- Email: [email protected]
The Lenz vector in the confined hydrogen atom problem1
9 October 1998
Chemical Physics Letters 295 Ž1998. 217–222
The Lenz vector in the confined hydrogen atom problem
1
Vladimir I. Pupyshev ) , Andrei V. Scherbinin Laboratory of Molecular Structure and Quantum Mechanics, Chemistry Department, Moscow State UniÕersity, Moscow, 119899, Russian Federation Received 13 July 1998; revised 24 August 1998
Abstract A general criterion for the degeneracy of states of the hydrogen atom placed in a spherical impenetrable box of radius R is formulated on the basis of the Lenz vector formalism. The reasons for the elimination of the Coulomb degeneracy typical of the conventional hydrogen atom problem and for the non-conservation of the Lenz vector are discussed. The explanation of the simultaneous degeneracy of some energy levels Že.g. 2s and 3d, 3s and 4d, etc. at R s 2 au. is given. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The quantum–mechanical model for an atom with a nucleus placed at the center of an impenetrable spherical box of radius R has been discussed since 1937, when it was considered for the hydrogen atom w1x. The interest in this field is stimulated by a large variety of possible applications of the model — from astrophysics to solid state theory Žsee Refs. w2–5x and the references therein.. The well-known quantum problem of the hydrogen atom corresponds to R ™ `, which allows the states of the confined atom Ž R - `. to be denoted by the same set of quantum numbers Ž n,l . as those used in the former case, taking into account the fact that the principal quantum number satisfies the inequality n 0 l q 1. As it turns out, both problems are essentially different. In particular, the radial wavefunctions of the confined atom are expressed via the confluent
hypergeometric function w2,5,8,9x, which is not reduced to elementary functions in general, except for some special cases, considered in Section 3.6. As a consequence, the corresponding eigenvalues Enl Ž R . are determined by a transcendental equation and there is no general formula for them. The specific feature of a conventional hydrogen atom problem is the presence of an additional Žapart from the energy and the angular momentum. integral of the motion, the so-called Lenz vector Žor the Runge–Lenz vector, known also as the Laplace integral in the classical Kepler problem., which results in ‘accidental’ n 2-fold degeneracy of all the energy levels with the same principal quantum number n Žthe so-called ‘Coulomb degeneracy’. w6,7x. In the confined atom problem, the Coulomb degeneracy is broken, and the energy levels corresponding to the same principal quantum number are ordered as follows w2x: E2 p Ž R . - E2 s Ž R . ;
) 1
Corresponding author. E-mail: [email protected] In commemoration of the 100th anniversary of V.A. Fock.
E3d Ž R . - E3 p Ž R . - E3s Ž R . ; . . .
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 9 6 1 - 0
Ž 1.1 .
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
218
for any finite R w8,9x. Hence, the Lenz vector is not a constant of the motion for finite R values. However, some other levels can become degenerate instead. In particular, a simultaneous degeneracy of the states Ž n,l . and Ž n q 1,l q 2., n F l q 2, is observed when R reaches the value Ž l q 1.Ž l q 2. au w8,9x. For example, at R s 2 au, E2 s s E3d ;
E3s s E4 d ; . . .
Ž 1.2 .
In the present work the following problems are discussed: Ži. why the Coulomb degeneracy is broken at finite R values and Žii. the reason for the regular picture of the accidental degeneracy of energy levels whose l numbers differ by 2. As will be shown below, any accidental degeneracy of energy levels in the spherically confined hydrogen atom problem is closely connected with some special properties of the Lenz vector, which is defined as: r 1 A s y Ž p = l y l = p. , Ž 1.3 . r 2 Žatomic units are used throughout this Letter., where r is the electron position vector, r s
2.1. The wavefunction c of a stationary state of the hydrogen atom in a spherical box of radius R is the solution of the Schrodinger equation, ¨
Ž 2.1 .
with the boundary conditions ŽBC.:
c Ž r . is bounded at r s 0 ; c Ž r . rs R s 0 .
c Ž r . s f Ž r . Yl Ž u , w . ,
Ž 2.3 .
where Yl s Yl l Ž u , w . s sin lu e i l w is the unnormalized spherical harmonic function. The radial wavefunction f Ž r . is a solution of the corresponding secondorder ordinary differential equation, bounded at r s 0 due to Ž2.2a. and such that: fŽ R . s 0
Ž 2.4 .
due to Ž2.2b.. Note that: Ži. f Ž r . A r l , r ™ 0, as f Ž r . is a regular solution w6x; Žii. f X Ž R . / 0 since f Ž r . is a non-trivial solution of a second-order linear differential equation and Ž2.4. holds. 2.2. The explicit expression for the operator Aqs A x q i A y in spherical coordinates Ž r, u , w . reads: Aqs pz lqq
xqi y r
q i Ž px q i p y . Ž l z q 1. ,
Ž 2.5a . or, equivalently, Aqs Ž x q i y .
ž
1 r
E y p2 y i r
/ ž
Er
q1
/Ž
px q i p y . .
Ž 2.5b . Here lqs l x q i l y is the standard ‘step-up’ operator of the angular momentum theory and p x q i p y s yie i w
2. Operators AkH
Hc s Ec ,
possible value of the angular momentum projection, namely
Ž 2.2a . Ž 2.2b .
