Constructing Hermitian Hamiltonians for spin zero neutral and charged particles on a curved surface : physical approach
aa r X i v : . [ qu a n t - ph ] F e b Constructing Hermitian Hamiltonians for spin zero neutral and charged particles on acurved surface : physical approach
M. S. Shikakhwa
Physics Group, Middle East Technical University Northern Cyprus Campus,Kalkanlı, G¨uzelyurt, via Mersin 10, Turkey
N.Chair
Department of Physics,University of Jordan,Queen Rania Street,Amman, Jordan
The surface Hamiltonian for a spin zero particle that is pinned to a surface by letting the thicknessof a layer surrounding the surface go to zero - assuming a strong normal force- is constructed. Thenew approach we follow to achieve this is to start with an expression for the 3D momentum operatorswhose components along the surface and the normal to the surface are separately Hermitian. Thenormal part of the kinetic energy operator is a Hermitian operator in this case. When this operatoris dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surfaceHamiltonian that contains the geometric potential term as expected. Hamiltonians for both a neutraland a charged particle in an electromagnetic field are constructed. We show that a Hermitiansurface and normal momenta emerge automatically once one symmetrizes the usual normal andsurface momentum operators. The present approach makes it manifest that the geometrical potentialoriginates from the term that is added to the surface momentum operator to render it Hermitian; thisterm itself emerges from symmetrization/ordering of differential momentum operators in curvilinearcoordinates. We investigate the connection between this approach and the similar approach ofJenssen and Koppe and Costa ( the so called Thin-Layer Quantization (TLQ)). We note that thecritical transformation of the wavefunction introduced there before taking the thickness of the layerto zero actually - while not noted explicitly stated by the authors- renders each of the surface andnormal kinetic energy operators Hermitian by itself, which is just what our approach does from theonset.
I. INTRODUCTION
The revival of the interest in the quantum mechanics of particles on surfaces and curves in the last decad is evidentlydue to the advance in technology that made it possible to fabricate nano-scale curved geometries like nano-spheres,nano-tubes and nano-wires...etc. A major and well-established approach for the problem introduced first by Jenssenand Koppe [1] and then elaborated on by Costa [2] is the so called thin layer quantization (TLQ). The idea is tofirst embed the 2D surface in a 3D layer of thickness d and then, by introducing a strong confining potential in thedirection normal to the surface to pin the particle to the surface. The part of the Hamiltonian containing the normaldegrees of freedom is then ignored on the ground that the transverse excitations for a sufficiently strong confiningpotential have a much higher energy than those at the surface, and so can be safely neglected in comparison tothe range of energies considered. Mathematically, this amounts to taking the limit d → d → Hermitian normal kinetic energy is singled out rightfrom the beginning. The reduction to the 2D surface Hamiltonian was achieved by taking the limit d → II. THE HERMITIAN SURFACE AND NORMAL MOMENTUM OPERATORS
