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Ch 39 The Wave Nature of Particles © 2005 Pearson Education

39.1 De Broglie Waves de Broglie wavelength of a particle © 2005 Pearson Education

Example 39.1 Find the speed and kinetic energy of a neutron (m=1.675*10e-27kg) that has a de Broglie wavelength λ=0.2nm, approximately the atomic spacing in many crystals. Compare the energy with the average translational kinetic energy of a gas molecules at room temperature. Find the speed and kinetic energy of a neutron (m=1.675*10e-27kg) that has a de Broglie wavelength λ=0.2nm, approximately the atomic spacing in many crystals. Compare the energy with the average translational kinetic energy of a gas molecules at room temperature.ANS: © 2005 Pearson Education

39.2 Electron Diffraction © 2005 Pearson Education

de Broglie wavelength of an electron © 2005 Pearson Education

39.3 Probability and Uncertainty © 2005 Pearson Education

Heisenberg uncertainty principle for position and momentum

Heisenberg uncertainty principle for energy and time interval © 2005 Pearson Education

Two slit interference

39.4 The Electron Microscope © 2005 Pearson Education

one-dimensional Schrödinger equation 39.5 Wave Functions and the Schrodinger Equation time-dependent wave function for a stationary state © 2005 Pearson Education

Wave packet

© 2005 Pearson Education Electrons and other particles have wave properties. The wavelength of the wave depends on the particle’s momentum in the same way as for photons. The state of a particle is described not by its coordinates and velocity components but rather by a wave function, which in general is a function of the three space coordinates and time. (See Example 39.1)

Diffraction of an electron beam from the surface of a metallic crystal provided the first direct confirmation of the wave nature of particles. If a non-relativistic electron has been accelerated from rest through a potential difference V ab, its wavelength is given by Eq. (39.5). Electron microscopes use the very small wavelengths of fast-moving electrons to make images with resolution thousands of times finer than is possible with visible-light microscopes. (See Examples 39.2 and 39.5) © 2005 Pearson Education

) It is impossible to make precise determinations of a coordinate of a particle and of the corresponding momentum component at the same time. The precision of such measurements for the x-components is limited by the Heisenberg uncertainty principle, Eq. (39.11); there are corresponding relations for the y- and z- components. The uncertainty ∆E in the energy of a state that is occupied for a time ∆t is given by Eq. (39.13). In these expressions, Ћ=h/2π. (See Examples 39.3 and 39.4) © 2005 Pearson Education

The wave function ψ(x, y, z, t) for a particle contains all of the information about that particle. The quantity, called the probability distribution function, determines the relative probability of finding a particle near a given position at a given time. If the particle is in a state of definite energy, called a stationary state, ψ(x, y, z, t) is a product of a functionψ that depends only on spatial coordinates and a function e -iEt/Ћ that depends only on time. For a stationary state, the probability distribution function is independent of time.

For a particle that moves in one dimension in the presence of a potential energy function U(x), the wave function for a stationary state of energy E satisfies the Schrodinger equation. More complex wave functions can be made by superposing stationary- state wave functions. These can represent particles that are localized in a certain region and still have wave properties, giving it both particle and wave aspects. (See Example 39.6) © 2005 Pearson Education

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