  We have recently proposed a Lagrangian in trace dynamics, to describe a
possible unification of gravity, Yang-Mills fields, and fermions, at the Planck
scale. This Lagrangian for the unified entity - called the aikyon - is
invariant under global unitary transformations, and as a result possesses a
novel conserved charge, known as the Adler-Millard charge. In the present
paper, we derive an eigenvalue equation, analogous to the time-independent
Schr\"{o}dinger equation, for the Hamiltonian of the theory. We show that in
the emergent quantum theory, the energy eigenvalues of the aikyon are
characterised in terms of a fundamental frequency times Planck's constant. The
eigenvalues of this equation can, in principle, determine the values of the
parameters of the standard model. We also report a ground state, in this theory
of spontaneous quantum gravity, which could characterise a non-singular initial
epoch in quantum cosmology.
