Relativistic shape invariant potentials

Transcription

1 Relativistic shape ivaiat potetials A. D. Alhaidai Physics Depatmet, Kig Fahd Uivesity of Petoleum & Mieals, Box 5047, Dhaha 36, Saudi Aabia Diac equatio fo a chaged spio i electomagetic field is witte fo special cases of spheically symmetic potetials. This facilitates the itoductio of elativistic extesios of shape ivaiat potetial classes. We obtai the elativistic specta ad spio wavefuctios fo all potetials i oe of these classes. The oelativistic limit epoduces the usual Rose- Möse I & II, Eckat, Pöschl-Telle, ad Scaf potetials. PACS umbe(s): Pm, Ge Exactly solvable potetials i oelativistic quatum mechaics fall withi distict classes of shape ivaiat potetials [ 8]. Each potetial i a give class ca be mapped ito aothe i the same class by a caoical tasfomatio of the coodiates [7,9 3]. The tasfomatio gives a coespodece map amog the potetial paametes, agula mometum, ad eegy. Usig the esultig paamete substitutio ad the boud states spectum of the oigial potetial oe ca easily ad diectly obtai the specta of all othe potetials i the class. Moeove, the eigestates wavefuctios ae obtaied by simple tasfomatios of those of the oigial potetial. It is vey temptig to seach fo the elativistic extesios of these classes ad obtai the elativistic specta of the boud states ad coespodig spio wavefuctios. I fact, oe such class has aleady bee established. Recetly, the Diac-Möse potetial was itoduced ad its boud states spectum ad spio wavefuctios wee obtaied [4]. Togethe with its two well-established pates, the Diac- Coulomb ad Diac-Oscillato [5] potetials, they complete oe elativistic class. I this lette, we cotiue these effots by itoducig the elativistic extesio of yet aothe class of shape ivaiat potetials that icludes Diac-Rose-Möse, Diac-Eckat, Diac- Pöschl-Telle, ad Diac-Scaf potetials. We obtai thei elativistic boud states specta ad spio wavefuctios. This is accomplished by followig the same pocedue that was used i efeece [4] fo the itoductio ad solutio of the Diac-Möse poblem. We stat by settig up the physical poblem of a chaged spio i spheically symmetic fou-compoet electomagetic potetial. Gauge ivaiace ad spheically symmety of the electostatic potetial is used to aive at the adial Diac equatio. Aftewads, we apply a uitay tasfomatio to Diac equatio such that the esultig secod ode diffeetial equatio becomes Schödige-like so that compaiso with exactly solvable oelativistic poblems is taspaet. Thus, the esultig simple coespodece map amog paametes of the two poblems gives the sought afte boud states spectum ad wavefuctio. I atomic uits (m = e =! = ) ad takig the speed of light c = α, the Hamiltoia fo a Diac spio i fou-compoet electomagetic potetial, ( A0, A " ), eads " " " " + αa0 iασ + ασ A H = " " " " iασ + ασ A + α A 0

