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Minimally relativistic Schrödinger equation and the linearly rising potential
Volume 109 A, number 6
3 June 1985
PHYSICS LETTERS
MINIMALLY RELATIVISTIC SCHRaDINGER AND THE LINEARLY RISING POTENTIAL
EQUATION
M. ZNOJIL Nuclear Physics Institute, Czechoslovak Academy of Sciences, 250 68 ke& Czechoslovakia
Received 11 February 1985; accepted for publication 30 March 1985
In place of the usual approximation T=
trnzC4
+ p2cy2
-
me2
=
p2/2m
- p4/8m3c2
+ 0(l/c4)
in the “minimally relativistic” Schradinger equation, we suggest to employ the alternative formula T= p2c2[(m2c4 + p2c2)“2 + mc2]-’
= p2(2m + p2/2mc2)-’
+O(l/c4).
Its merits are illustrated on the particular linear potentials V(x) = px +const, where a small O(ce3) modification of the coupling (if needed) enables us to construct the ground and first M ( = O(c3)) excited states 4(x) = eecx xpolynomial(x) exactly.
The phenomenological Schriidinger equation Ip2/2m t V(x)] $(x) = E$(x),
p2 = -Ii2 A,
(1)
may
be complemented by the various relativistic corrections (cf., e.g., 0 0 33 and 83 in ref. [l]). Whenever they prove to be small, eq. (1) (+ perturbations) provides a fair understanding of the whole physical system in question. Outside of the perturbation formalisms, an inclusion of the 0( 1/c2) corrections may often lead to nontrivial difficulties. For example, the particular kinematic corrections [p2/2m - p4/8m3c2
+ V(x)] G(x) = E+(x)
(2)
make the exact energies complex - the new “hamiltonian” ceases to be bounded from below. In this context, our present idea lies in a transition to the “equivalent” Pad&type kinetic energy operator,
(
P2
2m + p2/2mc2
+ V(x)
)
J/(x) = W(x) .
Besides an elimination of the spurious “energy widths” in (2), the new equation (3) will exhibit also 0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
an improved asymptotic behaviour in p 9 1 since the rigorous relativistic formula gives (m2c4 +p2c2)lj2
- mc2 = cp .
Furthermore, we may rewrite (3) as a secondorder differential equation p2(1 + c-2 V(x) - c-%)$(x)
+ V(x)$(x) = E#(x),
2m=l,
(4)
for a broad class of potentials, and reinterpret it finally as the “minimally relativistic” (MR) Schrijdinger equation P2X(X) + wx, a
x(x) = Ex(x) 3
x(x) = [l t c-2 V(x) - c-2E] $(x) ,
(1 + c-2E) V(x) - c--2@ B&E)=
1 tc-2
V(x)-c-2E
*
(5)
The original potential V(x) is replaced now by its effective, weakly energy dependent modification Wx, E). Since 251
Volume
109A, number
6
PHYSICS
W(x,E)=c2tE-c4/[c2-Et
V(x)]
3 June 1985
(6)
by definition, we may write, symbolically, p2tc2=c4/[c2-EtV(xo)]>O,
LETTERS
2Q,
Ho =f2ao
xoE(O,w),
so that E
+ V(xl),
Q, =2n td:+3/2,
x1 E(O,m).
