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Chapter 4 The Wavelike Properties of Particles • The energy carried by a particle is confined to a small region of space • The energy carried by a wave is distributed throughout space, but localized. In quantum mechanics there is a clear distinction from classical mechanics. Particles must somehow obey the rules previously established for waves → Wave Mechanics. 4.1 The de Broglie Hypothesis 𝑝=ℎ 𝜆 Just as a wave can have momentum (a particlelike property), a particle with momentum 𝑝 can exhibit wavelike properties (having wavelength 𝜆). 1. Because of the smallness of ℎ, only particles of atomic or nuclear size will exhibit the wave behavior. 2. The de Broglie wavelength 𝜆 reveals itself when a wavetype experiment is performed on it. The outcome of the experiment depends on this wavelength. 3. The de Broglie wavelength characterizes the wave-type behavior of particles, and is central to the quantum theory. 4.2 Experimental Evidence for de Broglie Waves The diffraction minima are located at 𝑎𝑠𝑖𝑛𝜃 = 𝑛𝜆 n = 1, 2, 3, . . . 1926 Clinton Davisson and Lester Germer (Bell Telephone Labs) The diffraction of electrons 𝜙 = 50° V = 54 volts The crystal surface acts like a diffraction grating. 𝑑𝑠𝑖𝑛𝜙 = 𝑛𝜆 n = 1, 2, 3, Double Slit Experiment with Particles The electrons from a hot filament were accelerated through 50 kV (𝜆 =5.4 pm) and then passed through a double slit of separation 2.0 µm and width 0.5 µm. Double-slit pattern for electrons. 𝑑Δ𝑦 𝜆 = 𝐷 Neutron Scattering and the double slit experiment Intensity pattern observed for double-slit interference with neutrons. Which slit does the particle pass? A meter measures which slit the electron passes through. As a result, no interference fringes are observed on the screen. Atomic Structure of Benzene deduced from neutron diffraction. Black circles are the carbon atoms, while the blue circles denote the hydrogen atoms. 4.3 Uncertainty Relationships for Classical Waves ∆𝑥 ≈ 𝜆∆𝜆~𝜀𝜆∆𝑥∆𝜆~𝜀𝜆9 ∆𝑡 ≈ 𝑇∆𝑇~𝜀𝑇∆𝑡∆𝑇~𝜀𝑇 9 Using the de Broglie Relationship: 𝑝 = < < Δ𝑝 = − ? Δ𝜆 then = = Likewise, using Einstein's relationship: 𝐸 = ℎ𝑓 So, ∆𝐸 = ℎ∆𝑓 then ∆𝐸 = ℎ − B C? ∆𝑓 = − ∆𝑇 4.4 Heisenberg Uncertainty Relationships where ℏ ≡ < 9G ∆𝑥∆𝑝~𝜀ℎ ∆𝑥∆𝑝 ≥ ℏ ∆𝐸∆𝑡~𝜀ℎ ∆𝐸∆𝑡 ≥ ℏ and 𝜀 = B 9G The larger the Δ𝑥, the smaller the Δ𝑝H . B C? ∆𝑇 ∆𝒙 ∆𝒑𝒙 Examples: (1) Can electrons reside in the nucleus ? (2) What is the lifetime of a 𝜌L meson (mass = 775𝑀𝑒𝑉/𝑐 9 , and Δ𝑀 = 149𝑀𝑒𝑉/𝑐 9 ) ? 4.5 Wave Packets The purpose of this section is try and construct a wave (wave function) that describes the position and motion of a particle. Two waves of slightly different wavelengths: 𝑦 𝑥 = 𝐴B cos 𝑘B 𝑥 + 𝐴9 cos 𝑘9 𝑥 𝑦 𝑥 = 𝐴B cos 2𝜋𝑥/𝜆B + 𝐴9 cos(2𝜋𝑥/𝜆9 ) 𝜋𝑥 𝜋𝑥 𝜋𝑥 𝜋𝑥 𝑦 𝑥 = 2𝐴𝑐𝑜𝑠 − 𝑐𝑜𝑠 + 𝜆B 𝜆9 𝜆B 𝜆9 If 𝜆B and 𝜆9 are close together, then: 𝑦 𝑥 = 2𝐴𝑐𝑜𝑠 𝛥𝜆𝜋𝑥 2𝜋𝑥 𝑐𝑜𝑠 𝜆bc 𝜆9bc 1. The first cosine term is the shaping envelope. 2. The second cosine term represents a wavelength 𝜆bc . 1. 2. 3. This results in a repeating “beat” pattern. See example. This does not localize the wave packet. How can we localize the “continuous” wave into a single wave packet? The book describes three different ways of doing this: 1. modulating the wave using the 𝑦 𝑥 = 2. 9d H 𝑠𝑖𝑛 e=GH 𝑐𝑜𝑠 =?f B H technique: 9GH =f modulating the wave using the Gaussian technique: 𝑦 𝑥 = e=GH g9 =?f 𝐴𝑒 ? cos 2𝜋𝑥 𝜆L 3. Adding together waves of differing amplitude and wavelength, but the wavelengths form a continuous rather than a discrete set of waves. 𝑦 𝑥 = 𝐴h cos 𝑘h 𝑥 → 𝐴 𝑘 𝑐𝑜𝑠 𝑘𝑥 𝑑𝑘 This is called a Fourier cosine transformation. See the example on my website: A better approximation of the shape of the wave packet can be found by letting 𝐴 𝑘 vary according to a Gaussian distribution: 𝐴 𝑘 = 𝐴L 𝑒 g igif ? /9 ei ? See the comparison of 𝛥𝑝 to 𝛥𝑥 shown in the previous plot earlier in this lecture: Using the Fourier cosine transform with the Gaussian 𝐴 𝑘 : 𝑦 𝑥 = 𝐴L 𝛥𝑘 2𝜋𝑒 eiH ? /9 cos 𝑘L 𝑥 4.6 The Motion of a Wave Packet A traveling wave: 𝐴cos(𝑘𝑥 − 𝜔𝑡) 1. Has a unique momentum 𝑝 = ℏ𝑘 2. Has an infinite extent 𝛥𝑥 = ∞ Let’s look at the structure of a traveling wave packet made up of two waves close in wave-number and close in frequency. 𝛥𝑘 𝛥𝜔 𝑘B + 𝑘9 𝜔B + 𝜔9 𝑦 𝑥, 𝑡 = 2𝐴 𝑥 − 𝑡 𝑐𝑜𝑠 𝑥 − 𝑡 2 2 2 2 1. The first term describes the overall shape of the waveform. 2. The second term represents the rapid variation of the wave within the envelopes described by the first term. The speed of the overall waveform is: 𝛥𝜔 𝑣mnLop = 2 = 𝛥𝜔 𝛥𝑘 𝛥𝑘 2 The speed of the underlying wave is: 𝜔B + 𝜔9 𝜔bc 𝜔 2 𝑣p<bqr = = = 𝑘B + 𝑘9 𝑘bc 𝑘 2 The Group Speed of a deBroglie Wave 𝑣mnLop = 𝑑𝜔 𝑑𝐸 ℏ 𝑑𝐸 = = 𝑑𝑘 𝑑𝑝 ℏ 𝑑𝑝 For a classical particle having only kinetic energy 𝐸 = 𝐾 = So, 𝑣mnLop = 𝑝9 2𝑚 𝑑𝐸 𝑝 = = 𝑣 𝑑𝑝 𝑚 Thus, 𝑣mnLop = 𝑣pbnuhvwr