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Ch 9 pages 446-451; 455-463PowerPoint Presentation

Ch 9 pages 446-451; 455-463

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- Planck’s theory of black body radiation proposed that energy is emitted by oscillators in discrete packets E=hn. 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:

Combining the two, if an atom absorbs energy, its electron will be promoted from the nth orbit to, say, the kth orbit, the frequency of the energy particle or photon emitted by the atom is given by:

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.

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 (“no”).

• Also find that for n ≥ no,

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.

F = energy needed to release e-

• Light apparently behaves as a particle.

The Photoelectric Effect

• For Na with F = 4.4 x 10-19 J,

what wavelength corresponds to no?

0

hn = F = 4.4 x 10-19 J

hc/l = 4.4 x 10-19 J

l = 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.

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:

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=hn with momentum, deBroglie hypothesized that electrons have wavelengths

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):

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?

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=2p/l is the propagation constant, w 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

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?

A matter wave packet can be localized to a single point in space if we superimpose an infinite number of waves with differing wavelengths l. For a stationary wave center at x=x0:

In other words, if we superimpose a finite number of waves with wavelengths varying between

The resulting wave packet has the form:

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.

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 Dp=2p0. Thus

The range of momentum 2p0 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 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:

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 Uncertainty Principle imposes a limit on the accuracy of measurements of the dynamical variables x and px

Together with the quantization condition E=nhn, Heisenberg’s Uncertainty Principle overturned the basic principles of classical mechanics.

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