Here H s y1r2 D y 1rr is the non-relativistic Hamiltonian of the hydrogen atom. Taking into account the spherical symmetry of the problem, it is sufficient to consider the states with the highest
ž
= sin u
cos u E
E q Er
r
Eu
i q
E
r sin u Ew
/
.
Ž 2.6 . Using Eqs. Ž2.5a., Ž2.5b. and Ž2.6., it is easy to check that if c has the form Ž2.3., then
x s Aq c s Ž l q 1 . f X q 1 y
ž
l Ž l q 1. r
/
f Ylq1 .
Ž 2.7 . Applying the operator Aq k times results in cumbersome relations which include higher-order
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
derivatives of the radial part f Ž r . of c . However, if one takes into account the fact that f Ž r . is a solution of the radial Schrodinger equation, then the explicit ¨ k expression for x s Aq c is reduced to the following simple one: k k x s Aq c s Aq Ž fYl .
s a l k Ž r , E . f X q b l k Ž r , E . f Ylqk ,
Ž 2.8 .
where a l k Ž r , E . and b l k Ž r , E . are polynomials in 1rr, as may be verified by induction. In the case k s 1, Eq. Ž2.7. gives:
a l1 s l q 1 ,
Ž 2.9a .
219
Hamiltonian of a free system and that of the system in a box are in fact different operators with different domains Žsee, e.g. Ref. w10x.. One should mention, however, that the other BC, Eq. Ž2.2a., can not be a source of any difficulty; indeed, we have f Ž r . A r l , r ™ 0 and from Eq. Ž2.7., the radial part of x s Aq c goes to zero at least as r l too; hence, x is bounded at the origin provided that the original wavefunction c is. Being at the same time a regular solution of Eq. Ž2.1. with the angular momentum value Ž l q 1., c therefore vanishes as r lq1 at the origin. This conclusion is k c , k ) 1. easily generalized for x s Aq
and in the case k s 2:
ž
a l 2 s Ž 2 l q 3. 1 y
Ž l q 1. Ž l q 2. r
/
.
Ž 2.9b .
For k ) 2, a l k Ž r , E . is a polynomial in E of order Ž k y 2.. 2.3. The components of the Lenz vector Ž1.3. are formally commutable with the Hamiltonian H w6,7x; the same is obviously true for the operator Aqs A x q i A y . It means that if c is a solution of Eq. Ž2.1., then x s Aq c is also a solution: H x s HAq c s Aq Hc s EAq c s E x .
Ž 2.10 .
Based on Eqs. Ž2.7. and Ž2.10., one would think that Aq plays the role of a ‘step-up’ operator, which increases the angular quantum number l by 1. As long as the free atom is considered, this is indeed true Žsee Section 2.6.; however, for the confined atom, this formal observation fails owing to the following fact: although c is a wavefunction of the system, satisfying both Eqs. Ž2.1., Ž2.2a. and Ž2.2b., x is definitely not. Indeed, Eq. Ž2.7. implies that:
xrs R s Ž l q 1 . f X Ž R . Ylq1
Ž 2.11 .
which is inconsistent with the BC Ž2.2b. Žrecall that f X Ž R . / 0; see remark Žii. at the end of Section 2.1.. In other words, x s Aq c is not from the domain of the Hamiltonian of the confined atom problem Ž2.1. – Ž2.2. and hence it cannot be a wavefunction of the system. Therefore, in the confined atom problem, the components of the Lenz vector are no longer commutable with the Hamiltonian as operators in the Hilbert space Ževen though their formal expressions still are.; this conclusion is not surprising since the
2.4. It follows from the previous discussion that the formal commutator relation Ž2.10. is insufficient to guarantee that x s Aq c is a wavefunction of the problem; it has been shown, in particular, how inconsistencies with the BC Ž2.2b. can arise. Another question is whether x ' 0. For this to be the case, according to Eq. Ž2.7., the radial part of c should satisfy the following equation:
Ž l q 1. f X q 1 y
ž
l Ž l q 1. r
/
fs0 ,
with the solution of the form f Ž r . s Cr l exp y
ž
r lq1
/
,
Ž 2.12 .
i.e. c is proportional to the wavefunction of the Ž l q 1,l .-state of the free atom Ž R s `. with energy w6,7x ` E s Elq1 sy
1 2 Ž l q 1.
2
.
Ž 2.13 .
In a similar manner, if A k , k ) 1, is considered k instead of Aq, then the equation x s Aq ' 0 implies that: ` E s Elqj sy
1 2Ž l q j .
2
Ž 2.14 .
