6.4: THE WAVE BEHAVIOR OF MATTER

Transcription

1 6.4: THE WAVE BEHAVIOR OF MATTER SKILLS TO DEVELOP To understand te wave particle duality of matter. Einstein s potons of ligt were individual packets of energy aving many of te caracteristics of particles. Recall tat te collision of an electron (a particle) wit a suf ciently energetic poton can eject a potoelectron from te surface of a metal. Any excess energy is transferred to te electron and is converted to te kinetic energy of te ejected electron. Einstein s ypotesis tat energy is concentrated in localized bundles, owever, was in sarp contrast to te classical notion tat energy is spread out uniformly in a wave. We now describe Einstein s teory of te relationsip between energy and mass, a teory tat oters built on to develop our current model of te atom. Te Wave Caracter of Matter Einstein initially assumed tat potons ad zero mass, wic made tem a peculiar sort of particle indeed. In 1905, owever, e publised is special teory of relativity, wic related energy and mass according to te following equation: c E = ν = λ = mc 2 (6.4.1) According to tis teory, a poton of wavelengt λ and frequency ν as a nonzero mass, wic is given as follows: E ν m = = = c 2 c 2 λc (6.4.2) Tat is, ligt, wic ad always been regarded as a wave, also as properties typical of particles, a condition known as wave particle duality (a principle tat matter and energy ave properties typical of bot waves and particles). Depending on conditions, ligt could be viewed as eiter a wave or a particle. In 1922, te American pysicist Artur Compton ( ) reported te results of experiments involving te collision of x-rays and electrons tat supported te particle nature of ligt. At about te same time, a young Frenc pysics student, Louis de Broglie ( ), began to wonder weter te converse was true: Could particles exibit te properties of waves? In is PD dissertation submitted to te Sorbonne in 1924, de Broglie proposed tat a particle suc as an electron could be described by a wave wose wavelengt is given by λ = mv (6.4.3) were is Planck s constant, m is te mass of te particle, and v is te velocity of te particle. Tis revolutionary idea was quickly con rmed by American pysicists Clinton Davisson ( ) and Lester Germer ( ), wo sowed tat beams of electrons, regarded as particles, were diffracted by a sodium cloride crystal in te same manner as x-rays, wic were regarded as waves. It was proven experimentally tat electrons do exibit te properties of waves. For is work, de Broglie received te Nobel Prize in Pysics in If particles exibit te properties of waves, wy ad no one observed tem before? Te answer lies in te numerator of de Broglie s 34 equation, wic is an extremely small number. As you will calculate in Example 6.4.1, Planck s constant ( J s) is so small tat te wavelengt of a particle wit a large mass is too sort (less tan te diameter of an atomic nucleus) to be noticeable. EXAMPLE 6.4.1: WAVELENGTH OF A BASEBALL IN MOTION Calculate te wavelengt of a baseball, wic as a mass of 149 g and a speed of 100 mi/. ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 1/6