2 whee α is the fie stuctue costat ad " σ ae the thee Pauli spi matices. I quatum electodyamics (the theoy of iteactio of chaged paticles with the electomagetic field), local gauge symmety implies ivaiace ude the tasfomatio ( A, 0 A " ) ( A, 0 A " ) + ( α Λ t, Λ " ) whee Λ ( t, " ) is a eal space-time fuctio. That is addig a 4-dimesioal gadiet of the gauge field Λ ( t, " ) to the electomagetic potetial will ot alte the physical cotet of the theoy. I the lab fame, gauge ivaiace implies that the geeal fom of the electomagetic potetial fo static chage distibutio with spheical symmety is ( A, 0 A " ) = ( αv ( ),0 " ) + ( 0, Λ " ( ) ) ( αv ( ), W ˆ ( ) ) whee V() is the electostatic potetial fuctio ad ˆ is the adial uit vecto. Obviously, W() is a gauge field that does ot cotibute to the magetic field. Howeve, fixig this gauge degee of feedom by takig W = 0 is ot the best choice. A alteative ad pope gauge fixig coditio, which is much moe fuitful, will be imposed as a costait i equatio (4) below. We will coside, howeve, a alteative couplig of the electomagetic potetial to " the chaged Diac paticle. The two off-diagoal tems ασ A " i the Hamiltoia H above ae " to be eplaced by ± iασ A ", espectively, esultig i the followig two-compoet adial Diac equatio κ d + α V( ) α + W( ) g () g () d = ε κ d α W() α () V() f() f() d whee ε is the elativistic eegy ad κ is the spi-obit couplig paamete defied as κ = ± (j + ½) fo l = j ± ½. Equatio () gives two coupled fist ode diffeetial equatios fo the two adial spio compoets. By elimiatig the lowe compoet we obtai a secod ode diffeetial equatio fo the uppe. The esultig equatio may tu out to be ot Schödigelike, i.e. it may cotai fist ode deivatives. We apply a geeal local uitay tasfomatio that elimiates the fist ode deivative as follows: g ( ) cos( ρ( x)) si( ρ( x)) φ( x) = q( x) ad f( ) = si( ρ( x)) cos( ρ( x)) θ( x) () The stated equiemet gives the followig costait: dq d ρ dx α V + cos( ρ) + αsi( ρ) ( W + κ q) + α + ε = costat η 0 (3) dx dq dx This tasfomatio ad the esultig costait ae the elativistic aalog of poit caoical tasfomatio i oelativistic quatum mechaics [7,9 3]. I this aticle, we coside the case of global uitay tasfomatio defied by qx ( ) = x ad dρ dx= 0. Substitutig these i the costait equatio (3) yields α κ W() = V() S (4) η = C + ε whee S si( ρ) ad C cos( ρ). The fist elatio i (4) is the gauge fixig coditio fo the electomagetic potetial. The tasfomatio defied above subject to the costait maps the adial Diac equatio () ito the followig:

3 S αc d C+ α V α + V φ() φ() α S d = ε S αc d α V C θ() θ() + + α S d which i tu gives a equatio fo the lowe spio compoet i tems of the uppe: α S αc d θ() = + V + φ() C+ ε α S d (5) Givig the followig Schödige-like d ode diffeetial equatio fo the uppe compoet: d α α dv ε + V + εv φ( ) 0 = (6) d T T d α whee T S C = ta( ρ). Noelativistic shape ivaiat potetials ca be divided i two classes based o the fom of thei eigefuctios. I the fist class, which icludes the Coulomb, Oscillato ad Möse potetials, the wavefuctios ae witte i tems of the Cofluet hypegeometic fuctios. The elativistic extesio of this class has aleady bee established [4,5]. I the secod class, which is of iteest to ou peset ivestigatio, the wavefuctios ae witte i tems of the hypegeometic fuctios. This class icludes Rose-Möse, Eckat, Pöschl- Telle, ad Scaf potetials. The algebaic expessios of these potetials ad thei popeties ae give i [3-5,7,8] ad efeeces theei. Specifically, we will coside the hypebolic athe tha the tigoometic fom of these potetials. Theefoe, i ou attempt to seach fo the elativistic extesio of these potetials we will coside expessios fo V() o W() which ae simple liea combiatios of sih(), sech(), tah(), etc. such that the oelativistic potetials ae epoduced i the limit. Ou use of the tems simple ad liea i the pevious statemet is due to the fact that these ae domiat popeties of the elativistic theoy. As examples: () Diac equatio is liea i the deivative wheeas Schödige equatio is quadatic; () the Diac-Oscillato potetial [5] is liea i the coodiate while the oscillato potetial is quadatic; (3) the Diac-Möse potetial [4] is x liea i the expoetial (i.e. of the fom e ) wheeas the oelativistic Möse potetial is of the fom ( e x ). Now, let us coside the case whee the potetial fuctio V( ) = Dtah( λ) with D ad λ beig eal paametes. Equatio (6) gives the followig secod ode diffeetial equatio fo the uppe spio compoet d αd αd αd ε λ εdtah( λ) φ( ) d T T cosh ( λ = ) T α We compae this with Schödige equatio fo the S-wave Rose-Möse I potetial [7] d AA ( + λ) + Btah( λ) + A E φ( ) 0 = (7) d cosh ( λ) whee A, B, ad λ ae eal costat paametes with λa > 0, ad E is the oelativistic eegy. The compaiso gives the followig coespodece betwee oelativistic ad elativistic paametes: 3