We start by considering a spin zero particle in the 3D space that is confined to a layer of arbitrary thickness d surrounding a surface S embedded in the space. The space is spanned by a special set of curvilinear coordinates[1, 2] { u i } , i = 1 .. u and u lying on the surface, ˆ u ( u , u ) the unit vector normal to the surface and u thecoordinate along that normal. This way, the position vector of the particle R ( u , u , u ) is given as : R ( u , u , u ) = r ( u , u ) + u ˆ u (1)The vectors u i ≡ ∂ R ∂u i = ∂ i R tangent to the coordinates are defined as usual, and for the coordinate system definedin Eq.(1) read: u a ≡ ∂ a R = ∂ a r + u ∂ a ˆ u = a a + u ∂ a ˆ u (2) u ≡ ∂ R = ˆ u (3)where we have defined the tangent vectors on the surface a a ≡ ∂ a r (4)and the indices a, b, c.. running over 1 , G ij = ∂ i R · ∂ j R assumes the form: G ij = (cid:18) G ab
00 1 (cid:19) (5) G ab can be expressed in terms of the surface metric g ab ≡ ∂ a r · ∂ b r = a a · a b . The two metric tensors are relatedas [1, 2] G ab = ( a a + u ∂ a ˆ u ) · ( a b + u ∂ b ˆ u ) (6)= g ab − u K ab + ( u ) K ac K cb (7)where K ab = K ab ( u , u ) is the symmetric curvature tensor of the surface defined as [18]: ∂ a ˆ u = − K ab a a (8)The components of K ab are the projections of the derivative of the normal along the surface tangent vectors a a . Thedeterminant of the metric G ≡ det G ij = det G ab can be calculated from Eq.(6) and reads : G ( u , u , u ) = g (1 − M u + (4 M + 2 K ) u + O ( u )) (9) g = g ( u , u ) = det g ab ; M and K are, respectively, the mean and the Gaussian curvatures of the surface defined as: M = 12 K aa (10)= g K + g K − g K g and, K = det K ab (11)= K K − ( K ) g We will also need √ G which from Eq.(9)reads: √ G = √ gγ = √ g (1 − M u + (4 M + 2 K ) u + O ( u )) (12)Note that the u -dependence in √ G lies exclusively in γ . To second order in u this latter reads : γ = (1 − M u + (4 M + 2 K ) u + O ( u )) = 1 − M u + Ku + O ( u ) (13)We now construct the Hermitian radial and surface momentum operators. The Hermicity of the 3D momentumoperator p = − i ~ ∇ should be preserved when it is expressed in general curvilinear coordinates where it reads p = − i ~ ∇ = − i ~ u i ∂ i (recall that u i ≡ ∂ i R ). Checking, we find : h Ψ | p Ψ i = h Ψ | p Ψ i + h Ψ | − i ~ √ G ∂ i ( √ G u i )Ψ i (14)where integration is over all space with the measure √ Gd u and the wavefunction was assumed to satisfy boundaryconditions that allows the surface term to be dropped . Hermicity of p demands the vanishing of the second term onthe r.h.s, i.e.: 1 √ G ∂ i ( √ G u i ) = 1 √ G ∂ ( √ G ˆ u ) + 1 √ G ∂ a ( √ G u a ) = 0 (15)where the tangent vectors u a and the normal unit vector ˆ u have been defined in Eq.(2). The above is in fact anidentity and can be easily proven. One just needs to note that (see [18]) ∂ i u i = Γ iik u i and Γ iij = √ G ∂ j √ G whereΓ ijk refer to Christoffel Symbol of the second kind. The 3D momentum operators explicitly expressed as a sum of thesurface and normal parts reads: p = p ′ + p (16)= − i ~ u a ∂ a + − i ~ ˆ u ∂ It is straightforward to check that neither the surface nor the normal momentum as they stand are Hermitian in3D space; only their sum is. Now adding (half) of the zero-valued expression in Eq.(15) multiplied by − i ~ and splitamong p ′ and p to the above p , we have : p = p ′ + p = p ′ H + p H (17)with the operators p ′ H and p H being now: p ′ H = p ′ − i ~ √ G ∂ a ( √ G u a ) = p ′ + i ~ √ G ∂ ( √ G ˆ u ) (18)= − i ~ ( u a ∂ a + ˆ u F ( G ))and, p H = p − i ~ √ G ∂ ( √ G ˆ u ) (19)= − i ~ ˆ u ( ∂ − F ( G ))and we have defined F ( G ) as: F ( G ) = − √ G ∂ ( √ G ) = M + (2 M − K ) u + O ( u ) (20)and noted that ∂ ˆ u = 0. The newly defined surface momentum p ′ H and normal momentum p H can be readilychecked to be Hermitian over the 3D space. Therefore, by adding a zero-valued quantity to the full 3D Hermitianmomentum operator, we managed to express it as the sum of Hermitian surface and normal momenta. This a keystep for the following analysis. III. HERMITIAN HAMILTONIAN FOR A SPIN ZERO NEUTRAL PARTICLE
