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A Gravitational Hydrogen Atom

Don Herbison-Evans ( donherbisonevans@yahoo.com )

(updated 24 November 2009)

SUMMARY

Using simplistic assumptions, Schrodinger's equation has beeen augmented using special relativity to investigate deeply bound states in the hydrogen atom which appear when gravitation is taken into account theoretically. The basic qualitative idea was that the relativistic increase in mass of a rapidly orbiting electron would allow the gravitational interaction to exceed the electrostatic interaction in some states.

The extra mass factor due to special relativity in the Schrodinger 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 1019 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.

Having a negative energy that is greater in magnitude than the rest masses of the component particles suggests that matter composed of atoms in these states would suffer a repulsive gravitational force from normal matter, and have a negative inertial mass.

The Peculiar States are of interest also because they are an infinitude of solutions, one for each 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.

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.

MATHEMATICAL DETAILS

Quantitatively we may set up the Schrödinger equation to find the characteristic energy of the system. Taking the approximations that the electron and proton are point masses and charges, that the proton is stationary, and that the mass of the electron is made greater than its rest mass by its own kinetic energy as in a simple approximation to Special Relativity [Schwartz, 2007], then, in CGS units:

V ≈ - { e2/(4.π.ε.r) + [GMm/r].[ 1 + T/(mc2) ] } 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 10-25 kg
M = rest mass of the proton = 1.8 x 10-21 kg
c = velocity of light = 3.0 x 810 mg/sg
ε = permitivity of free space = 1.4 x 10-17 F/m [Allen, 1964]. Then with E = total energy of the atom, we have E = T + V or E = T.[ 1 - GM/(c2.r) ] - (e2/(4.π.ε) + GMm)/r

Following the usual development (e.g [Houston, 1959]) we transform this using de Broglie's relationship:

T → -(h2/2m).Δ 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/(4.π.ε) + GMm)/r + E}.ψ = 0 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 If we abbreviate: A = GM/c2 = 1.2 x 10-52 cms
B = 2m(e2/(4.π.ε) + GMm)/h2 ≈ 2me2/h2 = 4.2 x 108 cms-1
C = 2mE/h2 then we obtain: { [ r-A ].Δ + B + C.r }.ψ = 0 or: { Δ + [ B + C.r ]/[ r-A ] }.ψ = 0 Writing: [ B + C.r ]/[ r-A ] = F/[ r-A ] + C.[ r-A ]/[ r-A ]
 we have F = AC + B Changing the dependent variable to s: s = r - A
 we obtain the usual form of the hydrogen atom energy equation: { Δ + F/s + C }.ψ = 0 Writing ψ in spherical coordinates as the product of a radial function R and an angular function W: ψ = R(s).W(θ,φ) 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 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 where v2 = -C Then: (d2Q/ds2) - (v - 1.s).(dQ/ds) + [ F/s - k(k+1)/s2 ].Q = 0 This equation is known to have two solutions, which at the origin behave as Q   ≈   sk   or   s-(k+1)

The normalisation condition requires a finite value of

  ∞   π   2π
  ∫     ∫     ∫ ψ.ψ*.r2.sin(θ).dφ.dθ.dr
  0    0    0 or   ∞   π   2π
  ∫     ∫     ∫ ψ.ψ*.(s+A)2.sin(θ).dφ.dθ.ds
-A    0    0 or   ∞
  ∫ Q2.e(-2s.v).(s+A)2.ds
-A 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.

In order to examine the solutions near the origin: let

Q = sk.P(s) where P(s) is a polynomial in s, and changing the variable by the substitutions t = 2s.v gives a form of Kummer's Equation [Abramowits & Stegun, 1972]: t.d2P/dt2 - (k+t).dP/dt + {(2k-F/v)/4}.P = 0 which has as a solution the Confluent Hypergeometric Function: P = H( (F/v-2k)/4, k, t) This reduces to a finite polynomial if k is non-zero, and (F/v-2k)/4 = -j,     j = 0,1,2,... or F/v = 2k - 4j or ( AC + B )/v = 2n,     n = k-2j = ...,-2,-1,0,1,2,... The general states are found by solving: Av2 + 2nv - B = 0 so v = [ -n ± (n2+AB)1/2]/A
  = (n/A).[ -1 ± (1 + AB/(2n2) + O(AB)2]
  ≈ B/2n     or     -2n/A

2mE/h2 = -[B/(2n)]2     or     -[2n/A]2

E = -(hB)2/(8mn2) J
   or
    -2(nh)2/(mA2) J

E = -2.2 x 10-18/n2 J
   or
    -2.8 x 10101n2 J

A unique 'Peculiar Solution' exists when n = 0 (i.e. when 2j = k). Then

AC + B = 0 so 2me2/h2 + ( GM/c2 )( 2mE/h2 ) ≈ 0 or e2 + ( GM/c2 )( E ) ≈ 0 or E = - e2c2/(GM) = - 1.87 x 1025 J = - 2.1 x 1018 kg

DISCUSSION

The solutions of the form

E = 2.2 x 10-18/n2 J are the classical states of the hydrogen atom.

The solutions of the form

E = 2.8 x 10101n2 J     =     3.1 x 1084n2 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, estimated at about 2.4 x 1052 kg [Behr, 2007].

The 'Peculiar Solution' with an energy level of -2.1 x 1018 kg is perhaps of most interest.

This solution 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 a supernova (1044 J [Woolsley, 2007]).

Having a negative energy that is greater in magnitude than the rest masses of the component particles suggests that matter composed of atoms in this state would appear to have negative gravitational and inertial masses. Thus they would suffer a repulsive gravitational force from normal matter, but having a negative inertial mass, would accelerate in the opposite direction to this applied force. Together, these mean that this Peculiar matter would still appear to be gravitationally attracted to a normal matter.

A more peculiar result for this Peculiar matter is that the negative mass would mean that a piece of normal matter would be repelled from it, but because the Peculiar matter has a negative inertial mass: the Peculiar matter would move towards the repelled normal matter. If they had equal mass moduli, they would both accelerate off together at a constant distance apart.

If the Peculiar matter was surrounded by pieces of normal matter, the Peculiar matter would move according to the average of the forces of the normal matter, but all the normal matter would be repelled radially away from the Peculiar matter.

ACKNOWLEDGEMENTS

Many thanks are due to friends and colleagues who examined initial drafts of this work and informed me of errors therein, particularly Michael Partridge. Thanks are also due to Gaye Stinson of 'Burwood Public Library', Sydney, 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, pp. 13-16.

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.

Woolsley, S.E., 2007. "Supernovae", in Volume 17, "Encyclopedia of Science and Technology", McGraw-Hill, New York, 10th Edition, pp. 745-748.

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