4 A= α D T B= Dε (8) E = ( ε ) α The well-kow oelativistic boud states spectum of equatio (7) is λ ( B λ) A E = ( A λ ) +, = 0,,, max < A λ (9) ( A λ ) The substitutio fomulas i (8) give the followig spectum fo this elativistic Diac- Rose-Möse I potetial αλ αd λ ε = + ( α DT) + ( αd λt ) αd λt whee = 0,,,..., ad max is the smallest itege satisfyig max α D max > λt ( αd λt) + ( αλ) Takig the oelativistic limit of this spectum with α 0 ε + α E T ατ epoduces the oelativistic spectum (9) with τ = D/A. The boud states wavefuctio of the oelativistic poblem [7] is mapped, usig (8), ito the followig uppe spio compoet wavefuctio ( β ) ( ) (, ) () ( ) + γ β ( ) γ β + γ β γ φ = R z + z P () z µν, whee P( ) is the Jacobi polyomial [6], R is the omalizatio costat, ad z = tah( λ) β = αd λt Dε λ γ = β Equatio (5) gives the lowe spio compoet i tems of the uppe as αλ S d θ() = + βz+ ( z ) φ() ε + C αλ dz Usig the diffeetial ad ecusio popeties of the Jacobi polyomials [6], we ca wite this explicitly as αλ β µ ν ( µν, ) ( µν, ) θ( ) = R( z) ( + z) ( D λ )( ε + C) P + ( β γ ) P ε + C whee µ = β + γ ad ν = β γ. If we ow take the alteative choice of potetial, V( ) = Dcoth( λ), ad go though the same steps above we aive at the elativistic extesio of Eckat potetial [3,7]. The boud states spectum ad spio wavefuctio fo this elativistic Diac-Eckat potetial ae listed i the Table. 4

5 To obtai the elativistic extesio of the othe potetials i this class we coside the case V = 0 which is equivalet to the idetity tasfomatio (i.e. ρ = 0) combied with the costait (4). Thus, Diac equatio () ow eads κ d α W + φ() φ() d = ε κ d α W θ() θ() + + d This gives the followig equatio fo the lowe spio compoet i tems of the uppe: α κ d θ() = W + + φ() (0) + ε d While, the uppe compoet solves the followig Schödige-like d ode diffeetial equatio d κκ ( + ) dw W ε + + W + κ φ( ) = 0 d d α () Now, all oelativistic potetials i this class ae solvable oly fo the S-wave poblem (i.e. l = 0), thus we estict ou aalysis to the case whee κ = 0. We stat by cosideig W( ) = Fcoth( λ) Gcsch( λ) with F, G, ad λ beig eal costat paametes ad λf > 0. With this potetial fuctio ad κ = 0, equatio () gives the followig secod ode diffeetial equatio fo the uppe spio compoet d F + G + λf cosh( λ) ε + G ( F + λ) + F φ( ) 0 = d sih ( λ) sih ( λ) α Compaig this with Schödige equatio fo the S-wave Rose-Möse II potetial [3,7] d A + B + λa cosh( λ) + B ( A+ λ) + A E φ( ) 0 = () d sih ( λ) sih ( λ) gives the followig coespodece betwee oelativistic ad elativistic paametes: A= F B= G (3) E = ( ε ) α The well-kow oelativistic boud states spectum of equatio () is λ A E = ( A λ ) +, = 0,,, max < A λ (4) The substitutio (3) esults i the followig elativistic spectum fo this Diac-Rose- Möse II potetial ( ) ε =± + α F α λ F λ (5) whee = 0,,,..., ad max is the lagest itege satisfyig max max F λ < ( F λ) + ( αλ) It is obvious that the oelativistic limit ( α 0 ) of (5) epoduces the spectum i (4). The boud states wavefuctio of the oelativistic poblem [3,7] is tasfomed, usig (3), ito the followig uppe spio compoet wavefuctio ( γ β) ( γ + β) ( γ β, γ β ) φ() = R( z ) ( z+ ) P () z whee 5

