Gravitation and the Hydrogen Atom
Don Herbison-Evans (donherbisonevans@yahoo.com)

SUMMARY
Schrödinger's equation has beeen augmented using special relativity to investigate theoretically the deeply bound states in the hydrogen atom which appear when gravitation is taken into account. The basic qualitative idea was to accommodate the relativistic increase in mass of a rapidly orbiting electron.
The extra mass factor due to special relativity in the Schrödinger equation can be approximated to include a term linear in the kinetic energy, so that the equation may be solved by regular methods. A change of radial variable avoids the singularity at the origin in normalisation which classically precludes deeply held states.
A set of states was discovered the energy of each of which exceed the mass of the observable universe.
A further set of Peculiar States were found with an energy of -2.1 x 108 Kilograms.
A single atom undergoing a transition from a conventional state to one of these states would release an amount of energy of order 1025 Joules.
The Peculiar States are of interest also because they are an infinitude of solutions, one for each integer value of angular momentum 'k'. In the simplistic model analysed here, these all had the same energy. However: more sophisticated models may be anticipated to predict splitting of this energy level. Transitions between such levels may then be sought in cosmic radiation which might indicate the actual existence of gravitational atoms.
 An atom in a Peculiar state would also have a negative energy rather greater than the positive energy equivalent to its mass, so would appear to have a negative mass. Perhaps gravitationally it would repel ordinary atoms.

INTRODUCTION
 This work describes the theoretical investigation into possible deeply bound states in the hydrogen atom which appear when gravitation is taken into account. The basic qualitative idea is that the mass of a rapidly orbiting electron is increased due to its kinetic energy [Schwartz, 2007]:

 u = m + T/c2 = m.[ 1 + T/(mc2) ]      (1)
 where
 u = mass of the moving electron
 m = rest mass of the electron
 T = kinetic energy of the electron
 c = velocity of light

MATHEMATICAL DETAILS
 Quantitatively we may set up the Schrödinger equation to find the characteristic energy of the system. In this simple exploratory study, we take the approximations that the electron and proton are point masses and charges, and that the electron moves around a stationary proton centred on the coordinate origin, then:
 V ≈ - { e
2/(4π.ε.r) + [GMm/r].[ 1 + T/(mc2) ] }      (2)
 where
 V = potential energy
 T = kinetic energy
 r = separation of the electron and the proton
 e = electron charge = 1.6 x 10-19 C
 G = universal constant of gravitation = 6.7 x 10-11 m3kg-1s-2
 m = rest mass of the electron = 9.1 x 10-31 kg
 M = rest mass of the proton = 1.7 x 10-27 kg
 c = velocity of light = 3.0 x 108 m/s
 ε = permitivity of free space = 8.9 x 10-12 F/m [Allen, 1964].

 
Then with E = total energy of the atom, we have
 E = T + V      (3)
 or
 E = T.[ 1 - GM/(c2.r) ] - [e2/(4π.ε) + GMm]/r      (4)

 To find the stationary states of the system (e.g [Houston, 1959]) we transform this using de Broglie's relationship:
 T → -(h2/2m).Δ      (5)
 where
 h = reduced Planck's constant = 1.1 x 10-34 J.s [Allen, 1964]
 Δ is the Laplacian operator

 Thus we obtain:
 { (h2/2m).[ 1 - GM/(c2.r) ]Δ + [e2/(4π.ε) + GMm]/r + E}.ψ = 0      (6)
 where
 ψ = wave function of the electron

 Rearranging this, we obtain:
 { [ 1 - GM/(c2.r) ]Δ + 2m[ e2/(4.π.ε) + GMm ]/(rh2) + 2mE/h2 }.ψ = 0      (7)

 We will abbreviate the coefficients as follows in the rest of this paper:
 A = GM/c2 = 1.34 x 10-54 m      (8)
 B = 2m[e2/(4π.ε) + GMm]/h2 ≈ -me2/(2π.ε.h2) = 7 x 1021 m-1      (9)
 C = 2mE/h2 m-2     (10)
 then we obtain:
 { [ r-A ].Δ + B + C.r }.ψ = 0      (11)

 or:
 { Δ + [ B + C.r ]/[ r-A ] }.ψ = 0      (12)

 Writing:
 [ B + C.r ]/[ r-A ] = F/[ r-A ] + C.[ r-A ]/[ r-A ]      (13)
 we have
 F = AC + B      (14)
Changing the dependent variable to s:
 s = r - A      (15)
 we obtain the usual form of the hydrogen atom energy equation:
 { Δ + F/s + C }.ψ = 0      (16)

Writing ψ in spherical coordinates as the product of a radial function R and an angular function W:
 ψ = R(s).W(θ,φ)      (17)
and assuming an angular momentum k about the point s = 0 (i.e. r = A), we can remove the angular dependence to obtain:
 (d2R/ds2) + (2/s).(dR/ds) + [ C + F/s - k(k+1)/s2 ].R = 0      (18)
 with
 k = 1,2,3,...

