact with respect to &(Tj, the spectrum o[Hz(y)] differs from o[Ho(y)] only in the point or “discrete” spectrum [6] _ Thus the Landau leveis in Ho(r) are now perturbed and shifted by the Coulomb potential and become “quasi-Landau” levels. In addition, below each quasi-landau level. there is a series of levels (doublets, if electron spinis considered) forming a Coulomb series. All the series but the first (IV = G) have a background of the continuous spectra of all the preceding Landau levels. (They are therefore quasistationary resonance states and possess finite widths.) The quasi-Landau levels themselves are branching points of the continuous spectrum (fig. 2). Applying now the dilatation transformation, r+ie icr, to H,(y), we have H,(y)
-j&(-y.
cr) =Ho(y_ ff) + ewie l?
Bound shales
Resnnonccs krlPasedl
Fig. 2. Effects of a dilatation transformation on the spectrum,
o[fi,],
Of an
atomic Zeeman hamiltonian Hz_ Tile bound
states (the Coulomb series below the fust (N = 0) quasi-
Landau threshold) are invariant to the transformation while the conthuua rotate about their respective quasi-bndau thresholds, exposing complex “resonance” Coulomb series (above the iv = 0 Landau threshold) in appropriate strips of the comples energy plrtne.
+ Wo(-Y, 01) 1 -2irUv2 Te
!aesonL7nces uudden,
(4)
The spectrum of the rotated hamiltonian, o[&(y, ar)] , is shown in fig- 2. Thus the point spectrum, including bound states (IV= 0 Coulomb series) and resonance
1 Coulomb senes), and the Landau thresholds are invariant, while the continua “rotate” about their thresholds by an angle 2a onto the lower half energy plane, “exposing” complex resonance eigenvalues for appropriate ranges ofo. To test the invariance of the bound states. we made a calculation of the energy of the ground state of the hydrogen atom in a magnetic field (y = 0.1) as a function of the rotational angle Q. The results given in table 1 indicate that it is the case. Perhaps the most useful aspect of applying the dilatation transformation to the Zceman problem, as suggested in fig. 2, is to facilitate the location of the positions and the widths of these “exposed” resonance states via analytic continuation. It should states @r>
Table 1 Test of the invariance of the ground state energy of the H atom in a rnagetic fie!d”) with respect to the dilatatron transformation, r + rer&, o! is the rotational angle. The ener,7 is obtained by the standard matrix linear variational method using 10s 1Od (rn = 0) Lagrerre type basis with exponents (Xs = 1.2, &d = 0.3) (cf. eq. (71), and is in harmony with the e.xact calculation of Cabib et al. [ 181 P (radian)
E&u)
0 0.2 0.4 0.6
-0.49032 -0.49032 -0.49032 -0.49032
a)Magrtetic field strength is -y = 0.1.
463
Volume 58, number 3 be noted that quasi-Landau field strensh and Sr atoms spectroscopy
CHELIICAL PHYSICS LJXLERS
the determination of the posirions of resonances as a function of magnetic has been recently accomplished for Ba thrrugh the use of two-photon laser [T] _
3. Dilatation transformation and the Stark-&man problem
When fast atomic or moiecular particles move in a strong magnetic field with velocity u, the electrons bound to the particles will experience besides the magnetic field B also an electric field E which is perpendicular to both B and U_Under the influence of electric field, the electrons can become detached by tunneling through the barrier. This process is known as the Lorenrz ionizaticn. Its possible technical applications (such as the buildup of piasma) have been pomted out a decade ago [g] _ The Stark-Zeeman effects are also important in solid state physics. Examples are the processes such as the magneto-optical absorption of Mott exciton [9] and the electron tunneling in semiconductors under the influence of crossed electric and magnetic fields :lO] _The behavior of atoms or molecules in intense electric and magnetic fields may be also important in astrophysics, as the e_xistence of superstrong magnetic fields in neutron stars and white dwarf.. has been weil.known [ll]Without loss of generality, we shall now deal with ‘LheprobIem of hydrogen-Iike system under the influence of crossed homogeneous electric and magnetic fields Assuming B 11z and E 11x, the StarkZeeman hamiitonian H,, can be written (in atomic units) as &(Y.
