This paper is divided into three parts. In the first, we demonstrate that all of quantum mechanics can be derived from the fundamental property that the propagation of a matter-wave packet is described by the same gravitational and kinematic time dilation that applies to a clock. We will do so in several steps, first deriving the Schroedinger equation for a non-relativistic particle without spin in a weak gravitational potential, and eventually the Dirac equation in curved space-time describing the propagation of a relativistic particle with spin in strong gravity. In the second part, we present interesting consequences of the above quantum mechanics: that it is possible to use wave packets as a reference for a clock, to test general relativity, and to realize a mass standard based on a proposed redefinition of the international system of units, wherein the Planck constant would be assigned a fixed value. The clock achieved an absolute accuracy of 4 parts per billion (ppb). The experiment yields the fine structure constant α=7.297 352 589(15)×10⁻³ with 2.0 ppb accuracy. We present improvements that have reduced the leading systematic error about 8-fold and improved the statistical uncertainty to 0.33 ppb in 6 hours of integration time, referred to α. In the third part, we present possible future experiments with atom interferometry: A gravitational Aharonov-Bohm experiment and its application as a measurement of Newton's gravitational constant, antimatter interferometry, interferometry with charged particles, and interferometry in space. We will give a review of previously published material when appropriate, but will focus on new aspects that have not been published before.