k with some j, 1 F j F k. Indeed, if Aq c ' 0, then there exists some j varying in the limits specified above, such that none of the functions c , Aq c , 2 jy1 j Aq c , . . . , Aq c is identically zero, while Aq c' k jy1 . . . ' Aq c ' 0. This means that Aq c is the wave-
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
220
function of the Ž l q j,l q j y 1.-state of the free hydrogen atom, which leads to Eq. Ž2.14.. 2.5. The results of this section can be summarized as follows. If c is a solution of Eq. Ž2.1. such that: Ža. c is bounded at the origin, Žb. c has a factorized form Ž2.3. and corresponds to the highest angular k momentum projection, then x s Aq c is also a solution of Eq. Ž2.1., which is bounded at the origin as well and corresponds to the angular quantum number l q k. In addition, x is not identically zero provided that Ž2.14. is not fulfilled. What is in question is the BC Ž2.2b. only. If x satisfies Ž2.2b., then x is also a wavefunction corresponding to the same energy but a different angular momentum value, i.e. the k energy level E is extra degenerate, Aq playing the role of a ‘step-up’ operator. 2.6. As an example, the well-known case R s ` can be considered, where the BC Ž2.2b. simply demands that c is vanishing at infinity rapidly enough to be square-integrable Žfor the discrete spectrum states.. In fact, c then decays exponentially and it is evident from Eqs. Ž2.7. – Ž2.11. that x s Aq c satisfies the BC at infinity straightforwardly. As a consequence, the accidental ‘Coulomb’ degeneracy occurs. Given below is a diagram featuring the ‘stepup’ property of the Aq operator which converts the wavefunctions of the mutually degenerated states as follows:
°1s ™ 0 Aq :
~2s ™ 2p
q1 ™ 0
¢
3s ™ 3pq1 ™ 3dq2 ™ 0 ...
Ž 2.15 .
Žsubscript denotes the angular momentum projection value.. Note that the states 1s, 2p, 3d, etc., are those with radial wavefunctions of the form Ž2.12., so that they are indeed from the kernel of Aq. Moreover, on the basis of the properties of Aq discussed above, it is possible to give a complete description of the discrete energy spectrum of the free hydrogen atom, taking into account that, for a given discrete spectrum energy value E, the possible angular momentum values are bounded from above. One should mention, however, that this is just a variety of the well-known approach to the quantum Kepler problem in terms of the Lenz vector w6,7x and
the four-dimensional angular momentum formalism derived by Fock many years ago w11x.
3. A general criterion for the degeneracy 3.1. This section is devoted to establishing rigorous necessary and sufficient conditions under which two given states Ž n,l . and Ž nX ,lX . of the encapsulated atom are degenerate: States Ž n,l . and Ž nX ,l q k ., k ) 0, of the encapsulated hydrogen atom are degenerate, i.e. correspond to the same energy E, for a given value of R if and only if the following two conditions are fulfilled simultaneously: ` E ) Elqkq1 sy
1 2 Ž l q k q 1.
2
;
Ž 3.1 .
a l k Ž R , E . s 0,
Ž 3.2 .
where a l k has been defined in Eq. Ž2.8.. 3.2. Proof: necessity. Let EnX ,lqk Ž R . be the energy of the Ž nX ,l q k .-state of the atom in a box of radius R; by the variational principle Žsee also Refs. w8,9,12x., it is a monotonically decreasing function of R that approaches the limiting value En`X s y1r2 nX 2 as R ™ ` w8,9,12x; on the other hand, the principal quantum number nX is not less than l q k q 1 Žsee Section 1., so that the following inequalities hold: E s EnX ,lqk Ž R . ) En`X s y
1 X2
2n
0y
1 2 Ž l q k q 1.
2
,
and Ž3.1. is proved. Now let c Ž r . s f Ž r . Yl Ž u , w . and w Ž r . s g Ž r . Ylqk Ž u , w . be the wavefunctions of the Ž n,l .and Ž nX ,l q k .-states, respectively. By the supposition, E is the energy of both states. The function k x s Aq c has the form x Ž r . s h Ž r . Ylqk Ž u , w ., according to Ž2.8. and comparison of Ž2.14. and Ž3.1. shows that x Žand hence h Ž r . . is not identically zero. The radial factor h Ž r . is a bounded solution of the radial Schrodinger equation with the same eigen¨ value E and the same angular quantum number l q k as g Ž r . . Therefore, h Ž r . and g Ž r . are two nontrivial solutions of the same second-order ordinary differential equation, which are both regular at the
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
singular point r s 0; hence, they are proportional Žsee, e.g. Ref. w13x. and from the BC Ž2.2b.,
x Ž r . rs R A w Ž r . rsR s 0 . Then Eq. Ž3.2. follows immediately from Eq. Ž2.8. as f X Ž R . / 0 Žsee remark Žii. in Section 2.1.. 3.3. Sufficiency. Let c be the wavefunction of the Ž n,l .-state with energy E; suppose that both Eqs. Ž3.1. and Ž3.2. hold. Then, by Eqs. Ž3.2. and Ž2.8., k x s Aq c satisfies the BC Ž2.2b.; from Eqs. Ž3.1. and Ž2.14. it follows that x is not identically zero. Being also a solution of Eq. Ž2.1., x is hence the wavefunction of a Ž nX ,l q k .-state degenerate in energy with the Ž n,l .-state, QED. 3.4. One simple consequence Žwhich in fact has been implicitly mentioned in Section 2. is straightforward: the states Ž n,l . and Ž nX ,l q 1. are never degenerated. Indeed, it is evident from Ž2.9a. that condition Ž3.2. can not be fulfilled for k s 1. This conclusion allows one to explain why the Coulomb degeneracy is lost when the atom is placed in a spherical box; it also shows that the inequalities Ž1.1. hold independently of R Žsee Section 1.. 3.5. The other interesting case is k s 2. Eq. Ž2.9b. shows that a l 2 Ž R . has a unique zero R l s Ž l q 1.Ž l q 2., independent of E. Therefore, if the radius of the box is equal to R l , then eÕery state Ž n,l . for which Ž3.1. holds, i.e. ` En l Ž R l . ) Elq3 sy
1 2 Ž l q 3.
