Notes for Class Meeting 17: The Origins of Quantum Mechanics

Transcription

1 Notes for Class Meeting 17: The Origins of Quantum Mechanics At the turn of the 20 th century, physics had reached a mature state. 1 on an excellent understanding of (1) Newtonian mechanics, (2) electromagnetism, and (3) statistical mechanics and thermodynamics. This was based Particles such as billiard balls and electrons (discovered in 1895) followed the laws of classical mechanics. Light, on the other hand, was understood as the result of oscillating electric and magnetic fields. It was a continuous wave phenomenon, exhibiting interference effects. Both classical mechanics and electromagnetism were deterministic. If one could completely specify the conditions at some time, then, in principle, one could predict the future with complete certainty. (We now know this is naïve because of chaos, but this was not known then.) However, there were a number of experimental phenomena that could not be explained by classical physics, such as the discrete emission spectra of atoms. The explanation of these phenomena led to a revolutionary new approach to physics: the invention of quantum mechanics. This revolution will modify our view of all of the characteristics of classical physics. We will see that (1) light, which is known to be a wave phenomenon, is composed of particles, (2) particles, such as billiard balls and electrons, have exactly the same properties as light, and (3) these two results will force us to abandon classical determinism, even in principle. Our view of time will need to be further modified. We will discover that time has a fuzzy character. In addition to understanding that the future is probabilistic rather than deterministic, we will see that time is strangely holistic in an inexplicable and frankly disturbing way. Time will seem even to play games with causality, however without ever creating a logical contradiction. The Quantum Nature of Light The first breakdown of classical physics came from an unlikely source: the study of light emission from a hot object, such as an oven. This is discussed well in March, Chapter 15, except that March understates the problem with the classical solution for the frequency spectrum of this light. The classical theory of statistical mechanics makes an unambiguous, but absurd, prediction that the brightness should increase as the frequency squared to infinite frequencies. Since Maxwell s theory of electricity and magnetism 1 In fact, Max Planck (whom we will get to shortly) was advised by his physics professor that he should not go into physics because almost everything is already discovered, and all that remains is to fill a few holes." Fortunately, both for Planck and physics, he did not take this advice. As an editor of Annalen der Physik, Planck published Einstein s paper on special relativity and became a champion of Einstein s work.

2 requires the energy of light to be proportional tot he frequency, it seems to imply that an infinite amount of energy will be radiated. Thus, classical physics predicts its own failure. This classical prediction is known as the Rayleigh-Jeans law and is shown in the drawing to the graph to the right along with the measured frequency spectra at three different temperatures. Max Planck, a professor at the University of Berlin, solved this problem in 1900 by deriving a formula from statistical mechanics that matched the observed spectra. However, to do so, he had to assume that light was emitted only in quanta of energy h!, where ν (the Greek letter nu) is the frequency and h is a new constant, now known as Planck s constant, which has the value of 6.6! 10 "34 Js. 2 The next step came in 1905 when Einstein explained the photoelectric effect: When light falls on a metal surface, electrons can be emitted from the surface. From classical physics, we would expect that (1) the intensity, but not the frequency of the light will matter, and (2) there will be a time lag between the start of the light and the emission of electrons, as it will take time to build up enough energy at the site of the atom that emits the electron. But the following were observed: (1) For light with a frequency below a critical value, no current is ever seen, and (2) there is no time lag between the start of the light and the emission of electrons. Einstein explained this effect by extending Planck s idea: not only was light emitted in quanta of h!, but it was also absorbed in quanta of h!. Einstein s explanation is quite simple. An electron is bound to an atom with a certain amount of potential energy call the work function, for which we can use the symbol φ. To release it, at least this much energy must be supplied the electron by a single quantum of light (because the probability of two quanta of light hitting the same atom at the exact same time is very small). Thus Einstein predicted that the kinetic energy of the released electron, K, would equal K = h! " #. (17.1) It would not be until 1916 that an experiment would confirm Eq. 17.1, showing that the constant h is the same as found by Planck. 3 2 Planck was awarded the Nobel Prize in 1918 for this work.

