Top 15 Steven Weinberg Quotes

My own conclusion is that today there is no interpretation of quantum mechanics that does not have serious flaws. This view is not universally shared. Indeed, many physicists are satisfied with their own interpretation of quantum mechanics. But different physicists are satisfied with different interpretations. In my view, we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is only a good approximation.

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.

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.

In this derivation Bohr had relied on the old idea of classical radiation theory, that the frequencies of spectral lines should agree with the frequency of the electron’s orbital motion, but he had assumed this only for the largest orbits, with large n. The light frequencies he calculated for transitions between lower states, such as n=2 → n=1, did not at all agree with the orbital frequency of the initial or final state. So Bohr’s work represented another large step away from classical physics.

Planck’s quantization assumption applied to the matter that emits and absorbs radiation, not to radiation itself. As George Gamow later remarked, Planck thought that radiation was like butter; butter itself comes in any quantity, but it can be bought and sold only in multiples of one quarter pound. It was Albert Einstein (1879–1955) who in 1905 proposed that the energy of radiation of frequency ν was itself an integer multiple of hν.

Perhaps the most important immediate consequence of Planck’s work was to provide long-sought values for atomic constants.

The principles of quantum mechanics are so contrary to ordinary intuition that they can best be motivated by taking a look at their prehistory.

The development of quantum mechanics in the 1920s was the greatest advance in physical science since the work of Isaac Newton. It was not easy; the ideas of quantum mechanics present a profound departure from ordinary human intuition. Quantum mechanics has won acceptance through its success. It is essential to modern atomic, molecular, nuclear, and elementary particle physics, and to a great deal of chemistry and condensed matter physics as well.

In the case where the universe does not recollapse, the proper distance to the is...<math>d_{MAX}(t) = a(t) \int_{0}^{r_{MAX}(t)} \frac{dr}{\sqrt{1-Kr^2}} = a(t)\int_{0}^{\infty} \frac{dt'}{a(t')}</math>... In the absence of a cosmological constant, <math>a(t)</math> grows like <math>t^{\frac{2}{3}}</math>, and the integral diverges, so there is no event horizon. But with a cosmological constant, <math>a(t)</math> will eventually grow as exp(<math>Ht</math>) with <math>H = H_0 \Omega^{1/2}_\Lambda</math> constant and... an event horizon... approaches... <math>d_{MAX}(\infty) = 1/H</math>. As time passes all sources of light outside our gravitationally bound will move beyond this... and become unobservable. The same is true for the quintessence theory... In that case <math>a(t)</math> eventually grows as exp(const <math>\times\, t^{2/{(2+\frac{\alpha}{2})}}</math>), so for any <math>\alpha \ge 0</math> the integral... [<math>d_{MAX}(t)</math>] converges.

[T]he distance at present is<math>d_{\mathrm{max}}(t_0) = \frac{1}{H_0} \int_{0}^{1} \frac{dx}{x^2 \sqrt{\Omega_\Lambda+\Omega_K x^{-2}+\Omega_M x^{-3}}}</math>...[T]here may have been a time before the radiation-dominated era in which there was nothing in the universe but , in which case the particle horizon distance would... be infinite. But as far as telescopic observations... [<math>d_{max}(t_0)</math>] gives the proper distance beyond which we cannot now see.

Modern cosmological theories can exhibit horizons of two... types. which limit the distances at which past events can be observed or at which it will ever be possible to observe future events. These are called Rindler s and s, respectively.

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.

[T]he property of being a geodesic is invariant under coordinate transformations... so the path of the photon or particle will... be a geodesic in any spatial coordinate system...

Consider... [the formula given by special relativity for the magnitude of the ]<math>P \equiv m_0 \sqrt{g_{ij}\frac{dx^i}{d\tau}\frac{dx^j}{d\tau}}</math>...where <math>d\tau^2 = dt^2 - g_{ij} dx^i dx^j</math>. [This holds because in] a locally inertial Cartesian coordinate system, for which <math>g_{ij} = \delta_{ij}</math>, we have <math>d\tau = dt\sqrt{1 - \mathbf {v}^2}</math> where <math>v^i = \frac{dx^i}{dt}</math>... [The <math>P</math>] is evidently invariant under arbitrary changes in the spatial coordinates, so we can evaluate it... in Robertson-Walker coordinates. ...[T]o save work ...adopt a spatial coordinate system in which the particle position is near the origin <math>x^i = 0</math>, where <math>\tilde{g}_{ij} = \delta_{ij} + \mathit0(\mathbf{x})</math>, and we can therefore ignore the purely spatial components of <math>\Gamma_{jk}^i</math> of the . General relativity gives [the momentum]... with a metric <math>g_{ij} = a^2(t)\delta_{ij}</math>...<math>P(t) \propto 1/a(t)</math>... for any non-zero mass, however small... Hence, although for photons both <math>m_0</math> and <math>d\tau</math> vanish... [the momentum relation] is still valid.

[T]o extend this to the geometry of spacetime... include a term... in the spacetime line element, with <math>a</math> now an arbitrary function of time (known as the Robertson-Walker scale factor):<math>d\tau^2 \equiv -g_{\mu\nu}(x) dx^\mu dx^\nu = dt^2-a^2(t)[d\mathbf{x}^2 + K \frac{(\mathbf{x} \cdot d\mathbf{x}^2)}{1-K\mathbf{x}^2}]</math>