A complete set of ladder operators for the hydrogen atom

Transcription

1 A copete set of adder operators for the hydrogen ato C. E. Burkhardt St. Louis Counity Coege at Forissant Vaey 3400 Persha Road St. Louis, MO J. J. Leventha Departent of Physics University of Missouri - St. Louis St. Louis, MO 63 Abstract Ladder operators capabe of converting one hydrogen ato eigenfunction into another by raising the anguar oentu quantu nuber are derived using ony eeentary techniques. The derivation is perfored using forais no ore sophisticated than that used to derive the properties of the ordinary anguar oentu adder operators in undergraduate quantu echanics courses. The properties of these operators, which consist of coponents of the quantu echanica Len vector, deonstrate the accidenta degeneracy of the hydrogen ato. It is shown that, starting fro the 0 eigenfunction for a given principa quantu nuber n the copete set of eigenfunctions for that n can be obtained. PACS nubers: Ca ; Fd ; Ge

2 It is we-known that the Keper/Couob potentias endow panetary orbits and hydrogen atos with specia properties, properties not present in systes subject to other centra potentias. A pure Keperian orbit is fixed in space, that is, it does not precess. Moreover, the tota energy of the syste depends ony on the vaue of the sei-ajor axis a and not on that of the sei-inor axis b. There exists, therefore, an infinity of possibe orbits, a having the sae energy and sei-ajor axis, but having different vaues for the sei-inor axis. Whie a is independent of the anguar oentu, b is deterined by the anguar oentu so the energy is independent of the anguar oentu. This cassica degeneracy is the resut of the sae syetry of the /r potentia that causes the ceebrated "accidenta degeneracy" of the hydrogen ato, the independence of the energy eigenvaues on the anguar oentu quantu nuber. This syetry is different fro the spatia syetry extant for any centra potentia. It is often referred to as a "dynaica syetry"[]. The spatia syetry causes the energy to be independent of, the quantu nuber corresponding to the -coponent of the anguar oentu. Cassicay, the dynaica syetry anifests itsef as an additiona constant of the otion, the Len vector A, which points aong the ajor axis of the eipse[]. This resuts in an orbit that is fixed in space. Quantu echanicay, A corresponds to an additiona operator  that coutes with the Haitonian Ĥ as shown by Paui in his andark paper[3]. Of course the agnitude of the anguar oentu and each of its coponents aso coute with Ĥ for any centra potentia. Athough these consequences of the dynaica syetry of the /r potentia are rarey discussed at the undergraduate eve, they provide insight into a variety of cassica and quantu concepts. In this paper we wi concentrate on the quantu echanica consequences of the cassica constant A. We wi show that it is possibe to derive a set of adder operators invoving coponents of  that transfors certain spherica hydrogen eigenstates into other spherica eigenstates. Spherica eigenstates are those that resut fro separation of the Schrödinger

3 equation in spherica coordinates and is characteried by the quantu nubers n (energy), (anguar oentu) and (-coponent of anguar oentu). The dynaica syetry akes it possibe to separate the Schrödinger in paraboic coordinates as we[4]. The fact that the Lˆ change ony the vaue of and neither nor n is a anifestation of the degeneracy associated with the spatia syetry. Because of this the Lˆ are adder operators for any centra potentia. The  and  adder operators effect state-to-state changes in the quantu nuber on ony spherica eigenfunctions for the Couob potentia, and, as wi be shown, ony on states for which  Â. Because and change they refect the accidenta degeneracy of the hydrogen ato in which the energy eigenvaues are independent of. In this paper we wi derive the properties of and and obtain the exact expressions for the actions of these operators without having to resort to advanced concepts[5]. The derivation wi be carried out using forais no ore sophisticated than that used to derive the properties   of Lˆ in ost undergraduate quantu echanics courses. It wi be seen that, given the 0 eigenfuntion for a particuar n, a eigenfunctions for that n can be obtained by judicious appication a cobination of Lˆ,  and Â. Using atoic units for which h e e where e is the eectronic charge and e the eectronic ass, the Len vector in cassica echanics is defined in atoic units as A p L ˆr ( ) where p is the inear oentu, L the anguar oentu and rˆ the unit vector in the r direction. The direction of A, toward apocenter or pericenter, is a atter of choice. We eect the definition in Equation ()[6]. The quantu echanica operator corresponding to A cannot be constructed by erey repacing each quantity with its corresponding quantu echanica 3