2 Given: mass and speed of object Asked for: wavelengt Strategy: A. Convert te speed of te baseball to te appropriate SI units: meters per second. B. Substitute values into Equation and solve for te wavelengt. Solution: Te wavelengt of a particle is given by λ = /mv. We know tat m = kg, so all we need to nd is te speed of te baseball: 100 mi v = ( )( )( )( ) B Recall tat te joule is a derived unit, wose units are (kg m )/s. Tus te wavelengt of te baseball is 1 60 min km mi 1000 m J s kg m 2 s 2 s λ = = = m (6.4.4) (0.149 kg) (44.69 m s) (0.149 kg ) (44.69 m s 1 ) (You sould verify tat te units cancel to give te wavelengt in meters.) Given tat te diameter of te nucleus of an atom is approximately m, te wavelengt of te baseball is almost unimaginably small. km EXERCISE 6.4.1: WAVELENGTH OF A NEUTRON IN MOTION Calculate te wavelengt of a neutron tat is moving at m/s. Answer: 1.32 Å, or 132 pm 3 As you calculated in Example 6.4.1, objects suc as a baseball or a neutron ave suc sort wavelengts tat tey are best regarded primarily as particles. In contrast, objects wit very small masses (suc as potons) ave large wavelengts and can be viewed primarily as waves. Objects wit intermediate masses, suc as electrons, exibit te properties of bot particles and waves. Altoug we still usually tink of electrons as particles, te wave nature of electrons is employed in an electron microscope, wic as revealed most of wat we know about te microscopic structure of living organisms and materials. Because te wavelengt of an electron beam is muc sorter tan te wavelengt of a beam of visible ligt, tis instrument can resolve smaller details tan a ligt microscope can (Figure 6.4.1). Figure 6.4.1: A Comparison of Images Obtained Using a Ligt Microscope and an Electron Microscope. Because of teir sorter wavelengt, igenergy electrons ave a iger resolving power tan visible ligt. Consequently, an electron microscope (b) is able to resolve ner details tan a ligt microscope (a). (Radiolaria, wic are sown ere, are unicellular planktonic organisms.) AN IMPORTANT WAVE PROPERTY: PHASE A wave is a disturbance tat travels in space. Te magnitude of te wave at any point in space and time varies sinusoidally. Wile te absolute value of te magnitude of one wave at any point is not very important, te relative displacement of two waves called te pase difference, is vitally important because it determines weter te waves reinforce or interfere wit eac oter. Figure 6.4.2A ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 2/6

3 sows an arbitrary pase difference between two wave and Figure 6.4.2B sows wat appens wen te two waves are 180 degrees out of pase. Te green line is teir sum. Figure 6.4.2C sows wat appens wen te two lines are in pase, exactly superimposed on eac oter. Again, te green line is te sum of te intensities. Figure 6.4.2: Pase. Two waves traveling togeter are displaced by a pase difference. If te pase difference is 0 ten tey lay on top of eac oter and reinforce. If te pase difference is 180 tey completely cancel eac oter out. For a review of pase aspects in sinusoids, ceck te mat Libretexts library. Standing Waves De Broglie also investigated wy only certain orbits were allowed in Bor s model of te ydrogen atom. He ypotesized tat te electron beaves like a standing wave (a wave tat does not travel in space). An example of a standing wave is te motion of a string of a violin or guitar. Wen te string is plucked, it vibrates at certain xed frequencies because it is fastened at bot ends (Figure 6.4.3). If te lengt of te string is L, ten te lowest-energy vibration (te fundamental (te lowest-energy standing wave) as wavelengt λ = L 2 λ = 2L (6.4.5) Higer-energy vibrations are called overtones (te vibration of a standing wave tat is iger in energy tan te fundamental vibration) and are produced wen te string is plucked more strongly; tey ave wavelengts given by λ = 2L n (6.4.6) were n as any integral value. Tus te resonant vibrational energies of te string are quantized. Wen plucked, all oter frequencies die out immediately. Only te resonant frequencies survive and are eard. By analogy we can tink of te resonant frequencies as being quantized. Notice in Figure tat all overtones ave one or more nodes (te points were te amplitude of a wave is zero), points were te string does not move. Te amplitude of te wave at a node is zero. ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 3/6

