Understanding Matter Waves and Quantization of Energy in Quantum Mechanics, Study notes of Chemistry

Download Understanding Matter Waves and Quantization of Energy in Quantum Mechanics and more Study notes Chemistry in PDF only on Docsity! 1 CHEM 20A/1 - winter 2001 LECTURE NOTES - sixth set MATTER WAVES. In the mid-1890's J. J. Thomson was able to create beams of electrons. He showed t hat these beams were composed of charged particles (electrons) which when subjected to ele ctric and magnetic field behaved exactly as expected for particles moving under the classical laws of mechanics. And indeed our view of matter is of material composed of small particles , electrons and nuclei within atoms, atoms combined to form molecules, and molecules aggre gated to form macroscopic material. But in the mid 1920's Thomson's so, and independentl y, Davissonn and Germer, showed that a beam of electrons behaved not like ballistically (lik e bullets or particles) but like waves. Thus it seems that matter, in this case electrons, behave s like particle in some experiments and wavers in other experiments. What is going on? W e have already seen that this wave-particle duality applies to electromagnetic radiation (whic h forms interference patterns, as expected for waves) but behaves like particles (photons) wh en impinging on a surface and breaking electrons loose out of the material (as in the photo-e lectric experiments).And now it seems that the same wave-particle duality exists for electron s, i.e., for matter. Indeed that is so, and it is very important. First we want to learn a bit about matter waves, also known as deBroglie waves. If a beam of electrons is focused onto a crystalline target, the electrons are scattered off the ato ms, and the scattered electrons can go in all directions. The scattered electrons can be collect ed on a screen at some distance from the target material, and one finds that at some position s on the screen, many scattered electrons strike while at others no electrons strike. This is c haracteristic of an interference pattern formed by a wave; for scattered materials the interfere nce pattern is called a diffraction pattern. One can interpret this experiment much like the Young 2-slit experiment with light, only here we effectively have many slits. We don't actua lly have slits through which the beam passes, but the scattered electrons originate on the ato ms much as the interfering light beam originates at the slits. With light waves the interferen ce pattern depends upon both the separation of the slits and the wavelength of the light; wit h the electrons the diffraction pattern depends upon the separation of the scatterers (the ato ms in the crystal) and a wavelength. Problem: For the two slit experiment the bright lines on a screen at a large distance R from the slits are positioned at an angle θ, where nλ = d sinθ (1) and d is the separation between slits, λ is the wavelength, and n is an integer. d << R, the distance R is measured from the midpoint between the slits and the screen, and th e angle θ is that between the horizontal line connecting the midpoint between the slits and the line connecting the midpoint between the slits and a bright spot on the screen. Be sure you understand this formula. Try to derive this formula, but don't be dismay ed if you can't. For electrons scattering off a crystal one might think of d as the distance between atoms, an d λ can be determined by the measured angles at which bright spots (spots where many elec trons strike) are located. But what is this wavelength that has been determined for electrons, other than something that makes the diffraction pattern intelligible? One can measure the velocity, u, of the incident electron beam; in fact, we can contro l the velocity of the the electrons by subjecting them to known electric fields (forces). And one finds that the measured wavelengths (as determined by the measured θ's substituted into Eq. (1) vary with the electron velocities; in fact, the λ is proportional to 2 u-1. In fact, one can do the same thing with a beam of neutrons, and it is found that λ is not only proportional to u-1, but to m-1, as well, where m is the mass of the particle. DeBroglie f ormulated these results as m = h mu (2) where the measured constant of proportionally, h, is the same Planck's constant as appears i n the Einstein expression, ε = hν, for the photon energy. We have added a subscript m to λ to indicate that it is a matter wave, and not an electromagnetic wave. This gives us some exp erimental insight into the meaning of λm, but we still might ask, "what is waving?" We addr ess this question below, but first we summarize some important uses for diffraction. DIFFRACTION. Equations such as (1) above relate the angle θ for bright spots in the diffraction patte rn to distances, d, between scatterers. If λ > d, no pattern is observed, and if λ << d, the brig ht spots are so close to each other that they cannot be distinguished. (Why?) But if λ is co mparable to. but smaller that d, i.e., λ ≤ d, then one expects good diffraction patterns. (Why? ) Since the wavelengths for photon (electromagnetic radiation), electron, and neutron beams can be controlled, by scattering these beams off crystalline material with scattering centers ( atoms) spaced a distance d apart, one can make use of such scattering experiments to determ ine the interatomic spacings (d) in materials. Important Problem. If the separation between atoms is d = 0.1 nm, the best diffraction patterns are obtained with wave lengths of slightly less than 0.1 nm. Calculate the corresponding energies for photons, electron, and neutrons (in eV's). Such experiments, known as x-ray diffraction, electron diffraction, and neutron diffrac tion experiments, are the primary technique for determing the structure of matter. Of course , the crystalline material consists of many scatterers (not just two) and there are many releva nt interatomic distances (not just d), and the formulas relating interatomic distances to positi ons of bright spots is far more complicated than Eq. (1), but the overall concept is that of the 2-slit interference experiment. Note that these expeiments depend upon the wave-nature of photons, electrons and neutrons. QUANTUM MECHANICS: WAVES & PROBABILITY. In our discussion of electromagnetic waves we discussed the relationship between w avefunctions and probability. The average value of the square of the "normalized" wave fun ction, |Ψ(r,t)|2, was taken as proportional to the probability density, the probability per unit v olume that a photon would be found at a point at position r. A probability is a positive fracti on that ranges from 0 to 1, the probability of finding a photon in a small volume δV about p oint r is the time averaged value of |Ψ(r,t)|δV, and when summed over all small volumes at a ll points r in space, average over time ∑|Ψ(r,t)|2δV = 1. (3) Problem. Be sure you understand all this. Look back at the discussion of probability density in the last set of notes. 