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This paper describes an easy and teaching way how quantum mechanics (QM) and general relativity (GR) can be brought together. The method consists of formulating Schrödinger’s equation of a free quantum wave of a massive particle in curved space-time of GR using the Schwarzschild metric. The result is a Schrödinger equation of the particle which is automatically subjected to Newtons’s gravitational potential.

The problem of synthesis of QM and GR has been the subject of much discussion among physicists in recent years. In this short paper, we try to tackle this question by subjecting the Schrödinger equation of a free quantum wave to the non-Euclidian geometry of space-time developed in the formalism of general relativity.

The motivation to do this is justified by the effort to find an easy and pedagogical way of understanding how the most important physical theories developed in the 20^{th} century, QM and GR, can be brought together in the limit of quantum particles that have extremely small masses compared to cosmological objects.

In doing so, we begin by writing down the well-known non-relativistic Schrödinger equation which describes a quantum particle of mass at rest

G is the constant of gravitation and M the mass which causes gravitation (e.g. mass of Earth).

We investigate stationary solutions of Equation (1) by using the ansatz

omitting reference to r for function

Equation (3) can be treated in complete analogy to the quantization of electron energies in an hydrogen atom, described in standard textbooks of quantum mechanics [

Going back to Equation (3), we initially consider a free quantum wave with

with plane-wave solution

Because of the radial-symmetric potential Equation (2), we switch to spherical coordinates rewriting

with spherical wave solution

Now let’s switch to the relativistic point of view.

Taking GR into account (e.g. [

Now, the following idea is discussed: embedding the QM-formalism of a free wave into space-time- formalism of GR, we can change Equation (5) in complete formal analogy by rephrasing

applied to

The right hand side of Equation (6) uses the relativistic momenta [

together with the well-known energy relation

where

kinetic energy T of a free particle in a flat space of Euclidian geometry (as in Equation (4)) but denotes the kinetic energy of this particle bounded in the space-time geometry of GR where the gravitational potential

Immediately, one deduces from Equation (7)

This equation describing a quantum wave in curved space-time of GR is our starting point for further conside- rations. This quantum wave is not free anymore because it is affected by the non-Euclidian geometry of space- time. We will see below that this is equivalent to a quantum wave described by a Schrödinger equation in Eu- clidian geometry where Newton’s gravitational potential is included (see Equation (17)).

As promised above the diagonal metric we use is the so-called inverse spherical Schwarzschild metric (e.g. [

If

The subscripts t and r of the wave function

Initially we would like to mention that in case of

In order to solve Equation (11) for

The LHS depends only on t, the RHS only on r. Therefore we can equalize each individual side with

From the LHS of Equation (12) we obtain

We consider

which can be rewritten by using Equation (13) and Equation (5) as

because

where we have moved

From the considerations above one can conclude that by embedding the Schrödinger equation of a free quantum wave (which is defined in Euclidian space) into curved space-time of GR (which is defined in non-Euclidian space) we obtain the Schrödinger equation of a quantum wave which is subjected to Newton’s gravitational potential. Moreover, it has been shown that Newton’s potential energy comes from the Schwarzschild metric of GR. The space-time geometry of GR applied to a free quantum wave causes Newton’s gravitational force to appear automatically in the Schrödinger equation. In this sense, QM and GR can be harmonized if the “Newtonian approximation” (defined through the ratio Schwarzschild radius/position coordinate to be much smaller than 1) is taken into consideration and they can be brought together without any difficulty.

I am grateful to M. Faber, F. Laudenbach and F. Hipp for many discussions and I. Glendinning for revising the manuscript.

Martin Suda, (2016) How Quantum Mechanics and General Relativity Can Be Brought Together. Journal of Modern Physics,07,523-527. doi: 10.4236/jmp.2016.76054