4 Figure 6.4.3: Standing Waves on a Vibrating String. Te vibration wit \(n = 1\) is te fundamental and contains no nodes. Vibrations wit iger values of n are called overtones; tey contain \(n 1\) nodes. Quantized vibrations and overtones containing nodes are not restricted to one-dimensional systems, suc as strings. A two-dimensional surface, suc as a drumead, also as quantized vibrations. Similarly, wen te ends of a string are joined to form a circle, te only allowed vibrations are tose wit wavelengt were r is te radius of te circle. De Broglie argued tat Bor s allowed orbits could be understood if te electron beaved like a standing circular wave (Figure 6.4.4). Te standing wave could exist only if te circumference of te circle was an integral multiple of te wavelengt suc tat te propagated waves were all in pase, tereby increasing te net amplitudes and causing constructive interference. Oterwise, te propagated waves would be out of pase, resulting in a net decrease in amplitude and causing destructive interference. Te non resonant waves interfere wit temselves! De Broglie s idea explained Bor s allowed orbits and energy levels nicely: in te lowest energy level, corresponding to n = 1 in Equation 6.4.7, one complete wavelengt would close te circle. Higer energy levels would ave successively iger values of n wit a corresponding number of nodes. SEISMIC SEICHES 2πr = nλ (6.4.7) Standing waves are often observed on rivers, reservoirs, ponds, and lakes wen seismic waves from an eartquake travel troug te area. Te waves are called seismic seices, a term rst used in 1955 wen lake levels in England and Norway oscillated from side to side as a result of te Assam eartquake of 1950 in Tibet. Tey were rst described in te Proceedings of te Royal Society in 1755 wen tey were seen in Englis arbors and ponds after a large eartquake in Lisbon, Portugal. Seice in Lake Geneva, Switcerland. A seice is te slosing of a closed body of water from eartquake saking. Swimming pools often ave seices during eartquakes. Image used wit permission (Prof. Brennan, Geneseo State Univ. of New York). Seismic seices were also observed in many places in Nort America after te Alaska eartquake of Marc 28, Tose occurring in western reservoirs lasted for two ours or longer, and amplitudes reaced as ig as nearly 6 ft along te Gulf Coast. Te eigt of seices is approximately proportional to te tickness of surface sediments; a deeper cannel will produce a iger seice. Still, as all analogies, altoug te standing wave model elps us understand muc about wy Bor's teory worked, it also, if pused too far can mislead. Figure 6.4.4: Standing Circular Wave and Destructive Interference. (a) In a standing circular wave wit \(n = 5\), te circumference of te circle corresponds to exactly ve wavelengts, wic results in constructive interference of te wave wit itself wen overlapping occurs. (b) If te ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 4/6

5 circumference of te circle is not equal to an integral multiple of wavelengts, ten te wave does not overlap exactly wit itself, and te resulting destructive interference will result in cancellation of te wave. Consequently, a standing wave cannot exist under tese conditions. As you will see, some of de Broglie s ideas are retained in te modern teory of te electronic structure of te atom: te wave beavior of te electron and te presence of nodes tat increase in number as te energy level increases. Unfortunately, is (and Bor's) explanation also contains one major feature tat we know to be incorrect: in te currently accepted model, te electron in a given orbit is not always at te same distance from te nucleus. Te Heisenberg Uncertainty Principle Because a wave is a disturbance tat travels in space, it as no xed position. One migt terefore expect tat it would also be ard to specify te exact position of a particle tat exibits wavelike beavior. A caracteristic of ligt is tat is can be bent or spread out by passing troug a narrow slit as sown in te video below. You can literally see tis by alf closing your eyes and looking troug your eye lases. Tis reduces te brigtness of wat you are seeing and somewat fuzzes out te image, but te ligt bends around your lases to provide a complete image rater tan a bunc of bars across te image. Tis is called diffraction. Tis beavior of waves is captured in Maxwell's equations (1870 or so) for electromagnetic waves and was and is well understood. Heisenberg's uncertainty principle for ligt is, if you will, merely a conclusion about te nature of electromagnetic waves and noting new. De Broglie's idea of wave particle duality means tat particles suc as electrons wic all exibit wave like caracteristics, will also undergo diffraction from slits wose size is of te order of te electron wavelengt. Tis situation was described matematically by te German pysicist Werner Heisenberg ( ; Nobel Prize in Pysics, 1932), wo related te position of a particle to its momentum. Referring to te electron, Heisenberg stated tat at every moment te electron as only an inaccurate position and an inaccurate velocity, and between tese two inaccuracies tere is tis uncertainty relation. Matematically, te Heisenberg uncertainty principle is greater tan or equal to Planck s constant divided by 4π: states tat te uncertainty in te position of a particle (Δx) multiplied by te uncertainty in its momentum [Δ(mv)] is greater tan or equal to Planck s constant divided by 4π: (Δx) (Δ [mv]) 4π (6.4.8) Because Planck s constant is a very small number, te Heisenberg uncertainty principle is important only for particles suc as electrons tat ave very low masses. Tese are te same particles predicted by de Broglie s equation to ave measurable wavelengts. If te precise position x of a particle is known absolutely (Δx = 0), ten te uncertainty in its momentum must be in nite: (Δ [mv]) = = = 4π (Δx) 4π (0) (6.4.9) Because te mass of te electron at rest ( m) is bot constant and accurately known, te uncertainty in Δ(mv) must be due to te Δv term, wic would ave to be in nitely large for Δ(mv) to equal in nity. Tat is, according to Equation 6.4.9, te more accurately we know te exact position of te electron (as Δx 0 ), te less accurately we know te speed and te kinetic energy of te electron (1/2 2 mv ) because Δ(mv). Conversely, te more accurately we know te precise momentum (and te energy) of te electron [as Δ(mv) 0], ten Δx and we ave no idea were te electron is. Bor s model of te ydrogen atom violated te Heisenberg uncertainty principle by trying to specify simultaneously bot te position (an orbit of a particular radius) and te energy (a quantity related to te momentum) of te electron. Moreover, given its mass and wavelike nature, te electron in te ydrogen atom could not possibly orbit te nucleus in a well-de ned circular pat as predicted by Bor. You will see, owever, tat te most probable radius of te electron in te ydrogen atom is exactly te one predicted by Bor s model. EXAMPLE 6.4.1: QUANTUM NATURE OF BASEBALLS Calculate te minimum uncertainty in te position of te pitced baseball from Example tat as a mass of exactly 149 g and a speed of 100 ± 1 mi/. Given: mass and speed of object Asked for: minimum uncertainty in its position Strategy: ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 5/6