6 z = cosh( λ) β = F λ γ = G λ Equatio (0) gives the lowe spio compoet i tems of the uppe as αλ ( d () ) ( θ = z γ βz+ z ) φ() ε + dz Agai, usig the diffeetial ad ecusio popeties of the Jacobi polyomials [6], we ca wite this explicitly as αλ ( µ ½) ( ν ½) γ ( µ, ν) θ() = R( z ) ( z+ ) z+ P () z ε + β + ( β + ½) γ ( µν, ) + P β + whee µ = γ β ad ν = γ β. Takig the alteative choice W( ) = Ftah( λ) + Gsech( λ) ad goig though the same steps above we aive at the elativistic extesio of Scaf potetial [3,7]. The boud states spectum ad spio wavefuctio fo this Diac-Scaf potetial ae listed i the Table. The table also lists Diac-Pöschl-Telle potetial W( ) = Ftah( λ) Gcoth( λ) which, i the oelativistic limit, epoduces the usual Pöschl-Telle potetial [7,8,9]. Fially, it is woth otig that it would be of pime elevace, as a futue developmet, to fid the geeal tasfomatios q(x) ad ρ(x) i () that map ay oe of these elativistic potetials ito othe membes of the class. Moeove, it might be possible that a exhaustive study of such tasfomatios may big about ew elativistic potetials that elage the class. A simila teatmet is called fo coceig the othe class of elativistic potetials that icludes Diac-Coulomb, Diac-Oscillato, ad Diac-Möse potetials. Ackowledgemets: The autho is gateful to D. M. S. Abdelmoem fo the ivaluable suppot i liteatue seach. 6

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8 Table Captio: Potetial fuctios V() ad W(), tasfomatio agle ρ, ad boud states spectum ε fo the five potetials. The table is cotiued to show explicitly the two-compoet adial spio wavefuctios φ () ad θ () fo each potetial. 8

9 Table V() W() ta( ρ ) ε Diac-Rose-Möse I Dtah( λ ) ( αds) tah( λ) κ DA Diac-Eckat Dcoth( λ) ( αds)coth( λ) κ DA αλ αd λ + + ( αd λt ) αd λt α ( α DT) αλ αd λ + + ( αd λt + ) αd λt + α ( α DT) Diac-Rose-Möse II 0 Fcoth( λ) Gcsch( λ) 0 Diac-Scaf 0 Ftah( λ) + Gsech( λ) 0 ( ) + α F α λ F λ ( ) + α F α λ F λ Diac-Pöschl-Telle 0 Ftah( λ) Gcoth( λ) 0 [ ] + α ( ) α λ ( ) λ+ G F G F 9

10 Table (cotiued) φ () θ () Diac-Rose-Möse I Diac-Eckat Diac-Rose-Möse II Diac-Scaf Diac-Pöschl-Telle φ = R z + z P z µ ν ( µ, ν) () ( ) ( ) () z = tah( λ ), µ = β + γ, ν = β γ ( )( ) β = αd λt, γ = ε D λ β φ = R z z+ P z µ ν ( µ, ν) () ( ) ( ) () z = coth( λ ), µ = β + γ, ν = β γ ( )( ) β = αd λt, γ = ε D λ β + φ = R z z+ P z ( µ + ½) ( ν + ½) ( µ, ν) () ( ) ( ) () z = cosh( λ), β = F λ, γ = G λ µ = β ½+ γ, ν = β ½ γ β γ ta ( µ, ν) = + φ () R ( z ) e P ( iz) z = sih( λ), β = F λ, γ = G λ µ = β ½" iγ, ν = β ½ + iγ φ = R z + z P z β γ ( µ, ν) () ( ) ( ) () z = cosh( λ), β = F λ, γ = G λ µ = β ½, ν = γ ½ θ αλ β θ () R µ ν = + ε + C ( )( ) ( ) ( µν, ) ( µν, ) D λ ε + C P + β γ P αλ β θ () R ( z ) ( z ) µ ν = + ε + C ( )( ) ( ) ( µν, ) ( µν, ) D λ ε + C P + γ β P αλ µ ½ ν ½ ( µν, ) () = R( z ) ( z+ ) z+ P () z ε + β + ( β + ½) γ ( µν, ) + P β + β + αλ ta ( ) (, ) ( ) γ z γ µ ν θ() = R + z e z P ( iz) ε + β + ( β + ½) + γ ( µν, ) i P ( iz) β + β γ αλ β γ ( µν, ) θ() R + + = + z P () z ε + β γ + ( β + ½)( γ + ½) ( µν, ) P β γ + γ 0