4 operator because the resut is a non-heretian operator. Paui recognied that  ust be defined as[3] Note that the quantu echanica operators ( pˆ Lˆ ) ( Lˆ pˆ ) [( pˆ Lˆ ) ( Lˆ pˆ )] rˆ ( ). It is easiy shown that  as defined in Equation () is indeed Heretian. It can aso be shown, but with consideraby ore A ˆ Hˆ. abor, that [, ] 0 The anguar oentu adder operators are defined as Lˆ L ˆ Lˆ i ˆ x L y ( 3) and, when operating on a spherica eigenfunction for any centra potentia, which we designate as n C, cause the foowing state-to-state conversion. ( )( ) n ; ( ) ( 4) L ˆ n C C where we have inserted a sei-coon between quantu nubers in the ket on the right hand side for carity. Equation (4) is, of course, aso vaid for spherica hydrogen ato eigenfunctions since the Couob potentia is a centra potentia. The operators  are defined as A ˆ i ˆ x A y (5) For convenience, a nuber of reations between the various operators are copied in Tabe I. Soe require engthy agebraic anipuations, but a are straightforward. We investigate the action of  on the spherica hydrogen eigenfunction n. We specify that this is a hydrogen eigenfunction by oitting the subscript C that was used to designate an eigenfunction for an arbitrary centra potentia. Using the reations contained in Tabe I we find that 4

5 which shows that { A n Lˆ { n Lˆ ( x i y ) n ( Lˆ ) n ( 6) ( ){ n ˆ is an eigenfunction of with eigenvaue Lˆ ( ). Siiary { ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ n A L A L A A L ) n { ( ) ( ) { n ( ) ( ) n; ( ) ( 7) ˆ L Note that the { A n n are not eigenfunctions of Â, but if ˆ is an eigenfunction of ˆL with eigenvaue  the ter with vanishes and Because { A n ( ) ( ) ( )( ) ( 8) ˆ is an eigenfunction of ˆL and with eigenvaues respectivey, we know that where D Lˆ ( )( ) and ( ) A ˆ D ; n ( ) ; ( ) ( 9) is a constant that depends on. But, to construct a copete set of eigenfunctions for a given n fro ony D n 00 it is necessary to evauate. We begin by foowing a siiar procedure to that used to evauate the constants when finding the action of Lˆ on the n. C We consider the atrix eeent n * ˆ ˆ A A ( D ) D ( 0) * ( D ) where the was obtained by operating to the eft with  and noting that the Heretian   conjugate of is. We specify for convenience, and by anaogy with the assuption epoyed in the Lˆ construction, that is rea. The operator  ay be expanded giving D  5

6 Lˆ Hˆ ( Lˆ Hˆ Hˆ ) Lˆ Hˆ ( ) Since the n are eigenfunctions of a operators on the right hand side except  we obtain n ( ) ( ) D n ˆ A ( ) where we have used Lˆ Lˆ Hˆ ( ) n ( 3) To copete the evauation of D we ust deterine the atrix eeent. In order to do this we first exaine the consequence of operating on  n with. The - coponent of the vector operator  ay be written A ˆ cosθ ( 4) where θ is the poar ange in spherica coordinates. Bearing in ind that the anguar parts of the n are erey the usua spherica haronics Y ( θ,φ ) n R n ( r) cosθy ( cosθ ) ( 5) where R n () r is the radia part of the hydrogen ato eigenfunction. Using a we-known recursion reation for the spherica haronics[4] we see that ( θ, φ ) ( )( ) ( )( 3) ( θ, φ ) ( )( ) ( )( ) cosθy Y Y φ ( θ, ) (6) 6

7 n ( )( ) ( )( 3) ( )( ) ( )( 3) ( ) ( )( ) ( )( ) Rn () r Y ( θ, φ ) Y ; ( )( ) ( )( ) ( θ, φ ) ( 7) ( ) ; Thus, the action of  on n is to raise and ower by unity whie eaving unchanged. In the present case, however, the resut is even siper because thus eiinating the second ter. We find then that ˆ A Y ( θ, φ ) ( 8) ( ) ; which shows that  operating on n is a raising operator for, but not for. Whie our interest at this tie is in the eigenvector n it shoud be noted that the second ter in Equation (7) aso vanishes for. In fact the coefficient of the first ter in Equation (7)  is the sae for. Thus, raises, but not for when operating on either n or n ;. Ared with Equation (8) we now exaine the atrix eeent A Using the coutator [, ] n ; n ; Lˆ ˆ we have ( ) ( ) ˆ ; A D ( 9) ( ) ;( ) n ( ) ( ) Lˆ Lˆ ; ; n ;( ) ; ( ) Lˆ ( 0) ( ) ( ) ; where L ˆ 0 and we have operated to the eft with Lˆ noting that the Heretian conjugate of Lˆ is Lˆ. We have then ( ) n ; ; D ( ) ( ) 7

