Planck’s theory of black body radiation proposed that energy is emitted by oscillators in discrete packets E=h. These packets, called photons, are treated as energy particles Bohr extended Planck’s hypothesis to provide an explanation for the stability of atoms and define energy levels for a hydrogenic atom: Summary of lecture 19 Combining the two, if an atom absorbs energy, its electron will be promoted from the n th orbit to, say, the k th orbit, the frequency of the energy particle or photon emitted by the atom is given by:
Wave-Particle Duality Up until now we have dealt with the first fundamentally different concept of quantum mechanics, quantization of energy. The second even more radical departure from classical principles (and every day’s intuition) is the dual nature of matter Several experimental observations suggested that light waves can have particle-light properties These observations can be summarized to say that energy can have particle-like properties
Waves vs. Particles We began our discussion by defining light in terms of wave-like properties. But Planck’s relationships suggest that light can be thought of as a series of energy “packets” or photons.
Wave-Particle Duality Light incident on a metal surface can, under some circumstances, eject electrons from the surface (photoelectric effects). If the light is below a certain frequency, no electrons are ejected. If the light is above a certain frequency, electrons will be emitted regardless of how low the intensity of the light is. The maximum kinetic energy of the ejected electrons is independent of the intensity of the light and dependent on its frequency The scattering of X-rays from carbon and other materials is explained by assuming the X-ray photons have particle-like collisions with atoms and electrons (Compton effect); that is to say, they have momentum though no mass. These observations can be summarized to say that energy can have particle-like properties.
The Photoelectric Effect Shine light on a metal and observe electrons that are released. Find that one needs a minimum amount of photon energy to see electrons (“ o ”). Also find that for ≥ o, number of electrons increases linearly with light intensity.
The Photoelectric Effect Finally, notice that as frequency of incident light is increased, kinetic energy of emitted e - increases linearly. = energy needed to release e - Light apparently behaves as a particle.
The Photoelectric Effect For Na with = 4.4 x 10 -19 J, what wavelength corresponds to o ? 0 h = = 4.4 x 10 -19 J hc = 4.4 x 10 -19 J = 4.52 x 10 -7 m = 452 nm
Particles as waves Electrons shine through a crystal and look at pattern of scattering. Diffraction can only be explained by treating electrons as a wave instead of a particle.
Wave-Particle Duality At about the same time, Davisson and Germer made the reciprocal observation; they diffracted electrons (which are of course particles of with a certain mass) against single crystals of nickel and observed a diffraction particle as would be produced by X-ray diffraction. Incidentally, electron diffraction is nowadays a major technique of structure determination Changing the electron speed they could change the momentum and they measured the diffraction pattern as a function of momentum; based on well-known classical wave diffraction theory and the experimental results, they calculated the wavelength associated with the electron and they discovered the following relationship:
Wave-Particle Duality This is called De Broglie’s wavelength and every particle (e.g. neutron, protons, etc.) has been experimentally demonstrated to have a characteristic De Broglie’s wavelength which depends on its momentum only. In 1925, deBroglie proposed an explanation for why an electron does not decay from a Bohr orbit. Recalling Planck’s hypothesis that radiation is quantized in particles of energy E=h with momentum, deBroglie hypothesized that electrons have wavelengths
Wave-Particle Duality If the wavelength of an electron wave orbiting a nucleus is an integral multiple of the length of the orbit, a standing electron wave results (see diagram below). deBroglie’s equation states that within a Bohr orbit: Using Bohr’s expression for the quantization of angular momentum:
Particle and Waves What is a standing wave? A standing wave is a motion in which translation of the wave does not occur. In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs. Note also that integer and half- integer values of the wavelength correspond to standing waves.
Particle and Waves Louis de Broglie suggests that for the e - orbits envisioned by Bohr, only certain orbits are allowed since they satisfy the standing wave condition. not allowed
De Broglie’s wavelength DeBroglie’s particle wave hypothesis can be used to obtain quantized energy expressions for very simple problems. Example: For a particle in a box, the length of the box L must equal an integral multiple of half DeBroglie’s wavelength (to have constructive interference between waves, otherwise there would be destructive interference and the waves would cancel out):
Wave-Particle Duality Both electrons and light waves collide with surfaces with finite momentum, although we normally associate momentum with particles. Both electrons and X-rays diffract off of surfaces, although we normally associate diffraction and interference with radiation waves. However, particles are localized in space, while waves are not. How do we treat particles as waves and viceversa?
Wave-Particle Duality In classical physics, radiation waves are represented by plane wave functions that are periodic in time and space. An example of a plane wave traveling in the x direction is Where A is the amplitude of the wave, k=2 / is the propagation constant, is the angular frequency If the wave function is independent of time, we have a stationary or standing wave For simplicity we show only the sine component below
Wave-Particle Duality A wave propagates through space at its wave velocity, which is viewed as the time it takes for the wave peak to shift by one wavelength. This diagram expresses the fact that waves are continuous functions in space and time, whereas particles are localized in space. How can a wave function represent a particle?
Wave-Particle Duality A matter wave packet can be localized to a single point in space if we superimpose an infinite number of waves with differing wavelengths. For a stationary wave center at x=x 0 : In other words, if we superimpose a finite number of waves with wavelengths varying between The resulting wave packet has the form:
Wave-Particle Duality As more waves are added the wave “packet” gets narrower. In the limit of an infinite number of waves covering all wavelength values, then we can localize a particle to a single point in space Using deBroglie’s expression, the width of the central lobe of the packet is: Associating a packet of wave to a particle has an unexpected consequence that leads to perhaps the most radical of all ideas of quantum mechanics.
Heisenberg’s principle From the relationship given above, when we use packet of waves to represent a particle localized within a certain space we must superimposes waves covering a range of p such that p=2p 0. Thus The range of momentum 2p 0 represents the uncertainty with which the momentum associated with the particle is known. Thus, if we measure how well localized is a certain particle and also its momentum, we cannot measure both with infinite precision; in other words the precision with which we can measure location and momentum is limited by the relationship:
Heisenberg’s principle Heisenberg’s observed that this is a best-case scenario. In general: This is the Heisenberg Uncertainty Principle, which limits our ability to define the position of a particle at a particular time. At best we can calculate the probability that a particle is located at a particular position at the time of a measurement A similar expression can be found for energy and time:
Heisenberg’s principle The Heisenberg Uncertainty Principle is perhaps that must controversial theory of modern times. The basic principles of classical mechanics can be summarized as follows: There is no limit to the accuracy with which dynamical variables (e.g. position, momentum, time, and energy) can be determined simultaneously, except the limit imposed by instruments of measurement. There is no restriction on the number of dynamical variables that can be measured simultaneously The velocity of a particle, and hence its kinetic energy, is a continuous function. There are no restrictions on the values that the energy may attain.
Heisenberg’s principle Heisenberg’s Uncertainty Principle imposes a limit on the accuracy of measurements of the dynamical variables x and p x Together with the quantization condition E=nh Heisenberg’s Uncertainty Principle overturned the basic principles of classical mechanics.