2
,
Ž 3.7 .
is degenerate with some Ž nX ,l q 2.-state, i.e. a simultaneous degeneracy takes place Žin Eq. Ž1.2., the degeneracy of s- and d-levels at R s 2 is shown as an illustration.. Note that the relation nX s n q 1 is a direct consequence of the above statement that the land Ž l q 1.-states cannot be degenerate in a box of finite radius. 3.6. One more interesting feature of the value Ž l q 1.Ž l q 2. is that it is at the same time the only radial node of the wavefunction c of the Ž l q 2,l .state of the free atom Že.g. 2s, 3p, etc... Hence, in a spherical box of radius R l s Ž l q 1.Ž l q 2., there
221
should exist an l-type atomic state with the same wavefunction c and the same energy. As c has no radial nodes except for the boundary sphere Žand the origin for l ) 0., it corresponds to the Ž l q 1,l .-state of the atom in a box, i.e. the ground state for the particular l value. This state is not degenerate with 2 any Ž nX ,l q 2.-state since the operator Aq converts c into zero Žsee the previous section.. The following diagram illustrates the simultaneous degeneracy of l- and Ž l q 2.-states in boxes of 2 radii R l , also showing how Aq acts on the corresponding wavefunctions Žcf. Ž2.15..: ls 0, Rs 2:
2 Aq :
° ~ ¢
1s ™ 0 2s ™ 3dq2 3s ™ 4dq2 ...
ls1, Rs 6:
° ~ ¢
2pq1 ™ 0 3pq1 ™ 4fq3
4pq1 ™ 5fq3 ...
Arbitrary l : R s Ž l q1 .Ž l q2 . :
½
< l q1, l : ™ 0 < n, l : ™ < nq1, l q2: , n) l q1
However, such a regular picture is not reproduced for higher values of k, for which the solution of Eq. Ž3.2. depends on E and hence, different states can be degenerate at different R values. In the other words, there is no reason to expect any simultaneous degeneracy of states whose l values differ by more than 2.
4. Conclusions The Lenz vector A has been shown to be a useful tool for understanding the picture of accidental degeneracies of the states of a hydrogen atom in a spherical impenetrable box. Indeed, any accidental degeneracy of energy levels in the problem is closely k , which conconnected with ‘step-up’ operators Aq vert one degenerate state into another. This formalism is similar to the conventional angular momentum algebra, especially in the ‘usual’ hydrogen atom problem Žsee the end of Section 2., where it turns out to be close to the approach developed in the classical work by Fock w11x Žsee Ref. w6x for details.. Contrary to the free hydrogen atom, in an encapsulated atom A is no longer an additional integral of motion since it is not commutable with the Hamiltonian of the boundary value problem Ž2.1. – Ž2.2. ŽSection 2.. The non-conservation of the Lenz vector is also evident from the following classical picture: when an electron, moving along the Kepler orbit around the nucleus, reaches the wall of the box, its
222
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
momentum p direction suddenly changes Želastic reflection., while the angular momentum l remains invariant due to the spherical symmetry of the boundary. Hence, A is not a constant of motion in general. It is interesting that only the direction of A varies, but not the length of the vector, which is a unique function of the energy and the angular momentum w7x. k operator formalism developed here gives The Aq an explicit interpretation for the structure of the energy spectrum of a spherically confined hydrogen atom, perhaps being more physically grounded than Žalthough mathematically equivalent to. the approach presented in our former works w8,9x, based on the use of some special properties of confluent hypergeometric functions Žthe so-called Gauss identities w14x..
Acknowledgements The authors are thankful to Prof. N.F. Stepanov for useful discussions. The financial support from the Russian Foundation for Fundamental Research ŽProject 98-03-33232a. and the Program ‘Universities of
Russia – Fundamental Research’ is greatly appreciated. References w1x A. Michels, J. De Boer, A. Bijl, Physica 4 Ž1937. 981. w2x P.W. Fowler, Mol. Phys. 53 Ž1984. 865. w3x P.O. Froman, S. Ingve, N. Froman, J. Math. Phys. 28 Ž1987. ¨ ¨ 1813. w4x Y.P. Varshni, J. Phys. B 30 Ž1997. L589. w5x W. Jaskolski, Phys. Rep. 271 Ž1996. 1. ´ w6x L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon, Oxford, 1981. w7x A. Bohm, Quantum Mechanics: Foundations and Applications, Springer, New York, 1986. w8x A.V. Scherbinin, V.I. Pupyshev, A.Yu. Ermilov, Izv. Akad. Nauk. Ser. Fiz. ŽRuss. Phys. Bull.. 61 Ž1997. 1779. w9x A.V. Scherbinin, V.I. Pupyshev, A.Yu. Ermilov, Physics of Clusters, World Scientific, Singapore, 1997, p. 273. w10x V.I. Pupyshev, A.V. Scherbinin, N.F. Stepanov, J. Math. Phys. 38 Ž1997. 5626. w11x V.A. Fock, Zs. f. Phys. 98 Ž1935. 145. w12x M.A. Nunez, Int. J. Quantum Chem. 50 Ž1994. 113. ´ w13x W. Wasow, Asymptotic Expansions for Ordinary Differential Equation, Wiley, New York, 1965. w14x H. Bateman, A. Erdelyi, Higher Transcendental Functions, ´ vol. 1, McGraw–Hill, New York, 1953.