This inequality resembles the Dirac discrete-spectrum
restriction [l] (remember that m = l/2 here). Our restriction is less restrictive for asymptotically growing potentials, but such forces are not suitable for direct use in the Klein-Gordon or Dirac equations at all. Here, eq. (6) converts such potentials into the asymptotically constant effective force FV(x,E’) which is quite comfortable from the purely formal point of view. As an illustrative example, let us pick up one of the most frequently used quark-antiquark interactions
b, = (n t 1)1’2(n t X t 3/2)‘i2,
iz = 0, 1, ... ) and may be solved in terms of analytic continued fractions [4]. For the particular couplings [5] A=-4g(MtItl),
M=O,l,...,
(11)
the full MR Schriidinger equation (10) degenerates, for the first M t 1 states, to the (M + l)dimensional set of equations
PI W)=w
,
(7)
and describe the corresponding semi-algebraic solution of our MR Schriidinger equation (5). Formally, after a change of variables x = r2,
x(x) = ,I ‘$(r/fi)
9
(8)
and resealing, we get the well-known eigenvalue problem [3] Wu + b2/(1 HO
+gr2)M) = 0 9
p=c3/[2(M+l+1)], (9)
(10)
[(1+gT)Ho+A7‘lcp=0,
,
[c2/2(Mtltl)]STS-EI)=O,
I KM+ l)/Qol 1’2
g = p/[2c(c2 - E)] ,
withJ=21+1/2andl=-l,O(onedimension)or I= 0, 1, ... (three dimensions). In the harmonicoscillator basis, this equation acquires the tridiagonal algebraic form
(13)
while the secular equation reads det{c21-
= -d2/dr2 t J(JZ+ l)/r2 + r2 ,
h = c2/(E - c2),
In the present notation, condition (11) determines simply the M-dependent “permissible” coupling constants
s=
(14)
\ (M/Q#'~
2
and determines therefore the first M t 1 real binding energies in the variational-type way. In a realistic situation with the potential V(x) = jIix, we may satisfy (13) approximatively, with a negligible error jIi- P = 0(l/c3). Then, we may also construct immediately the “low-lying” elementary solutions of the MR equation (3),
PHYSICS LETTERS
Volume 109A, number 6
xl+l e-CX X polynomial(x)
W) = -
1 f&c2
- E/c2
’
(15)
practice, these solutions will represent the whole relevant part of the spectrum since M = O(c3) is very large. In the non-relativistic limit c * 00,the indefinite character of Jl(x) (= 0 X -) is compatible also with a nonexistence [6] of elementary solutions of the nonrelativistic Schriidinger equation (1) with the linearly rising potential (7). In
3 June 1985
References [l] V.B. Beresteckii, E.M. Lifshitz and L.P. Pitaevskii, Relativisticheskaya kvantovaya teoria, Vol. 1 (Nauka, Moscow, 1968). [ 21 C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 169. [3] R.N. Choudhury and B. Mukhejee, J. Phys. Al6 (1983) 403 1, and references therein. [4] M. Znojil, J. Phys. Al6 (1983) 293. [5] R.R. Whitehead, A. Watt, G.P. Flessas and M.A. Nagarajan, J. Phys. Al5 (1982) 1217. [6] M. Znojil, J. Phys. Al6 (1983) 279.
253
3 June 1985
PHYSICS LETTERS
MINIMALLY RELATIVISTIC SCHRaDINGER AND THE LINEARLY RISING POTENTIAL
EQUATION
M. ZNOJIL Nuclear Physics Institute, Czechoslovak Academy of Sciences, 250 68 ke& Czechoslovakia
Received 11 February 1985; accepted for publication 30 March 1985
In place of the usual approximation T=
trnzC4
+ p2cy2
-
me2
=
p2/2m
- p4/8m3c2
+ 0(l/c4)
in the “minimally relativistic” Schradinger equation, we suggest to employ the alternative formula T= p2c2[(m2c4 + p2c2)“2 + mc2]-’
= p2(2m + p2/2mc2)-’
+O(l/c4).
Its merits are illustrated on the particular linear potentials V(x) = px +const, where a small O(ce3) modification of the coupling (if needed) enables us to construct the ground and first M ( = O(c3)) excited states 4(x) = eecx xpolynomial(x) exactly.
The phenomenological Schriidinger equation Ip2/2m t V(x)] $(x) = E$(x),
p2 = -Ii2 A,
(1)
may
be complemented by the various relativistic corrections (cf., e.g., 0 0 33 and 83 in ref. [l]). Whenever they prove to be small, eq. (1) (+ perturbations) provides a fair understanding of the whole physical system in question. Outside of the perturbation formalisms, an inclusion of the 0( 1/c2) corrections may often lead to nontrivial difficulties. For example, the particular kinematic corrections [p2/2m - p4/8m3c2
+ V(x)] G(x) = E+(x)
(2)
make the exact energies complex - the new “hamiltonian” ceases to be bounded from below. In this context, our present idea lies in a transition to the “equivalent” Pad&type kinetic energy operator,
(
P2
2m + p2/2mc2
+ V(x)
)
J/(x) = W(x) .
Besides an elimination of the spurious “energy widths” in (2), the new equation (3) will exhibit also 0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
an improved asymptotic behaviour in p 9 1 since the rigorous relativistic formula gives (m2c4 +p2c2)lj2
- mc2 = cp .
