Chapter 4: The Wave Nature of Matter

Transcription

1 Chapter 4: The Wave Nature of Matter q We have seen in Chap. 3 that EM radiation displays both wave properties (classical description) and particle properties (quantum description) q Matter is described in classical mechanics as a point particle. Rigid bodies are just a collection of point particles - in relativistic mechanics, massive objects are treated as point particles - the point-particle description is very successful (for macroscopic objects)

2 This concept of wave-particle duality for EM radiation lead Louis de Broglie to postulate in his 1923 Ph.D. dissertation that matter would also have both particle and wave behaviors For EM radiation - l > D Þ wave property - l < D Þ particle property where D is the ``dimension of an ``experimental apparatus Now for matter - l < D Þ particle property - l > D Þ wave property

3 We know that a photon has a momentum (as confirmed by the Compton effect) given by h h p = Þ l = l p de Broglie postulated that the same relation holds for a massive particle. That is it has a wavelength, which we call the de Broglie wavelength, given by l de Broglie = h = p h mv - here v is the speed of the particle (not that of reference frame S ) - v is usually non-relativistic. We can write the de Broglie wavelength for a relativistic particle as l = de Broglie g v h mv

4 However, in this course we will primarily be developing a non-relativistic version of quantum mechanics A simple example Find the de Broglie wavelengths (wavelength of a matter wave) of (a) a 46-g golf ball with a velocity of 30 m/s and (b) an electron with a velocity of 10 7 m/s. Solution (a) v<<c, g=1 l = m h g v = 6.63x10-34 J.s (0.046 kg)(30 m/s) 4.8x10 - the physical dimension of a golf ball is D~2 cm~10-2 m - We do not see wave aspects to its behavior - Further, we can t resolve a length of m = -34 m

5 (b) v<<c,g= > non-relativistic -34 h 6.63x10 J.s l = = m v (9.1x10 kg)(10 m/s) e - The radius of the hydrogen atom is 0.53x10-10 m. So, for an atom, l»d. - An electron s behavior in an atom is wave-like, not particle-like. The Davisson-Germer Experiment 0.73x10 Originally, the concept of the de Broglie wavelength of a massive object was considered just theoretical speculation In 1927, the de Broglie wavelength for an electron was measured by Davisson and Germer (accidently!) = -10 m

6 Davisson and Germer were conducting electron scattering (54 ev) off of bulk nickel A detector was used to detect the scattered electrons for various angles Classical physics predicts that the electrons would scatter in all directions with the scattered intensity having - moderate dependence on scattering angle - less dependence on incident electron energy q Classical predictions were confirmed for bulk nickel - random surface spacing of atoms and roughness q However, air leaked into the target chamber and the surfaced oxidized

7 The nickel was baked at high temperature in an effort to remove the oxidation and then the experiment repeated A different result was observed the reflection intensity displayed an interference pattern as a function of incident electron energy It was later realized that baking the nickel crystallized the surface The explanation can be obtained by analogy to crystal x-ray diffraction a well know tool for studying the structure of matter In x-ray diffraction, the effective path difference is 2dsinq where d is the atom spacing and q the incident x-ray angle with the surface The two paths constructively interfere when 2dsinq is a integer multiple of the wavelength giving Bragg s Law

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9 2 d sinq = nl n = 1,2,3,! In x-ray diffraction Bragg s law is used to calculate the spacing between the atoms in a crystal In the Davisson-Germer experiments, electrons diffract resulting in a wave phenomenon Ultimately, Davisson and Germer measured the de Broglie wavelength of an electron The same effect is observed for neutron scattering