Chemical Physics Letters 295 Ž1998. 217–222
The Lenz vector in the confined hydrogen atom problem
1
Vladimir I. Pupyshev ) , Andrei V. Scherbinin Laboratory of Molecular Structure and Quantum Mechanics, Chemistry Department, Moscow State UniÕersity, Moscow, 119899, Russian Federation Received 13 July 1998; revised 24 August 1998
Abstract A general criterion for the degeneracy of states of the hydrogen atom placed in a spherical impenetrable box of radius R is formulated on the basis of the Lenz vector formalism. The reasons for the elimination of the Coulomb degeneracy typical of the conventional hydrogen atom problem and for the non-conservation of the Lenz vector are discussed. The explanation of the simultaneous degeneracy of some energy levels Že.g. 2s and 3d, 3s and 4d, etc. at R s 2 au. is given. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The quantum–mechanical model for an atom with a nucleus placed at the center of an impenetrable spherical box of radius R has been discussed since 1937, when it was considered for the hydrogen atom w1x. The interest in this field is stimulated by a large variety of possible applications of the model — from astrophysics to solid state theory Žsee Refs. w2–5x and the references therein.. The well-known quantum problem of the hydrogen atom corresponds to R ™ `, which allows the states of the confined atom Ž R - `. to be denoted by the same set of quantum numbers Ž n,l . as those used in the former case, taking into account the fact that the principal quantum number satisfies the inequality n 0 l q 1. As it turns out, both problems are essentially different. In particular, the radial wavefunctions of the confined atom are expressed via the confluent
hypergeometric function w2,5,8,9x, which is not reduced to elementary functions in general, except for some special cases, considered in Section 3.6. As a consequence, the corresponding eigenvalues Enl Ž R . are determined by a transcendental equation and there is no general formula for them. The specific feature of a conventional hydrogen atom problem is the presence of an additional Žapart from the energy and the angular momentum. integral of the motion, the so-called Lenz vector Žor the Runge–Lenz vector, known also as the Laplace integral in the classical Kepler problem., which results in ‘accidental’ n 2-fold degeneracy of all the energy levels with the same principal quantum number n Žthe so-called ‘Coulomb degeneracy’. w6,7x. In the confined atom problem, the Coulomb degeneracy is broken, and the energy levels corresponding to the same principal quantum number are ordered as follows w2x: E2 p Ž R . - E2 s Ž R . ;
) 1
Corresponding author. E-mail: [email protected] In commemoration of the 100th anniversary of V.A. Fock.
E3d Ž R . - E3 p Ž R . - E3s Ž R . ; . . .
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 9 6 1 - 0
Ž 1.1 .
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
218
for any finite R w8,9x. Hence, the Lenz vector is not a constant of the motion for finite R values. However, some other levels can become degenerate instead. In particular, a simultaneous degeneracy of the states Ž n,l . and Ž n q 1,l q 2., n F l q 2, is observed when R reaches the value Ž l q 1.Ž l q 2. au w8,9x. For example, at R s 2 au, E2 s s E3d ;
E3s s E4 d ; . . .
Ž 1.2 .
In the present work the following problems are discussed: Ži. why the Coulomb degeneracy is broken at finite R values and Žii. the reason for the regular picture of the accidental degeneracy of energy levels whose l numbers differ by 2. As will be shown below, any accidental degeneracy of energy levels in the spherically confined hydrogen atom problem is closely connected with some special properties of the Lenz vector, which is defined as: r 1 A s y Ž p = l y l = p. , Ž 1.3 . r 2 Žatomic units are used throughout this Letter., where r is the electron position vector, r s
2.1. The wavefunction c of a stationary state of the hydrogen atom in a spherical box of radius R is the solution of the Schrodinger equation, ¨
Ž 2.1 .
with the boundary conditions ŽBC.:
c Ž r . is bounded at r s 0 ; c Ž r . rs R s 0 .
c Ž r . s f Ž r . Yl Ž u , w . ,
Ž 2.3 .
where Yl s Yl l Ž u , w . s sin lu e i l w is the unnormalized spherical harmonic function. The radial wavefunction f Ž r . is a solution of the corresponding secondorder ordinary differential equation, bounded at r s 0 due to Ž2.2a. and such that: fŽ R . s 0
Ž 2.4 .
due to Ž2.2b.. Note that: Ži. f Ž r . A r l , r ™ 0, as f Ž r . is a regular solution w6x; Žii. f X Ž R . / 0 since f Ž r . is a non-trivial solution of a second-order linear differential equation and Ž2.4. holds. 2.2. The explicit expression for the operator Aqs A x q i A y in spherical coordinates Ž r, u , w . reads: Aqs pz lqq
xqi y r
q i Ž px q i p y . Ž l z q 1. ,
Ž 2.5a . or, equivalently, Aqs Ž x q i y .
ž
1 r
E y p2 y i r
/ ž
Er
q1
/Ž
px q i p y . .
Ž 2.5b . Here lqs l x q i l y is the standard ‘step-up’ operator of the angular momentum theory and p x q i p y s yie i w
2. Operators AkH
Hc s Ec ,
possible value of the angular momentum projection, namely
Ž 2.2a . Ž 2.2b .
Here H s y1r2 D y 1rr is the non-relativistic Hamiltonian of the hydrogen atom. Taking into account the spherical symmetry of the problem, it is sufficient to consider the states with the highest
ž
= sin u
cos u E
E q Er
r
Eu
i q
E
r sin u Ew
/
.