3 The final experimental proof that light is made up of particles came in 1923 when the American physicist Arthur Compton scattered x-rays from atomic electrons and showed that the frequency of the x-rays changed by exactly the amount required by the conservation of energy and momentum. 4 The Bohr Atom In 1910, the British physicist Ernst Rutherford established that atoms had to be solar system like in the sense that most of the mass and the positive charge had to be concentrated in a very small space (which we call the nucleus of the atom), with electrons circling around the nucleus. 5 From the point of view of classical physics, this was strange in two regards: (1) Accelerating electrons, i.e., electrons moving in circles, radiate energy in the form of light. Thus, it was possible to calculate that in microseconds the electrons would lose all of their energy and collapse onto the nucleus. (2) Atoms were known to emit light at only a few discrete frequencies, whereas classical electromagnetism would require them to emit light over a continuous spectrum of frequencies. In regard to the discrete emission frequencies, in 1885, a Swiss schoolteacher, Johann Balmer discovered a formula for the four known visible frequencies of hydrogen:! = 3.3 " n 2 # 4 4n 2 Hz, n = 3, 4,5 (17.2) Balmer suggested that other lines be looked for with similar formulas. They were found:! = 3.3 " n 2 # 9 9n 2 Hz, n = 4,5,6 (17.3) by the German physicist Louis Paschen in 1908 in the infrared part of the spectrum, and! = 3.3 " n 2 # 1 n 2 Hz, n = 2,3,4 (17.4) by the Harvard physicist Theodore Lyman 6 in the period 1906 to 1914 in the far ultraviolet part of the spectrum. In 1913 the Danish physicist Niels Bohr proposed a model that could account for the observed spectra and the results of Rutherford s experiment. Bohr postulated: (1) Electrons move in discrete circular orbits under the attraction of the electric force. 3 Einstein was awarded the Nobel Prize in 1921 primarily for this work. The Nobel Committee indicated that if relativity proved to be correct, the prize would include that too. 4 Compton was awarded the Nobel Prize in 1927 for this work. 5 You can read about his experiment in March, Chapter 14, if you wish. However, it will not be a major concern of ours in this course. Rutherford had won the Nobel Prize in Chemistry two years earlier, which probably accounts for his not winning another Nobel Prize for this discovery. 6 Lyman was Chair of the Harvard Department of Physics from 1907 to He was responsible for raising the funds to build the Research Laboratory of Physics, which was renamed in his honor at his retirement in 1947.

4 (2) The orbits are fixed by the requirement that the angular momentum, i.e., mvr, is an integer times h/2π, for which we use the special symbol!. (3) When an electron moves from a higher energy orbit E i to a lower energy orbit E f, it emits a photon with h! = E i " E f. These rather ad hoc assumptions do two remarkable things. First, they agree with the emission spectra. The (potential plus kinetic) energy of the electron is given by E n =! R n 2 (17.5) where n equals number of! of angular momentum and R, called the Rydberg energy in honor of the Swedish spectroscopist, is 2 R = m # e 2 & 2! 2 $ % 4!" 0 ' ( = 13.6 electron volts, (17.6) where e is the electric charge of the electron. (Eq is derived in the appendix below for those who are interested in seeing it.) The zero of potential energy is taken to be where an electron is infinitely far from the proton. The electron volt (ev) is a more convenient unit of energy to use when dealing with atoms. It is the energy an electron gains when accelerated through an electric potential of one volt. One electron volt = 1.6! 10 "19 J. (In units of electron volts, h = 4.14! 10 "15 ev-s and! = 6.6! 10 "16 ev-s.) Thus, the emitted photons will have energies # h! = E i " E f = "R 1 2 n " 1 & % 2 $ i n ( f ' = R n 2 2 i " n f, (17.7) n 2 2 i n f which is just the Balmer form. The second remarkable thing is that for very large n, where the quantum jumps become quite small between n and n 1, the frequency of the photon equals the revolution frequency of the electron, the result expected in electromagnetic theory. This is called the correspondence principle: In the limit of large n, quantum mechanics must approach classical physics. 7 Finally, I must warn you that in spite of getting the measured experimental results, Bohr s model will prove to be incorrect in many respects. For example, the real hydrogen atom is 7 For the sake of simplicity, I am afraid that I have been a historical revisionist here. Actually Bohr set the orbit spacing with the correspondence principle and then discovered that this led to each orbit having n! units of angular momentum.

5 three-dimensional, not two dimensional, there are no orbits, and the angular momentum of the states is more complicated each state can have from 0 to (n 1)! units of angular momentum. However, the Bohr model is pedagogically useful, and we will continue to use it for a while. 8 Appendix: Derivation of Eq Assuming the electron orbits the proton in a circle under the attractive force given by Coulomb s Law (Eq. 9.2), we need to equate it to the force required to maintain the electron s orbit (Eq. 3.2), mv 2 = e2 1 r 4!" 0 r = k e2, where k =, (17.8) 2 2 r 4!" 0 where we have introduced k just to make the equations less cumbersome. Multiplying Eq by r/2, we obtain 1 2 mv2 = k 2r, (17.9) where the right hand side of Eq is the kinetic energy and the left hand side turns out to be minus half the potential energy (see the discussion of binding energy in Class Notes 5). Thus the total energy of the electron (the sum of the kinetic energy and the potential energy) is E = 1 2 mv2! k r =! k 2r. (17.10) Solving Eq for v gives The Bohr condition is v = k mr. (17.11) mvr = n!, (17.12) and substituting Eq into Eq gives kmr = n!! r = n2! 2 km. (17.13) Finally, substituting Eq into Eq yields E =! k km 2r n 2! =! $ e 2 ' 2 & % 4"# ) 0 ( 2 m 2n 2! 2. (17.5) 8 Bohr was awarded the Nobel Prize in 1922 for this model. It would be four years later that Werner Heisenberg and Erwin Schrödinger would invent the full theory of quantum mechanics and thus discover the correct description of the hydrogen atom. As you might have guessed at this point, both Heisenberg and Schrödinger were awarded Nobel Prizes, Heisenberg in 1932 and Schrödinger in 1933.