The Wave Nature of Particles

Goals

• To study particles as waves, de Broglie waves

• To consider particles as waves with electron diffraction and electron microscopy

• To state and explain the difficult conceptual concept of the Heisenberg uncertainty principle

• To study the general tool for description of a system—wave functions

• To consider the wide-ranging coverage and implications of the wave description for an electron in chemical systems, the Schrödinger equation

Introduction• At the turn of the 20th century,

Albert Einstein helped lead science to light as a particle and wave–particle duality. The next logical step was not far behind. De Broglie, Heisenberg, and eventually Schrödinger developed a formalism to treat the particle (an electron) as a wave spawning the new adventure, quantum theory.

• The electron micrograph of a fly’s foot depends on wave-interference properties of a fundamental atomic particle, the electron.

De Broglie waves• If you asked the baseball

pitcher about the wavelength of his fastball, he’d likely send you off to deep in the outfield. But, despite the seeming conundrum, macroscopic objects do have wave properties, and there is particle–wave duality. The part that’s so hard to see is the wavelengths of de Broglie waves. For fastballs and rifle bullets, they have wavelengths smaller than atomic nuclei by orders of magnitude. The genesis of the idea is shown in Figure 39.2 at right.

Electron diffraction sets a foundation for microscopy

• Heated filaments and electrostatic lenses can create and manipulate beams of electrons for experimentation. Refer to Figure at the bottom of the slide.

• Graphs of the results for electron scattering are shown in Figure at right.

Probabilities and electron interference• Electrons flowing through a slit will form interference patterns like those shown in

the Figure.

Heisenberg’s Uncertainty Principle (HUP)

Heisenberg stated that you can’t know the position and momentum of a particle simultaneously. The principle applies in other applications such as spectral-line width and laser-pulse duration, preventing high-resolution femtosecond spectroscopy.

•There is HUP between momentum and position along the same space dimension.

•There is also HUP between energy and time.

Two-slit interference

•The predicted maximum “grows in” as the quantity of measurements increases.

Electron microscopy• The instrument depends on

electrons as waves interfering.

Schrödinger’s wave equation

• The process presumes that a proper operator working on an equation which rightly describes the potential and kinetic components of a system will return eigenvalues and eigenvectors of the system.

• For example, a vibrational Schrödinger equation can be built upon Hooke’s Law. When the Hamiltonian operator operates on the wave equation, eigenvalues will come back as the ascending energies of sequentially higher vibrational levels.

Wave packets• A wave packet can be built by superposition of a large number of similar oscillating

waves with different wave numbers but similar amplitudes. This package of waves can be set into motion to … circumvent the uncertainty principle. The packet samples a problem from many different starting points so the outcome will contain elements of the answer, each from a different wave.

• Fourier analysis is necessary to “unpack the components of a wave packet” to discern the pieces of the puzzle … one by one. Figure below illustrates a wave packet.