Chapter 3 Wave Properties of Particles De Broglie

Chapter 3: Wave Properties of Particles De Broglie Waves photons cphys 351: 1

Example 3. 1 Find the de Broglie wavelengths of a 46 g golf ball with a velocity of 30 m/s and an electron with a velocity of 107 m/s. Example 3. 2 Find the kinetic energy pf a proton whose de Broglie wavelength is 1. 00 fm (10 -15 m), which is roughly the proton diameter. cphys 351: 2

Waves of what? “normal” waves are a disturbance in space carry energy from one place to another often (but not always) will (approximately) obey the classical wave equation matter waves disturbance is the wave function Y(x, y, z, t ) probability amplitude Y probability density p(x, y, z, t ) =|Y|2 cphys 351: 3

wave properties: phase velocity does not describe particle motion cphys 351: 4

Generic wave properties cphys 351: 5

phase and group velocities simple plane wave inadequate to describe particle motion problems with phase velocity and infinite wave train represent particle with wave packet (wave group) simplified version: superposition of two waves of slightly different wavelength -if wave velocity is independent of wavlength, each wave (and thus the packet) travel at the same speed -if wave velocity is depends upon wavlength, each wave travels at a different speed, in turn different from the wave packet speed. cphys 351: 6

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de Broglie waves for massive particles cphys 351: 8

Example 3. 3 An electron has a de Broglie wavelength of 2. 00 pm Find its kinetic energy, as well as the phase and group velocity of the waves. cphys 351: 9

electron gun Particle Diffraction: The Davisson-Germer experiment scattering of electrons from annealed surface (single crystal) classically, diffuse scattering waves produce constructive/destructive interference ala x-ray diffraction electron detector smaller wavelength => finer resolution as in electron microscope Example: 54 e. V electrons are scattered off of a surface with a strong maximum at an angle of 50 o with respect the incoming beam of electrons. If the spacing between the atomic planes is. 091 nm, what is the wavelength of the electrons from diffraction theory? What is the de Broglie wavelength of the electrons? cphys 351: 10

Particle in a box L Examples: electron in 0. 10 nm box, neutron in 1. 00 fm box, Gallis in room cphys 351: 11

The Uncertainty Principle: limits on probabilities with wave packets probability density |Y|2 maximum near center of wave packet (or near “average”) non-zero near maximum=> uncertainty in position Dx combination of several wavelengths => uncertainty in wave number => uncertainty in momentum Dp uncertainty principle: decreasing Dx (Dp)will eventually drive up Dp (Dx). It is impossible to know both the exact position and exact momentum of an object at the same time. cphys 351: 12

Wave function as a superpopsition of cosine waves: at a particular instant in time cphys 351: 13

Example 3. 6: A measurement establishes the position of a proton with an accuracy of +/-. 001 nm. Find the uncertainty in the proton’s position 1. 00 s later. assume v<<c. Uncertainty principle II: measurement as interaction observe a particle by bouncing photons off of the particle BUT: this uncertainty is an intrinsic limit, not an artifact of cphys 351: 14 measurement!!!

Applications of the uncertainty principle Example 3. 7: A typical atomic nucleus is about 5 fm in radius. Use the uncertainty principle to estimate a lower limit for the energy of an electron confined to the nucleus. Example 3. 8: A a hydrogen atom is about. 053 nm in radius. Use the uncertainty principle to estimate a lower limit for the energy of an electron confined to the atom cphys 351: 15

Energy-Time uncertainty Example 3. 9: An “excited” atom gives up its excess energy by emitting a photon of a characteristic frequency. The average time between the excitation of the atom and the emission of the photon is 10. 0 ns. What is the inherent uncertainty in the frequency of the photon? Chapter 3 problems: 2, 3, 4, 5, 7, 9, 16, 17, 22, 24, 27, 28, 35, 37, 38 cphys 351: 16