n = %(7-r) +fjc,
(5)
where H,(y) is the atomic Zeeman hamiltonian given in eq_ (I), andf= eEq#R, is the reduced electric field strength. cf= 1 corresponds to an electric field stren,$h of 5.1 X log V/cm.) Unlike the Zeeman potential. the Stark operator fi is non-analytic and not relatively compact with respect to the atomic hamiltonian. Thus the spectrum a[&,], similar to the dc Stark probiem, has no discrete bound states and has a CO~INIOUS spectm running from -to +m on the real axis. The bound 464
1 Ocrober 1978
states (namely, the Coulomb series below the first Landau level) in H,(y) are all Stark shifted and broadened and move down to the lower half complex energy plane and become resonance states [12] _ In the following we shall discuss the determination of these resonance states via the use of dilatation transformation. It has been shown in section 2 that dilatation transformation is applicable to the atomic Zeeman hamiltonian H,(y). IIowever. its applicability to the StarkZeeman problem is not so obvious as the Stark potentialfSr is notoriously not dilatation analytic [13]. Nevertheless, by studying the numetfcul range of the dc Stark hamiltonian, Cerjan et al. f3] showed recently that the dilatation transformation does indeed shift the continuous spectrum in a way as to expose the resonances. Also BrEndas and Froelich [14j have recently justified the use of dilatation transformation in the dc Stark problem by the use of extended virial theorem. The applicability of the dilatation transformation to the Zeeman and to the Stark problems suggests that the method may thus be extended to the combined Stark-Zeeman probIem. To illustrate the utility of the dilatation transformation in the Stark-Zeeman problem, we carry out a calculation of the position and the width of the ground state of the hydrogen atom in crossed electric and magnetic fields. The “rotated” Stark-Zeeman hamiltonian, (6)
will be “discretized” Iaguerre type basis
by use of the orthonormal
@Wn = Gr@~)~*l
exp(-W2)
XL?%W)
%?a@, 9)
0)
C,&) being a normalization constant. A typical structure of the Zeeman-Stark hamiltonian in matrix representation is shown in fig. 3_ It consists of diagonal angular momentum blocks Lm (where L = S, P, D, - . _etc., and m is the magnetic quantum number) and off-diagonal dipole coupling blocks- of types X and Y (type X couples L, to (L f I),,+ 1, whereasrype Y couples L, to (L + llm+l) due to Stark potential and off-diagonal quudrupok coupling blocks of type 2 (which coupies L, to (L f 2)& due
Volume 58, number 3
1 October 1978
CHEMICAL PKYSICS LETTERS -o_so~
(
,
)
)
'I ‘y-0
-0.51
-
T
9 %
w
n” t 5E
f
In
Fig. 3- Structureof the atomic Stark-Zeeman hamiltoni&. See text for notation_
s= "
to the quadratic diamagnetic interaction_ it is interes-
2 fn
-OS2
-
ii 1 z
ting to note that this Stark-Zeeman bloch structure (fig. 3), if the quadratic Zeeman coupling terms 2 are ignored, is identical in form to that of the ytmi-energy operator proposed by Chu in the treatment of multiphoton ionization via circularly polarized radiation [S] . Indeed the similarity of these two processes has been pointed out by Bunkin and Prokhorov [ 15 ] some 14 years ago. The shifted and broadened hydrogenic “Is” resonance state can be determined via an eigenvalue analysis* of the complex non-hermitean matriv shown in fig. 3. Convergence of such calculations as a function of the number of basis functions and the number of angular momentum blocks L, as weIl as the rotational angle or has been performed. Similar to previoils studies [2-S], the a-trajectory appears to have a “kink”, indicating the approaching of near converged results. Fig_ 4 shows the Stark-Zeeman shifts and widths of the “1s” resonance as a function of electric and magnetic field strengths. We choose f = 0.05 and 0-l for the electric field and varied the ratio r/“from 0 to l_ The shifts and widths at r = 0 (i.e. dc Stark effect) are in harmony with earlier calculations 13,161, whereas the results for y + 0 are new results. We note that for a f=ed electric field strength f, the width *If only one root is sought, the method of inverse_itemt.ion may be used instead of diagonabation of the whole matrix. See refk 13-51.
-o/53-
Fig- 4. The Stark-Zeeman level si;ifts and the field ionization widthsof rhe groundstate of the hydrogen atom in crossed electric and magnetic tields. fand y are respectively the reduced electric and magnetic field strength in atomic units. The size of basis used is about 15sl5plSdlSf to 20s20p20d20f, depending on the field strengths. The optimum rotation angle is about CY= 0.40 radian. The Laguerre exponents we used are hs = 1.2, Xp = 0.6: Ad = 0.3, and hf = 0.25 _
decreases with increasing magnetic field strength, indicating the stabilizing effect of the masetic fieldz _
4. Discussion We have shown that the dilatation trrtlsformation technique provides us with a viable and convenient tool for the studies of resonance states in external ~Similar results were obtained by Drukarev and hlonozon 1171 who studied the energy and decay probability of a particle in a model (zero-radius-well-zpprotimation)
potential.
Volume 58. number 3
CHEMICAL PHYSICS -RS
magnetic and electric fields. We have determined the hydrogen& level shifts and decay widths in crossed electric and magnetic fields via L’discretization of the continuum and the use of dilatation ana!yticity_ To extend our present work to superstrong (7 %- 1) magnetic field and to iV-body problems. we add two fmal remarks here_ (1) The J._aguerre type basis used in this work is o&y appropriate for moderate strong magnetic tieIds, say y 2 1_ For the case such as neutron stars, -y Z+ 1, one has to take into account explicitly the spherical symmetry of the Coulomb field and the cylindrical symmetry of the magnetic field- l_!seful references can be found in the review [ 1 ! ] _ (2) To extend the method to the IV-body problem involving magnetic fields, an analysis of the separation of the center of mzss may be required [6] Further work in these directions is in progressAcknowledgment The author is grateful to I. Herbst for a useful discussion and to B. Simon for providing the preprints of ref. [6] _This research was supported in part by the National Science Foundation under grant no. PHY 74-09408-A02. Part of this research was carried out at the Center for Astrophysics, Harvard University through the partial support of U-S- Department of Energy under contract no. EY-76-S-02-2887.
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