Ž 2.6 . Using Eqs. Ž2.5a., Ž2.5b. and Ž2.6., it is easy to check that if c has the form Ž2.3., then
x s Aq c s Ž l q 1 . f X q 1 y
ž
l Ž l q 1. r
/
f Ylq1 .
Ž 2.7 . Applying the operator Aq k times results in cumbersome relations which include higher-order
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
derivatives of the radial part f Ž r . of c . However, if one takes into account the fact that f Ž r . is a solution of the radial Schrodinger equation, then the explicit ¨ k expression for x s Aq c is reduced to the following simple one: k k x s Aq c s Aq Ž fYl .
s a l k Ž r , E . f X q b l k Ž r , E . f Ylqk ,
Ž 2.8 .
where a l k Ž r , E . and b l k Ž r , E . are polynomials in 1rr, as may be verified by induction. In the case k s 1, Eq. Ž2.7. gives:
a l1 s l q 1 ,
Ž 2.9a .
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Hamiltonian of a free system and that of the system in a box are in fact different operators with different domains Žsee, e.g. Ref. w10x.. One should mention, however, that the other BC, Eq. Ž2.2a., can not be a source of any difficulty; indeed, we have f Ž r . A r l , r ™ 0 and from Eq. Ž2.7., the radial part of x s Aq c goes to zero at least as r l too; hence, x is bounded at the origin provided that the original wavefunction c is. Being at the same time a regular solution of Eq. Ž2.1. with the angular momentum value Ž l q 1., c therefore vanishes as r lq1 at the origin. This conclusion is k c , k ) 1. easily generalized for x s Aq
and in the case k s 2:
ž
a l 2 s Ž 2 l q 3. 1 y
Ž l q 1. Ž l q 2. r
/
.
Ž 2.9b .
For k ) 2, a l k Ž r , E . is a polynomial in E of order Ž k y 2.. 2.3. The components of the Lenz vector Ž1.3. are formally commutable with the Hamiltonian H w6,7x; the same is obviously true for the operator Aqs A x q i A y . It means that if c is a solution of Eq. Ž2.1., then x s Aq c is also a solution: H x s HAq c s Aq Hc s EAq c s E x .
Ž 2.10 .
Based on Eqs. Ž2.7. and Ž2.10., one would think that Aq plays the role of a ‘step-up’ operator, which increases the angular quantum number l by 1. As long as the free atom is considered, this is indeed true Žsee Section 2.6.; however, for the confined atom, this formal observation fails owing to the following fact: although c is a wavefunction of the system, satisfying both Eqs. Ž2.1., Ž2.2a. and Ž2.2b., x is definitely not. Indeed, Eq. Ž2.7. implies that:
xrs R s Ž l q 1 . f X Ž R . Ylq1
Ž 2.11 .
which is inconsistent with the BC Ž2.2b. Žrecall that f X Ž R . / 0; see remark Žii. at the end of Section 2.1.. In other words, x s Aq c is not from the domain of the Hamiltonian of the confined atom problem Ž2.1. – Ž2.2. and hence it cannot be a wavefunction of the system. Therefore, in the confined atom problem, the components of the Lenz vector are no longer commutable with the Hamiltonian as operators in the Hilbert space Ževen though their formal expressions still are.; this conclusion is not surprising since the
2.4. It follows from the previous discussion that the formal commutator relation Ž2.10. is insufficient to guarantee that x s Aq c is a wavefunction of the problem; it has been shown, in particular, how inconsistencies with the BC Ž2.2b. can arise. Another question is whether x ' 0. For this to be the case, according to Eq. Ž2.7., the radial part of c should satisfy the following equation:
Ž l q 1. f X q 1 y
ž
l Ž l q 1. r
/
fs0 ,
with the solution of the form f Ž r . s Cr l exp y
ž
r lq1
/
,
Ž 2.12 .
i.e. c is proportional to the wavefunction of the Ž l q 1,l .-state of the free atom Ž R s `. with energy w6,7x ` E s Elq1 sy
1 2 Ž l q 1.
2
.
Ž 2.13 .
In a similar manner, if A k , k ) 1, is considered k instead of Aq, then the equation x s Aq ' 0 implies that: ` E s Elqj sy
1 2Ž l q j .
2
Ž 2.14 .
k with some j, 1 F j F k. Indeed, if Aq c ' 0, then there exists some j varying in the limits specified above, such that none of the functions c , Aq c , 2 jy1 j Aq c , . . . , Aq c is identically zero, while Aq c' k jy1 . . . ' Aq c ' 0. This means that Aq c is the wave-
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
220
function of the Ž l q j,l q j y 1.-state of the free hydrogen atom, which leads to Eq. Ž2.14.. 2.5. The results of this section can be summarized as follows. If c is a solution of Eq. Ž2.1. such that: Ža. c is bounded at the origin, Žb. c has a factorized form Ž2.3. and corresponds to the highest angular k momentum projection, then x s Aq c is also a solution of Eq. Ž2.1., which is bounded at the origin as well and corresponds to the angular quantum number l q k. In addition, x is not identically zero provided that Ž2.14. is not fulfilled. What is in question is the BC Ž2.2b. only. If x satisfies Ž2.2b., then x is also a wavefunction corresponding to the same energy but a different angular momentum value, i.e. the k energy level E is extra degenerate, Aq playing the role of a ‘step-up’ operator. 2.6. As an example, the well-known case R s ` can be considered, where the BC Ž2.2b. simply demands that c is vanishing at infinity rapidly enough to be square-integrable Žfor the discrete spectrum states.. In fact, c then decays exponentially and it is evident from Eqs. Ž2.7. – Ž2.11. that x s Aq c satisfies the BC at infinity straightforwardly. As a consequence, the accidental ‘Coulomb’ degeneracy occurs. Given below is a diagram featuring the ‘stepup’ property of the Aq operator which converts the wavefunctions of the mutually degenerated states as follows:
°1s ™ 0 Aq :
~2s ™ 2p
q1 ™ 0
¢
3s ™ 3pq1 ™ 3dq2 ™ 0 ...
