Einstein showed in his seminal paper on radiation that molecules with a quantum-theoretical distribution of states in thermal equilibrium are in dynamical equilibrium with the Planck radiation. The method he used assigns coordinates fixed with respect to molecules to derive the A and B coefficients, and fixed relative to laboratory coordinates to specify their thermal motion. The resulting dynamical equilibrium between quantum mechanical and classically defined statistics is critically dependent upon considerations of momentum exchange. When Einstein’s methods relating classical and quantum mechanical statistical laws are applied to the level of the single quantum oscillator they show that matrix mechanics describes the external appearances of an atom as determined by photon-electron interactions in laboratory coordinates, and wave mechanics describes an atom’s internal structure according to the Schrödinger wave equation. Non-commutation is due to the irreversibility of momentum exchange when transforming between atomic and laboratory coordinates. This allows the “rotation” of the wave function to be interpreted as the changing phase of an electromagnetic wave. In order to describe the momentum exchange of a quantum oscillator the Hamiltonian model of atomic structure is replaced by a Lagrangian model that is formulated with equal contributions from electron, photon, and nucleus. The fields of the particles superpose linearly, but otherwise their physical integrity is maintained throughout. The failure of past and present theoretical models to include momentum is attributed to the overwhelming requirement of human visual systems for an explicit stimulus.

Einstein did not contribute significantly to the quantum mechanics we are familiar with, but he did contribute to its theoretical foundations. In a seminal theory on black body radiation, he used two physically distinct sets of coordinates, a classical set to describe the continuity of heat energy with Maxwell Boltzmann statistics and an atomic set to describe discrete energy states required by the Planck radiation law. His methods are extended here to reveal that matrix mechanics is an incomplete description of atomic structure. In addition to the discrete energy states of the diagonal matrix elements, classically derived components of kinetic energy are necessary. We conclude that the diagonalized matrix elements and eigenvalues of the Schrödinger equation both describe characteristics of the steady states, but the physical variables they describe are different, because they are measured with respect to different coordinate systems. Einstein’s assumptions lead to a resolution of the measurement problem by introducing statistical laws at the microscopic level.

In our investigation of the time evolution of energy states Einstein’s interpretation from more than a century ago is found to have greater internal consistency than that of non-relativistic quantum mechanics. We choose to align more closely with his methods by abandoning the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a relativistically correct Lagrangian method. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, matrix mechanics, the function of the simplest quantum mechanical system an electron cyclotron, chaos theory, and evolutionary biology.

Einstein published four papers in 1905 that revolutionized our understanding of the fundamental concepts of space, time, mass, and energy. We draw attention here to an idea in one of the 1905 papers that revolutionizes our understanding of quantum mechanics, but its significance was never understood. We use it to derive classical interpretations of matrix and wave mechanics, resolve the “collapse” problem, and demonstrate that the standard model is deficient in its physical interpretation of Nature.

Matrix mechanics describes the spectral emissions of hydrogen gas, but quantum mechanics concerns the emission and absorption of radiation by a single atom. To obtain the spectral emissions of a single hydrogen atom we reformulate matrix mechanics by introducing thermal energy in infinitesimal increments and registering the spectral lines one by one until a complete spectrum is obtained. Wave mechanics is reformulated from a conservative to a non-conservative system by linking the output of matrix mechanics to the input of wave mechanics. To describe the gradual increase and decrease of energy by a quantum system we integrate between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the conditions necessary for Hamilton’s principle. The Lagrangian method is shown to be superior for analyzing quantum theory because it describes quantum systems relativistically as evolutions in time.

We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.