6 A. Rearrange te inequality tat describes te Heisenberg uncertainty principle (Equation 6.4.8) to solve for te minimum uncertainty in te position of an object (Δx). B. Find Δv by converting te velocity of te baseball to te appropriate SI units: meters per second. C. Substitute te appropriate values into te expression for te inequality and solve for Δx. Solution: A Te Heisenberg uncertainty principle (Equation 6.4.8) tells us tat. Rearranging te inequality gives 34 (Δx)(Δ(mv)) = /4π (6.4.10) Δx ( )( ) 4π 1 Δ(mv) B We know tat = J s and m = kg. Because tere is no uncertainty in te mass of te baseball, Δ(mv) = mδv and Δv = ±1 mi/. We ave 1 mi Δν = ( )( )( )( )( ) = m/s 1 60 min 1 min 60 s km mi 1000 m km (6.4.11) C Terefore, J s 4 (3.1416) Inserting te de nition of a joule (1 J = 1 kg m /s ) gives Δx ( )( ) 2 2 Δx kg m 2 s 4 (3.1416) ( s 2 ) 1 (6.4.12) (0.149 kg) ( m s 1 ) Tis is equal to inces. We can safely say tat if a batter misjudges te speed of a fastball by 1 mi/ (about 1%), e will not be able to blame Heisenberg s uncertainty principle for striking out. 1 s (0.149 kg ) ( m ) (6.4.13) Δx 7.92 ± m (6.4.14) EXERCISE Calculate te minimum uncertainty in te position of an electron traveling at one-tird te speed of ligt, if te uncertainty in its speed is ±0.1%. Assume its mass to be equal to its mass at rest. Answer m, or 0.6 nm (about te diameter of a benzene molecule) Summary An electron possesses bot particle and wave properties. Te modern model for te electronic structure of te atom is based on recognizing tat an electron possesses particle and wave properties, te so-called wave particle duality. Louis de Broglie sowed tat te wavelengt of a particle is equal to Planck s constant divided by te mass times te velocity of te particle. λ = mv Te electron in Bor s circular orbits could tus be described as a standing wave, one tat does not move troug space. Standing waves are familiar from music: te lowest-energy standing wave is te fundamental vibration, and iger-energy vibrations are overtones and ttps://cem.libretexts.org/textbook_maps/general_cemistry_textbook_maps/map%3a_cemistry%3a_te_central_science_(brown_et_al.)/06._ 6/6