The Hamiltonian for a spin zero particle confined to the curved surface and otherwise free will now be constructed.The approach is based on the intuitive argument that if one starts from the full Hamiltonian in the 3D space spannedby the coordinate system given in Eq.(1) and then confine the particle to the surface by introducing a strong confiningpotential (force) along the direction normal to the surface, then the excitation along this direction will need an infiniteenergy and so the dynamics is essentially along the surface. This amounts to freezing the normal degree of freedomand dropping it from the Hamiltonian which is achieved by setting d to zero and dropping the differential operatorswith respect to the normal variable u . The critical point of the present approach is to drop the Hermitian normalmomentum so that one is left with a Hermitian surface Hamiltonian. While this intuition might seem obvious andtrivial, blindly dropping the normal degrees of freedom without observing for Hermicity has led to the reporting ofnon-Hermitian surface Hamiltonians in the literature [19].Therefore, the essential starting point is the expression ofthe 3D momentum operator as a sum of the two Hermitian surface and normal momentum operators as in Eq.(17).The free particle Hamiltonian in 3D with the momentum operator given by Eq.(17) reads : H = p m = 12 m ( p H + p ′ H + p ′ H · p H + p H · p ′ H ) (21)Delicate calculation gives the following expressions for each term of the above Hamiltonian:12 m ( p ′ H · p H + p H · p ′ H ) = − ~ m ∂ F ( G ) (22) p ′ H m = − ~ m ( 1 √ G ∂ a √ GG ab ∂ b − F ( G )) (23)So, the Hamiltonian, Eq.(21), becomes: H = p H m + − ~ m ( 1 √ G ∂ a √ GG ab ∂ b + ∂ F ( G ) − F ( G )) (24)The last two terms in the bracket on the r.h.s. of the above expression for the Hamiltonians are generated by theextra terms that were added to p ′ H and p H to render each Hermitian. Evidently, if one expands p H m - which we willnot do !- identical terms with opposite signs will be emerge leading to full cancellation of these terms and the usualexpression for the 3D free particle Hamiltonian will be restored. Now, using the explicit expressions of G ab and F ( G )given, respectively, by Eqs.(6) and (20)we can easily verify that as u → √ G ∂ a √ GG ab ∂ b | u → → √ g ∂ a √ gg ab ∂ b (25) F ( G ) | u → → M (26) ∂ F ( G ) | u → → M − K (27)Therefore, the dropping of the normal degree of freedom from the Hamiltonian, Eq.(24), will be achieved by droppingthe normal Hermitian momentum operator p H and taking the limit u →
0, i.e. employing Eqs.(25)-(27). Theresulting Hermitian surface Hamiltonian is : H s = − ~ m ( 1 √ g ∂ a √ gg ab ∂ b ) − ~ m ( M − K ) (28)This is the well-known expression of the surface Hamiltonian of the TLQ derived first by Costa [2]. The first termis the Laplace-Beltrami operator at the surface and the last two terms are the geometric potential or the geometrickinetic energy. IV. HAMILTONIAN FOR A CHARGED SPIN ZERO PARTICLE IN AN ELECTROMAGNETIC FIELD