Ž 2.15 .
Žsubscript denotes the angular momentum projection value.. Note that the states 1s, 2p, 3d, etc., are those with radial wavefunctions of the form Ž2.12., so that they are indeed from the kernel of Aq. Moreover, on the basis of the properties of Aq discussed above, it is possible to give a complete description of the discrete energy spectrum of the free hydrogen atom, taking into account that, for a given discrete spectrum energy value E, the possible angular momentum values are bounded from above. One should mention, however, that this is just a variety of the well-known approach to the quantum Kepler problem in terms of the Lenz vector w6,7x and
the four-dimensional angular momentum formalism derived by Fock many years ago w11x.
3. A general criterion for the degeneracy 3.1. This section is devoted to establishing rigorous necessary and sufficient conditions under which two given states Ž n,l . and Ž nX ,lX . of the encapsulated atom are degenerate: States Ž n,l . and Ž nX ,l q k ., k ) 0, of the encapsulated hydrogen atom are degenerate, i.e. correspond to the same energy E, for a given value of R if and only if the following two conditions are fulfilled simultaneously: ` E ) Elqkq1 sy
1 2 Ž l q k q 1.
2
;
Ž 3.1 .
a l k Ž R , E . s 0,
Ž 3.2 .
where a l k has been defined in Eq. Ž2.8.. 3.2. Proof: necessity. Let EnX ,lqk Ž R . be the energy of the Ž nX ,l q k .-state of the atom in a box of radius R; by the variational principle Žsee also Refs. w8,9,12x., it is a monotonically decreasing function of R that approaches the limiting value En`X s y1r2 nX 2 as R ™ ` w8,9,12x; on the other hand, the principal quantum number nX is not less than l q k q 1 Žsee Section 1., so that the following inequalities hold: E s EnX ,lqk Ž R . ) En`X s y
1 X2
2n
0y
1 2 Ž l q k q 1.
2
,
and Ž3.1. is proved. Now let c Ž r . s f Ž r . Yl Ž u , w . and w Ž r . s g Ž r . Ylqk Ž u , w . be the wavefunctions of the Ž n,l .and Ž nX ,l q k .-states, respectively. By the supposition, E is the energy of both states. The function k x s Aq c has the form x Ž r . s h Ž r . Ylqk Ž u , w ., according to Ž2.8. and comparison of Ž2.14. and Ž3.1. shows that x Žand hence h Ž r . . is not identically zero. The radial factor h Ž r . is a bounded solution of the radial Schrodinger equation with the same eigen¨ value E and the same angular quantum number l q k as g Ž r . . Therefore, h Ž r . and g Ž r . are two nontrivial solutions of the same second-order ordinary differential equation, which are both regular at the
V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
singular point r s 0; hence, they are proportional Žsee, e.g. Ref. w13x. and from the BC Ž2.2b.,
x Ž r . rs R A w Ž r . rsR s 0 . Then Eq. Ž3.2. follows immediately from Eq. Ž2.8. as f X Ž R . / 0 Žsee remark Žii. in Section 2.1.. 3.3. Sufficiency. Let c be the wavefunction of the Ž n,l .-state with energy E; suppose that both Eqs. Ž3.1. and Ž3.2. hold. Then, by Eqs. Ž3.2. and Ž2.8., k x s Aq c satisfies the BC Ž2.2b.; from Eqs. Ž3.1. and Ž2.14. it follows that x is not identically zero. Being also a solution of Eq. Ž2.1., x is hence the wavefunction of a Ž nX ,l q k .-state degenerate in energy with the Ž n,l .-state, QED. 3.4. One simple consequence Žwhich in fact has been implicitly mentioned in Section 2. is straightforward: the states Ž n,l . and Ž nX ,l q 1. are never degenerated. Indeed, it is evident from Ž2.9a. that condition Ž3.2. can not be fulfilled for k s 1. This conclusion allows one to explain why the Coulomb degeneracy is lost when the atom is placed in a spherical box; it also shows that the inequalities Ž1.1. hold independently of R Žsee Section 1.. 3.5. The other interesting case is k s 2. Eq. Ž2.9b. shows that a l 2 Ž R . has a unique zero R l s Ž l q 1.Ž l q 2., independent of E. Therefore, if the radius of the box is equal to R l , then eÕery state Ž n,l . for which Ž3.1. holds, i.e. ` En l Ž R l . ) Elq3 sy
1 2 Ž l q 3.