Furthermore, we may rewrite (3) as a secondorder differential equation p2(1 + c-2 V(x) - c-%)$(x)
+ V(x)$(x) = E#(x),
2m=l,
(4)
for a broad class of potentials, and reinterpret it finally as the “minimally relativistic” (MR) Schrijdinger equation P2X(X) + wx, a
x(x) = Ex(x) 3
x(x) = [l t c-2 V(x) - c-2E] $(x) ,
(1 + c-2E) V(x) - c--2@ B&E)=
1 tc-2
V(x)-c-2E
*
(5)
The original potential V(x) is replaced now by its effective, weakly energy dependent modification Wx, E). Since 251
Volume
109A, number
6
PHYSICS
W(x,E)=c2tE-c4/[c2-Et
V(x)]
3 June 1985
(6)
by definition, we may write, symbolically, p2tc2=c4/[c2-EtV(xo)]>O,
LETTERS
2Q,
Ho =f2ao
xoE(O,w),
so that E
+ V(xl),
Q, =2n td:+3/2,
x1 E(O,m).
This inequality resembles the Dirac discrete-spectrum
restriction [l] (remember that m = l/2 here). Our restriction is less restrictive for asymptotically growing potentials, but such forces are not suitable for direct use in the Klein-Gordon or Dirac equations at all. Here, eq. (6) converts such potentials into the asymptotically constant effective force FV(x,E’) which is quite comfortable from the purely formal point of view. As an illustrative example, let us pick up one of the most frequently used quark-antiquark interactions
b, = (n t 1)1’2(n t X t 3/2)‘i2,
iz = 0, 1, ... ) and may be solved in terms of analytic continued fractions [4]. For the particular couplings [5] A=-4g(MtItl),
M=O,l,...,
(11)
the full MR Schriidinger equation (10) degenerates, for the first M t 1 states, to the (M + l)dimensional set of equations
PI W)=w
,
(7)
and describe the corresponding semi-algebraic solution of our MR Schriidinger equation (5). Formally, after a change of variables x = r2,
x(x) = ,I ‘$(r/fi)
9
(8)
and resealing, we get the well-known eigenvalue problem [3] Wu + b2/(1 HO
+gr2)M) = 0 9
p=c3/[2(M+l+1)], (9)
(10)
[(1+gT)Ho+A7‘lcp=0,
,
[c2/2(Mtltl)]STS-EI)=O,
I KM+ l)/Qol 1’2
g = p/[2c(c2 - E)] ,
withJ=21+1/2andl=-l,O(onedimension)or I= 0, 1, ... (three dimensions). In the harmonicoscillator basis, this equation acquires the tridiagonal algebraic form
(13)
while the secular equation reads det{c21-
= -d2/dr2 t J(JZ+ l)/r2 + r2 ,
h = c2/(E - c2),
In the present notation, condition (11) determines simply the M-dependent “permissible” coupling constants
s=
(14)
\ (M/Q#'~
2
and determines therefore the first M t 1 real binding energies in the variational-type way. In a realistic situation with the potential V(x) = jIix, we may satisfy (13) approximatively, with a negligible error jIi- P = 0(l/c3). Then, we may also construct immediately the “low-lying” elementary solutions of the MR equation (3),
PHYSICS LETTERS
Volume 109A, number 6
xl+l e-CX X polynomial(x)
W) = -
1 f&c2
- E/c2
’
(15)
practice, these solutions will represent the whole relevant part of the spectrum since M = O(c3) is very large. In the non-relativistic limit c * 00,the indefinite character of Jl(x) (= 0 X -) is compatible also with a nonexistence [6] of elementary solutions of the nonrelativistic Schriidinger equation (1) with the linearly rising potential (7). In
3 June 1985
References [l] V.B. Beresteckii, E.M. Lifshitz and L.P. Pitaevskii, Relativisticheskaya kvantovaya teoria, Vol. 1 (Nauka, Moscow, 1968). [ 21 C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 169. [3] R.N. Choudhury and B. Mukhejee, J. Phys. Al6 (1983) 403 1, and references therein. [4] M. Znojil, J. Phys. Al6 (1983) 293. [5] R.R. Whitehead, A. Watt, G.P. Flessas and M.A. Nagarajan, J. Phys. Al5 (1982) 1217. [6] M. Znojil, J. Phys. Al6 (1983) 279.
253