 We seek solutions for R that decay exponentially to zero as s → ∞ , so let:
 R = Q(s).e-s.v      (19)
 where
 v2 = -C
 Then:
 (d2Q/ds2) - 2(v - 1.s).(dQ/ds) + [ (F-2v)/s - k(k+1)/s2 ].Q = 0      (20)

 This equation is known to have two solutions, which at the origin behave as
 Q   ≈   s  or   s-(k+1)      (21)

 The normalisation condition requires a finite value of
   ∞   π   2π
   ∫     ∫     ∫ ψ.ψ*.r2.sin(θ).dφ.dθ.dr     ( 22)
   0    0    0
 or a finite value of
   ∞   π   2π
   ∫     ∫     ∫ ψ.ψ*.(s+A)2.sin(θ).dφ.dθ.ds      (23)
 -A    0    0
 which requires a finite value of
   ∞
   ∫ Q2.e(-2s.v).(s+A)2.ds      (24)

 -A
 In order to examine the solutions near the origin of s: let
 Q = sk.P(s)      (25)
 where P(s) is a polynomial in s. The solutions of the form sk are those of conventionally assigned to the hydrogen atom, but the solutions of the form s-(k+1) are usually rejected because the singularity at r = 0 impedes normalisation of ψ. In the current case, the singularity is not at the lower limit of the integral, and these solutions may have some physical meaning.
 Changing the variable by the substitutions
 t = 2s.v      (26)
 gives a form of Kummer's Equation [Abramowits & Stegun, 1972]:
 t.d2P/dt2 - (k+t).dP/dt + {(2k-F/v)/4}.P = 0      (27)
 which has as a solution the Confluent Hypergeometric Function:
 P = H( (F/v-2k)/4, k, t)      (28)
 This reduces to a finite polynomial if k is non-zero, and
 (F/v-2k)/4 = -j,     j = 0,1,2,...      (29)
 or
 F/v = 2k - 4j      (30)

 or
 ( AC + B )/v = 2n,     n = k-2j = ...,-2,-1,0,1,2,...      (31)

 The general states are found by solving:
 Av2 + 2nv - B = 0      (32)
 so
 v = [ -n ± (n2+AB)1/2]/A
   = (n/A).[ -1 ± (1 + AB/(2n2) + O(AB)2]   ≈ B/2n     or     -2n/A      (33)
 2mE/h2 = -[B/(2n)]2     or     -[2n/A]2      (34)
 E = -(B2.h2)/(8m.n2) J    or     -2(n2.h2)/(m.A2) J      (35)
 E = -(1/n2).(m.e4)/(32π22.h2) J    or     -n2.(2h2.c4)/(G2.m.M2) J      (36)
 E = -2.2 x 10-18/n2 J    or     -1.3 x 1070n2 J      (37)

 A unique 'Peculiar State' exists when n = 0 (i.e. when 2j = k). Then
 AC + B = 0      (38)
 so
 2me2/(2π.ε.h2) + ( G.M/c2 )( 2m.E/h2 ) ≈ 0      (39)
 or
 e2/(2π.ε) + ( G.M/c2 )( E ) ≈ 0      (40)
 or

 E = - e2.c2/(2π.ε.G.M)      (41)
     = - 1.87 x 1025 J ≡ - 2.1 x 108 kg ≡ - 1.2 x 1038 Mev

DISCUSSION
 The solutions of the form
 E = -2.2 x 10-18/n J
 are the classical states of the hydrogen atom.
 The solutions of the form
 E = -1.3 x 1070n2 J     =     -1.5 x 1053n2 kg
 are new deep states. The energies of these, even for the lowest state with n=1, rather exceed the mass of the observable universe, whch has been estimated at about 2.4 x 1052 kg [Behr, 2007].
 The 'Peculiar State' with an energy level of -2.1 x 108 kg is perhaps of most interest.
 This state is actually an infinitude of solutions for all integer values of angular momentum 'k'. In the simplistic model analysed here, these all had the same energy. However: more sophisticated models may be anticipated to predict splitting of this energy level. Transitions between such levels may then be sought in cosmic radiation which might indicate the actual existence of gravitational atoms.
 The energy level of the Peculiar State would mean that a single atom undergoing a transition from a conventional state to this state would release an amount of energy of order 1025 J. This may be compared with a 10 Mt hydrogen bomb (1017 J [De Volp, 2007]), and the solar output: 4 x 1026 J/s [Allen, 1964]).
 The state has a negative energy far in excess of the positive energy of the rest masses of the constituent particles. It would appear to be an object with a negative mass. It would presumably gravitationally repel and be repelled by normal matter. This would certainly account for the absence of any reports of such objects.

ACKNOWLEDGEMENTS
 Many thanks are due to friends and colleagues who examined initial drafts of this work and explained to me some of errors therein, particularly Michael Partridge. Thanks are also due to Gaye Stinson of 'Burwood Public Library', Sydney, Professor John Crossley of Monash University, and many staff of the 'Faculty of Engineering and Information Technology' at the 'University of Technology, Sydney' for their assistance and support.

REFERENCES
 Abramowitz, M., and Stegun, I.A. (eds.), 1972, "Handbook of Mathematical Functions", Dover, New York, 9th Printing, p. 504.

 Allen, C.W., 1964, "Astrophysical Quantities", Athlone Press, London, 2nd Edition.

 Behr, B.B., 2007. "Universe", in Volume 19, "Encyclopedia of Science and Technology", McGraw-Hill, New York, 10th Edition, pp. 80-89.

 De Volp, A., 2007. "Hydrogen Bomb", in Volume 8, "Encyclopedia of Science and Technology", McGraw-Hill, New York, 10th Edition, pp. 712-713.

 Houston, W.V, 1959, "Principles of Quantum Mechanics", Dover, New York, pp. 58-82.

 Schwartz, H.M., 2007. "Relativistic Mechanics", in Volume 15, "Encyclopedia of Science and Technology", McGraw-Hill, New York, 10th Edition, pp. 330-332.
 
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 (written 11 August 1960, last updated 26 May 2015)