... I found that I was unable to explain the foundations of quantum mechanics in a way that I found entirely satisfactory. - Steven Weinberg

In 1929 Hubble announced... a "roughly linear" relation between and distance. ...His data points ...did not really support a linear relation. But in the early 1930s he had measured redshifts and distances out to the , with a redshift <math>z \eqsim 0.02</math>, corresponding to... 7,000 km/sec and a linear relation... was evident. The conclusion... the universe is really expanding. ...At the time of writing, the largest... <math>z=6.96</math>.
It may eventually become possible to measure the expansion rate <math>H(t) \equiv \dot{a}(t)/a(t)</math> at times <math>t</math> earlier than the present, by observing the change in very accurately measured redshifts of individual galaxies over times as short as a decade.

Heisenberg’s starting point was the philosophical judgment, that a physical theory should not concern itself with things like electron orbits in atoms that can never be observed. This is a risky assumption, but in this case it served Heisenberg well.

To start, we will consider a single particle moving in three space dimensions under the influence of a general central potential. Later we will specialize to the case of a Coulomb potential, and work out the spectrum of hydrogen. One other classic problem, the harmonic oscillator, will be treated at the end of this chapter.