2
,
Ž 3.7 .
is degenerate with some Ž nX ,l q 2.-state, i.e. a simultaneous degeneracy takes place Žin Eq. Ž1.2., the degeneracy of s- and d-levels at R s 2 is shown as an illustration.. Note that the relation nX s n q 1 is a direct consequence of the above statement that the land Ž l q 1.-states cannot be degenerate in a box of finite radius. 3.6. One more interesting feature of the value Ž l q 1.Ž l q 2. is that it is at the same time the only radial node of the wavefunction c of the Ž l q 2,l .state of the free atom Že.g. 2s, 3p, etc... Hence, in a spherical box of radius R l s Ž l q 1.Ž l q 2., there
221
should exist an l-type atomic state with the same wavefunction c and the same energy. As c has no radial nodes except for the boundary sphere Žand the origin for l ) 0., it corresponds to the Ž l q 1,l .-state of the atom in a box, i.e. the ground state for the particular l value. This state is not degenerate with 2 any Ž nX ,l q 2.-state since the operator Aq converts c into zero Žsee the previous section.. The following diagram illustrates the simultaneous degeneracy of l- and Ž l q 2.-states in boxes of 2 radii R l , also showing how Aq acts on the corresponding wavefunctions Žcf. Ž2.15..: ls 0, Rs 2:
2 Aq :
° ~ ¢
1s ™ 0 2s ™ 3dq2 3s ™ 4dq2 ...
ls1, Rs 6:
° ~ ¢
2pq1 ™ 0 3pq1 ™ 4fq3
4pq1 ™ 5fq3 ...
Arbitrary l : R s Ž l q1 .Ž l q2 . :
½
< l q1, l : ™ 0 < n, l : ™ < nq1, l q2: , n) l q1
However, such a regular picture is not reproduced for higher values of k, for which the solution of Eq. Ž3.2. depends on E and hence, different states can be degenerate at different R values. In the other words, there is no reason to expect any simultaneous degeneracy of states whose l values differ by more than 2.
4. Conclusions The Lenz vector A has been shown to be a useful tool for understanding the picture of accidental degeneracies of the states of a hydrogen atom in a spherical impenetrable box. Indeed, any accidental degeneracy of energy levels in the problem is closely k , which conconnected with ‘step-up’ operators Aq vert one degenerate state into another. This formalism is similar to the conventional angular momentum algebra, especially in the ‘usual’ hydrogen atom problem Žsee the end of Section 2., where it turns out to be close to the approach developed in the classical work by Fock w11x Žsee Ref. w6x for details.. Contrary to the free hydrogen atom, in an encapsulated atom A is no longer an additional integral of motion since it is not commutable with the Hamiltonian of the boundary value problem Ž2.1. – Ž2.2. ŽSection 2.. The non-conservation of the Lenz vector is also evident from the following classical picture: when an electron, moving along the Kepler orbit around the nucleus, reaches the wall of the box, its
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V.I. PupysheÕ, A.V. Scherbininr Chemical Physics Letters 295 (1998) 217–222
momentum p direction suddenly changes Želastic reflection., while the angular momentum l remains invariant due to the spherical symmetry of the boundary. Hence, A is not a constant of motion in general. It is interesting that only the direction of A varies, but not the length of the vector, which is a unique function of the energy and the angular momentum w7x. k operator formalism developed here gives The Aq an explicit interpretation for the structure of the energy spectrum of a spherically confined hydrogen atom, perhaps being more physically grounded than Žalthough mathematically equivalent to. the approach presented in our former works w8,9x, based on the use of some special properties of confluent hypergeometric functions Žthe so-called Gauss identities w14x..
Acknowledgements The authors are thankful to Prof. N.F. Stepanov for useful discussions. The financial support from the Russian Foundation for Fundamental Research ŽProject 98-03-33232a. and the Program ‘Universities of
Russia – Fundamental Research’ is greatly appreciated. References w1x A. Michels, J. De Boer, A. Bijl, Physica 4 Ž1937. 981. w2x P.W. Fowler, Mol. Phys. 53 Ž1984. 865. w3x P.O. Froman, S. Ingve, N. Froman, J. Math. Phys. 28 Ž1987. ¨ ¨ 1813. w4x Y.P. Varshni, J. Phys. B 30 Ž1997. L589. w5x W. Jaskolski, Phys. Rep. 271 Ž1996. 1. ´ w6x L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon, Oxford, 1981. w7x A. Bohm, Quantum Mechanics: Foundations and Applications, Springer, New York, 1986. w8x A.V. Scherbinin, V.I. Pupyshev, A.Yu. Ermilov, Izv. Akad. Nauk. Ser. Fiz. ŽRuss. Phys. Bull.. 61 Ž1997. 1779. w9x A.V. Scherbinin, V.I. Pupyshev, A.Yu. Ermilov, Physics of Clusters, World Scientific, Singapore, 1997, p. 273. w10x V.I. Pupyshev, A.V. Scherbinin, N.F. Stepanov, J. Math. Phys. 38 Ž1997. 5626. w11x V.A. Fock, Zs. f. Phys. 98 Ž1935. 145. w12x M.A. Nunez, Int. J. Quantum Chem. 50 Ž1994. 113. ´ w13x W. Wasow, Asymptotic Expansions for Ordinary Differential Equation, Wiley, New York, 1965. w14x H. Bateman, A. Erdelyi, Higher Transcendental Functions, ´ vol. 1, McGraw–Hill, New York, 1953.