8 This equation uniquey deterines ˆ in ters of the constant. Note that without A D Equation (8) additiona ters in Equation () woud be possibe as ong as they were orthogona to n ;( ) ;. We have then ˆ A D ( ) ( ) ; ( ) Inserting Equation () into Equation (), soving for ( ) D and taking the square root we have D n ( ) [ n ( ) ] ( 3) ( 3) It is iportant to note that, athough the operator  was present in our anaysis of the action of  on n, the ony property of that was used was that it is the heretian conjugate is Â.   The action of on an eigenfunction was not needed. Aso, because was obtained by D D ) taking the square root of ( there is no inforation on the sign of. In accordance with convention it is the inus sign that is retained[5]. D It is tepting to assue that the action of  on a spherica eigenfunction is as a owering operator. In fact, for a state-to-state conversion it does ower the -coponent of the anguar oentu, but it raises the tota anguar oentu as wi be shown. In a anner identica to that epoyed for { A ˆ n we find that { A n of Lˆ with eigenvaue (. Evauation of ) Lˆ { A ˆ n ˆ is an eigenfunction reveas it to be { n { ( ) ( ) { n ( )( ) ;( ) ( 4) Lˆ ˆ is an eigenfunction of with eigenvaue so that { A n ˆL ( )( ) if. Note that 8

9 this is the ony way to ake the ter containing  vanish because is prohibited. We concude therefore that ( ) D ( ) ; ( ) ( 5) A ˆ ; D can be evauated in a anner anaogous to that epoyed to evauate, but it can aso be D D ( θ, φ ) ( ) { ( θ, φ ) * Y evauated in ters of by aking use of the fact that Y. In the bra and ket notation used here this becoes n; ( ) ( ) n ( 6) which we use to evauate the copex conjugate of the atrix eeent A ˆ. We have * { ( ) Now, the eft hand side of Equation (7) can be written The right hand side of Equation (7) ay be written * { ; ˆ ˆ ( ) ( ) A A ( ) ( 7) { ; ( ) ; ( ) * * { D ( ) ;( ) D ( ) ;( ) A ˆ * { n; ( ) ; ( ) n; ( ) n; ( ) D n; ( ) ( ; )[ ( ) ] ( 9) D ( )( ) ; ( ) ( ) D ( ; )( ) where we have used ( ) Equations (8) and (9) we see that * { ( 8) in Equation (9). Coparing the right hand sides of D D n ( ) [ n ( ) ] ( 30) ( 3) 9

10 Notice that had we evauated using the sae technique epoyed to evauate we D woud not have obtained the inus sign in Equation (30). D The properties of the raising and owering operators are suaried in Tabe II. Whie the anguar oentu adder operators change ony the quantu nuber a of the raising operators consisting of coponents of the Len vector operator change the quantu nuber. This is a consequence of the accidenta degeneracy of the hydrogen ato in which the energy eigenvaues are independent of. Exaination of Tabe II shows that it is possibe to construct the entire set of hydrogen ato eigenfunctions for a given n if ony the 0 eigenfunction for that n is known. Figure iustrates two possibe sequences of appications of the adder operators required to generate the copete set of n 3 spherica eigenfunctions. ACKNOWLEDGEMENT The authors woud ike to thank their coeagues, Joseph F. Baugh and Ta-Pei Cheng for hepfu discussions. 0

11 REFERENCES. L. I. Schiff, Quantu Mechanics (McGraw-Hi; New York; third editio 968). P 34.. T. P. Hee, C. E. Burkhardt, M. Ciocca and J. J. Leventha, "Cassica View of the Stark Effect in Hydrogen Atos", A. J. Phys. 60, 34 (99). 3. W. Paui, "On the Hydrogen Spectru fro the Standpoint of the New Quantu Mechanics" The Engish transation is in: Sources of Quantu Mechanics edited by B. L.d Van Der Waerden (Dover; New York; 967) pp H. A. Bethe and E. E. Sapeter, Quantu Mechanics of One- and Two-Eectron Atos (Springer-Verag; Beri 957). pp L. C. Biedenharn, J. D. Louck and P. A. Carruthers, Anguar Moentu in Quantu Mechanics: Theory and Appication (Addison-Wesey; Reading, MA; 98). Pp H. Godstein, Cassica Mechanics (Addison-Wesey; Reading, MA; second editio 980).

12 FIGURE CAPTIONS Figure. Two possibe sequences of appications of the adder operators that generate the copete set of n 3 spherica eigenfunctions starting with 300.

13 Figure. Two possibe sequences of appications of the adder operators that generate the copete set of n 3 spherica eigenfunctions starting with

14 Tabe I. List of soe reations invoving the quantu echanica operators used in this work. ( ) ( ) [ Lˆ i, Lˆ j ] ilˆ k [ Lˆ, ] [ Lˆ i, j ] i k [ i, j ] ilˆ k Hˆ ˆ ( ˆ A L ) Hˆ 4

15 Tabe II. The resut of operations with the adder operators on the specified spherica eigenfunctions of the hydrogen ato. Operation L n Resut n; ˆ ( )( ) ( ) [ ] A n ( ) n 3 ( ) n ; ( ) ; n ; ; ( ) [ n ( ) ] n ; ( ) ; ( ) n ( 3) A ( ) n ( ) n ( 3) ; A ˆ ; ( ) ( ) n ( ) n ( 3) n ; ; [ ] ( ) ( ) [ ] ( ) ( ) 5