The surface Hamiltonian for a spin zero charged particle coupled to an electromagnetic field can be constructedalong exactly the same lines as that of the neutral particles. A bit of extra care needs to be taken in the calculations.In 3D general curvilinear coordinates, the Hamiltonian for a spin zero particle of charge q coupled to a scalar potential V ( u ) and a vector potential A ( u ) expressed in terms of the kinematic momentum Π = ( p − q A ( u ) ) reads: H = Π m + qV (29)In the coordinates at hand the vector potential is given as: u i A i = u a A a + ˆ u A (30)It is crucial here to note that the operator p appearing in Π is ( see Eqs. (17)-(19)) the sum of the Hermitian normaland surface momenta. Therefore, the normal and surface components of Π are given explicitly as: Π H = p H − q ˆ u A = − i ~ ˆ u ( ∂ − F ( G ) − iq ~ A ) (31) Π H = p ′ H − q u a A a = − i ~ ( u a ( ∂ a − iq ~ A a ) + ˆ u F ( G )) (32)Thus, the Hamiltonian, Eq.(29), becomes H = 12 m ( Π H + Π ′ H ) + qV (33)The various terms in this Hamiltonian can be calculated in a straightforward manner just as in the spin zero case andread: 12 m ( Π ′ H · Π H + Π H · Π ′ H ) = − ~ m ∂ F ( G ) (34)which is the same expression as in the corresponding term in the spin zero case, Eq.(22). Also,Π ′ H m = p ′ H m + i ~ q m ( 1 √ G ∂ b ( √ GG ab A a )) + i ~ qm G ab A a ∂ b + q m G ab A a A b (35)Substituting for p ′ H from Eq.(23) and expanding the second term in Eq.(35) above, we have the explicit form of theHamiltonian, Eq.(33): H = Π H m − ~ m ∂ F ( G ) − ( ~ m ) 1 √ G ∂ a √ GG ab ∂ b + ~ m F ( G ) (36)+ i ~ q m (cid:18) ( ∂ b G ab ) A a + 1 √ G ( ∂ b ( √ G ) G ab A a + G ab ( ∂ b A a ) (cid:19) + i ~ qm G ab A a ∂ b + q m G ab A a A b + qV In the limit u →
0, in addition to Eqs,(25)-(27) we can easily verify the following :( ∂ b G ab ) A a ( u , u , u ) | u → → ( ∂ b g ab ) A a ( u , u ) (37)1 √ G ( ∂ b ( √ G ) G ab A a ( u , u , u ) | u → → √ g ( ∂ b ( √ g ) g ab A a ( u , u ) (38) G ab ( ∂ b A a ( u , u , u )) | u → → g ab ( ∂ b A a ( u , u , u ) | u =0 ) (39) G ab A a ( u , u , u ) ∂ b | u → → g ab A a ( u , u ) ∂ b (40)At this point, we carry a gauge transformation that eliminates the normal A ( u , u , u ) component of the vectorpotential. It is always possible to find such a gauge transformation [20]. The consequence of this is : Π H = p H − q ˆ u A → p H = − i ~ ˆ u ( ∂ − F ( G )) (41)which is just the expression of the Hermitian normal momentum in the absence of coupling to the electromagneticfield, Eq.(19). The Hamiltonian, Eq.(36, upon setting u and p H to zero and using the sets of equations (25)-(27)and (37)-(40))reduces to H | u → ,P H → ≡ H ems = − ~ m ∇ ′ + i ~ q m ∇ ′ · A ′ ( u , u , u ) | u =0 + i ~ qm A ′ ( u , u ) · ∇ ′ − ~ m ( M − K )+ qV + q m | A ( u , u ) | (42)where we have used the surface divergence notation ∇ ′ · A ′ ( u , u , u ) | u =0 ≡ a b ∂ b · A a a a ( u , u , u ) | u =0 to denote √ g ( ∂ b ( √ gg ab A a ( u , u , u ) | u =0 ), with ∇ ′ ≡ a b ∂ b ; A ′ ≡ A b a b and a b = g bc a c . We have also used the nota-tion g ab A a ∂ b = A ′ · ∇ ′ and ∇ ′ denotes the surface Laplacian. i.e. the Laplace-Beltrami operator, Eq.(25). TheHamiltobian, Eq.(42), was first derived by Ferrari and Coughi [4]. Finally, we note that we can express this surfaceHamiltonian compactly as usual in terms of a covariant derivative D ′ ≡ ( ∇ ′ − iq ~ A ′ ( u , u , u )) as: H ems = − ~ m ( ∇ ′ − iq ~ A ′ ( u , u , u )) · ( ∇ ′ − iq ~ A ′ ( u , u , u )) | u =0 − ~ m ( M − K ) + qV (43)= − ~ m D ′ · D ′ | u =0 − ~ m ( M − K ) + qV V. SYMMETRIZATION OF MOMENTUM OPERATORS, HERMICITY AND ORIGIN OF THEGEOMETRIC POTENTIAL
