Quantum Physics

896 Chapter 27 Quantum PhysicsSUMMARYTake a practice test by logging intoPhysicsNow at www.cp7e.com and clicking on the Pre-Testlink for this chapter.27.1 Blackbody Radiation and Planck’sHypothesisThe characteristics of blackbody radiation can’t beexplained with classical concepts. The peak of a blackbodyradiation curve is given by Wien’s displacementlaw; max T 0.289 8 10 2 m K [27.1]where max is the wavelength at which the curve peaksand T is the absolute temperature of the object emittingthe radiation.Planck first introduced the quantum concept whenhe assumed that the subatomic oscillators responsiblefor blackbody radiation could have only discreteamounts of energy given byE n nhf [27.2]where n is a positive integer called a quantum numberand f is the frequency of vibration of the resonator.27.2 The Photoelectric Effect and theParticle Theory of LightThe photoelectric effect is a process whereby electronsare ejected from a metal surface when light is incidenton that surface. Einstein provided a successful explanationof this effect by extending Planck’s quantum hypothesisto electromagnetic waves. In this model, light isviewed as a stream of particles called photons, each withenergy E hf, where f is the light frequency and h isPlanck’s constant. The maximum kinetic energy of theejected photoelectrons isKE max hf [27.6]where is the work function of the metal.27.3 X-Rays27.4 Diffraction of X-Rays byCrystalsX-rays are produced when high-speed electrons aresuddenly decelerated. When electrons have been acceleratedthrough a voltage V, the shortest-wavelength radiationthat can be produced is[27.9]eVThe regular array of atoms in a crystal can act as a diffractiongrating for x-rays and for electrons. The conditionfor constructive interference of the diffracted raysis given by Bragg’s law:2d sin m (m 1, 2, 3, . . .) [27.10]Bragg’s law bears a similarity to the equation for the diffractionpattern of a double slit.27.5 The Compton EffectX-rays from an incident beam are scattered at variousangles by electrons in a target such as carbon. In such ascattering event, a shift in wavelength is observed for thescattered x-rays. This phenomenon is known as theCompton shift. Conservation of momentum and energyapplied to a photon–electron collision yields thefollowing expression for the shift in wavelength of thescattered x-rays: min hc0 hm e c[27.11]Here, m e is the mass of the electron, c is the speed oflight, and is the scattering angle.27.6 The Dual Nature of Lightand MatterLight exhibits both a particle and a wave nature. DeBroglie proposed that all matter has both a particle anda wave nature. The de Broglie wavelength of any particleof mass m and speed v is h p h mv[27.14]De Broglie also proposed that the frequencies of thewaves associated with particles obey the Einstein relationshipE hf.27.7 The Wave Function(1 cos)In the theory of quantum mechanics, each particle isdescribed by a quantity called the wave function.