5 ation for small energy gaps between quantum states, i.e., small values of Em-En, require low frequency photons, whereas excitation over large energy gaps require high frequency photo ns. Many systems are found primarily in their ground states, so that ** absorption carries them from the ground state to various excited states, but this is not always the case. In some cases the allowed energy levels En form a continuum, i.e., the energy gaps are und etectably small. Usually the energies of a free particle, be it a free electron (in contrast to one bound to a nucleus) or a free atom (in contrast to one bound within a molecule), exist in co ntinuum states. One thus finds that the absorption spectrum of ground state H atoms consi sts of a series of absorption lines but that above some very high frequency, νo, all frequen cies are absorbed, i.e., the line spectrum becomes a continuum spectrum. The discrete sp ectral lines are indicative of the quantized energy gaps for the electron bound to the nucleus, but for frequencies above νo the electron has absorbed so much energy that it breaks free, i. e., it is ionized. Problem. Why is the ionization energy equal to hνo? The process above is called photo-ionization. Why? Problem. If the absorbed frequency is 2hνo, what is the kinetic energy of the ionized electron. How does this relate to the explanation of the photoelectric effect as described by Einstein? Problem. The photo-dissociation of a molecule occurs when a photon is absorbed with sufficient energy to break the chemical bond holding the molecule together. Discuss some of the qualitative features of the spectrum of a molecule. Quite generally, chemical reactions can be initated by radiation. Each chemical system has its specific wave functions, Ψk(r), and corresponding allo wed energies, Ek. Thus each chemical system has its own specific allowed energy gaps, Em -En, and its own characteristic absorption line spectrum. By identifying the spectral lines (i. e., the absorption frequencies) one can identify the substance; this is one of the best tools fo r identifying chemical substances. Systems that are in excited quantum states often emit a photon so as to return to a lo wer state, usually the ground state. Such a process is called emission. The photo emitted ca n only have frequencies that correspond to the allowed energy gaps in the system. (Why?) Thus the emission spectrum, the plot of number of photons emitted versus frequency, mus t also be a line spectrum, and it too is characteristic of the particular substance generating the spectrum. The compositions of the sun and stars have been studied by way of their emissi on spectra. Problem: Modify Eq. (6) for emission. SCHROEDINGER EQUATION. The allowed energies, or at least the allowed energy gaps, for each system can be det ermined experimentally by means of the spectra. But we are interested in more than this; we are interested in knowing why a particular system has a particular set of discrete energies w ith specifc energy gaps. To do this we must first obtain the potential energy of the system; we have already shown how to write down the exact potential energy for atoms and molecul es. One then has to assign the appropriate masses to each particle, i.e., to the electrons and nuclei; this is easily done. One then has to "solve" the equations of motion; in this case the 6 equation of motion is not Newton's equation but Schroedinger's wave equation. We will not even write down this equation here, but it can be written, and in simple cases solved exactly; in complicated cases it can be solved approximately. here we will discuss some of the "solu tions" to the Schroedinger equation and leave for other courses the discussion of obtaining t he solutions. By solving the Schroedinger equation one obtains the complete set of wave function s, Ψk(r), and corresponding discrete energies, Ek, for each system. The Ψk(r)'s and Ek's are different for each system because the Schroedinger equation is different for each system. E ach system may involve different numbers of electrons and nuclei, which means that each sy stem is characterized by a different potential energy; this gives rise to different Schroedinger equations and different solutions, i.e., different Ψk(r)'s and Ek's. One wants to know the Ψk(r)'s because, as mentioned above, they contain full information a bout the system. But the task of obtaining them is very non-trivial. One feature of the Schroedinger equation is that includes Planck's constant, h. It is for this reason, as indicated earlier, that Planck's constant appears in all quantum mechanica l expressions. CORRESPONDENCE PRINCIPLE. One might wonder why in our daily lives we live by the rules of Newton but in the at omic world objects live by the rules of Schroedinger? The answer is that everyone and ever ything lives by the rules of Schroedinger but that for big objects (macroscopic objects) the r ules of Schroedinger become identical to those of Newton. This means that the effect of co ntributions dependent upon Planck's constant, h, become negligibly small as the system incr eases in size. This also means that the quantization becomes increasingly negligible, i.e., the energy gaps become negligible. Note that although quantum mechanics turns into classical mechanics as the system gets larger, classical or Newtonian mechanics does not turn into q uantum mechanics as the system gets smaller; thus although Schroedinger's equation is vali d (if useless) for large systems, Newtons' equations are not valid for small systems. Problem. Explain all this. QUANTUM WAVE FUNCTIONS. particle in box. nodes. etc.