Investigating Eqs.(22)-(24) and (28), noting Eqs.(25)-(27),we see that the geometric potential − ~ m ( M − K ) hasits origin in the F ( G ) term that was introduced into p ′ and p to render them Hermitian. We now look more closelyinto this term and try to clarify its meaning. Consider p ′ H given in Eq.(18) and note that it can be symmetrized in ∂ a : p ′ H = − i ~ ( u a ∂ a + 12 √ G ∂ a ( √ G u a )) (44)= − i ~ ( u a ∂ a + 12 √ G ∂ a √ G u a − √ G √ G u a ∂ a )= − i ~ √ G ( √ G u a ∂ a + ∂ a √ G u a ) = − i ~ √ G {√ G u a , ∂ a } + where in the expression ∂ a √ G u a the derivative acts to everything to its right, and { , } + denotes the anti-commutator.So, the term added to render p ′ Hermitian is nothing but a recipe for symmetrizing the derivatives in the specificmanner given in Eq.(44)in order to construct the Hermitian operator p ′ H . This means that the geometric potentialis the outcome of symmetrization or ordering of the derivatives in the momentum operators. It is easy to symmetrize p and p separately and construct the Hermitian version for each of them in the same manner [21]. p H can besymmetrized in exactly the same way: p H = − i ~ ˆ u ( ∂ + 12 √ G ∂ ( √ G )) (45)= − i ~ √ G ( ˆ u √ G∂ + ∂ √ G ˆ u ) = − i ~ √ G {√ G ˆ u , ∂ } + Using the above symmetrized expressions for these operators, we can write the 3D momentum operator in a novelsymmetrized form as: p = p ′ H + p H = − i ~ √ G ( {√ G u a , ∂ a } + + {√ G ˆ u , ∂ } + ) (46)Such symmetrized forms of the momentum can be a good starting point for further calculations sometimes ( see thelast paragraph below). the limit u → p H = − i ~ γ { γ ˆ u , ∂ } + | u =0 (47) p ′ H | u → = − i ~ √ g {√ g u a , ∂ a } + (48)This last expression for p ′ H is the symmetric version of what is some times called the geometric momentum at thesurface in the literature [22]. One can express the surface Hamiltonian, Eq,(28), employing Eq.(23) in the limit u → H s = p ′ H m − ~ m (2 M − K ) (49)The above form of the surface Hamiltonian, with p ′ H m symmetrized can be quite convenient in some derivations. In [21]it was used to derive the centripetal force on a spin zero particle on the surface of a sphere and a cylinder. Keepingtrack of the symmetrization all the way, we obtained a symmetrized expression of this force. VI. CONNECTION WITH THE THIN LAYER QUANTIZATION
Here we compare and contrast our approach with the TLQ procedure[1, 2]. In the TLQ, one also starts with theSchr¨odinger equation for a particle in a 3D layer of thickness d surrounding a surface S , with the space spanned bythe special coordinate system given by Eq.(1). Eventually the limit d → d is set to zero. The authors of [1] call this the dangerous term. It turns out that in the limit d → − ~ m ( G − ∂ a G ab G ∂ b ψ + G − ∂ G ∂ ψ ) = Eψ (50)The expression in the brackets on the l.h.s is just the 3D Laplacian split into surface part; the first term, and the normalpart; the second term. The wavefunction ψ is normalized in 3D space with the measure d u √ G as R d u √ G | ψ | = 1.Note that non of the two parts of the Laplacian is Hermitian by itself , only their sum is. When the transformation ψ = γ − χ (51)is invoked ( γ defined in [1] is the square of γ defined in Eq.(13) in the present work) , the normal part in the l.h.s ofEq.(50) becomes γ − ( ∂ χ + U χ ), with U being the sought for finite term which does not vanish in the limit d → R d u √ G | ψ | = 1 demands that R d u √ g | χ | = 1. The transformed Schr¨odinger equation for χ is now [1]: − ~ m ( γ g − ∂ a g ab g ∂ b γ − χ + ∂ χ + U χ ) = Eχ (52)The differential operator ∂ of the normal variable can be easily checked to be Hermitian with the measure d u √ g .The surface differential operator, i.e. the first term in Eq.(52) should also be so since the whole l.h.s. should beHermitian. Therefore, the transformation, Eq.(51), serves to render each of the differential operators with respect tothe normal and surface parts in the Hamiltonian Hermitian by itself. This is the connection between the TLQ andour approach presented in this work. It is all about having differential operators for the normal and surface degreesof freedom that are separately Hermitian before setting d to zero. We do this right from the start whereas in TLQ,it takes one to carry out the transformation in Eq.(51) to achieve this. TLQ proceeds by splitting the wavefunctionas χ = χ s ( u , u ) χ ( u ) so that the Schr¨odinger equation separates into normal and surface equations. In the limit d → U tothe geometric potential thus one gets the same expression for the surface Hamiltonian as the one obtained by thepresent approach, Eq.(28). VII. SUMMARY AND CONCLUSIONS
We suggest a new physics-based idea to construct the surface Hamiltonian for a spin zero particle within a layer ofthickness d around a surface when it is pinned to the surface as the thickness of the layer is shrunk to zero. The ideais based on starting right from the beginning with the 3D momentum operator p (which is the Laplacian in generalcurvilinear coordinates) that has each of its component operators parallel and normal to the surface Hermitian byitself. This is shown to be achieved naturally by symmetrizing the derivatives in these directions as in Eqs.(44) and(45). This symmetrization is shown to merely adding a zero-valued quantity to p so that it is effectively the same.The kinetic energy operators p m has its normal and surface parts Hermitian, too. The surface Hamiltonian withthe geometric potential term is obtained by taking the limit d → ? ]. The present approach, however, makes itmanifest that the geometric potential that appears in the surface Hamiltonian originates from the symmetrization ofthe momenta and the ordering of their differential operators. We have also demonstrated that the manipulations inthe TLQ leading to the construction of the surface actually - though not noted explicitly by the authors- serve torender both parts of the kinetic energy operator; the normal and the parallel to the surface, Hermitian by itself beforesetting the thickness of the layer to zero; Hermicity, which is the physical criterion observed by our approach is alsoat the center of TLQ. [1] H. Jensen and H. Koppe, Ann. Phys. , 586(1971)[2] R. C. T. da Costa, Phys. Rev. A , 1982 (1981).[3] Dingkun Lian, Liangdong Hu and Quanhui Liu, Ann. Phys. (Berlin) 1700415 (2018).[4] G. Ferrari and G.Coughi, Phys. Rev. Lett. , 240403 (2008).[5] B. Jensen and R. Dandoloff, Phys. Rev. A , 052109 (2009).[6] B. Jensen and R. Dandoloff, Phys. Rev. A , 049905(E) (2010).[7] Carmine Ortix and Jeroen van den Brink, Phys. Rev. B . 113406 (2011).[8] M. V. Entin, L. I. Magarill, Phys. Rev. B , 085330 (2001).[9] M-H. Liu et. al., Phys. Rev. B , 085307 (2011).[10] T-C. Cheng, J-Y. Chen, and C-R. Chang, Phys. Rev. B , 214423 (2011).[11] J-Y. Chang, J-S. Wu, and C-R. Chang Phys. Rev. B , 174413 (2013).[12] T.Kosugi, J. Phys. Soc. Jpn. , 073602 (2011).[13] K-C Chen and C-R. Chang, SPIN , 1340006 (2013).[14] Y.-L. Wang, H. Jiang and H.-S. Zong, Phys. Rev. A , 022116 (2017)).[15] M.S.Shikakhwa and N.Chair, Phys.Lett.A ,1985 (2016).[16] M.S.Shikakhwa and N.Chair, Phys.Lett.A ,2876 (2016).[17] M.S.Shikakhwa and N.Chair, Eur.J.Phys., ,015402 (2017).[18] PA.Kelly, Mechanics Lecture Notes: An introduction to Solid Mechanics. Available fromhttp://homepages.engineering.auckland.ac.nz/ ∼ pkel015/SolidMechanicsBooks/index.html.[19] A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. B, , 343 (1993).[20] S.Weinberg, The Quantum Theory of Fields Vol.II, Cambridge: Cambridge University Press (1996).[21] M.S.Shikakhwa, Physica E: Low-dimensional Systems and Nanostructures, , 249-252 (2019).[22] Q. H. Liu, C. L. Tong, and M. M. Lai, J. Phys. A40