To illustrate the matter waves of the electron in an hydrogen atom, the square of the local amplitude is plotted. The dark spot in the middle of the surface indicates that this point does not move. The mode (f) may be obtained by gently shaking the glass with the right frequency again different directions of motion are possible, and, similar to the above, there are two linearly independent modes, the superposition of which will yield all the others, in particular also waves going round clockwise or counterclockwise (g). It has rotational symmetry and therefore there is no other possible orientation. The mode of figure (e) can be excited by falling drops. This can happen in an arbitrary direction and therefore there are infinitely many standing waves of that kind, all with the same oscillation frequency. In the simplest mode in a glass of water, which can easily be excited, the liquid is swashing to and fro. The following pictures are idealized and in slow motion. Water waves have the advantage to be slow and the wave motion can clearly be seen. If we restrict the considerations to small amplitudes, we may neglect that. Water waves are more difficult to describe than those which we are interested in. In the case of vibrating metal plates, the nodal lines are known from classroom demonstrations as Chladni figures. Instead of nodes the normal modes exhibit nodal lines. Important properties of three-dimensional waves cannot be seen on strings vibrating membranes show somewhat more. The motion is so fast that it cannot be resolved by the eye, one sees a sequence of nodes and antinodes. The picture below shows how a string vibrates in the lowest three normal modes. The superposition of different normal modes is heard as superposition of ground- and overtones. These are called normal modes or eigenmodes. It is always possible to describe the motion of the string as a superposition of simple modes which have the peculiar property that all parts of the string move sinusoidally with the same frequency and phase. The exact way how a guitar's string vibrates depends on the spot where it has been plucked. Some simple examples will be used to demonstrate general properties of oscillating systems, standing waves in particular. For us it is important to realize that the electron forms some kind of standing wave. This is described in detail in all textbooks on quantum mechanics. Its possible wavefunctions can be obtained as solutions of the Schrödinger equation. Its nucleus carries one unit of positive elementary charge and thus binds only one electron to it. The hydrogen atom is the simplest of all atoms. Wavefunctions are used to calculate observable quantities in particular, the probability to find the (pointlike) particle in some volume is given by the squared value of the wavefunction integrated over the volume. Thus the electrons bound by electric force to an atomic nucleus (which contains almost all of the atom's mass) must be considered to be waves. The fact that these apparently contradictory attributes are compatible in matter waves and also in light (photons) is hard to understand, but all experimental data point out that this is the case. Waves always have some spatial extension, while one may imagine the elementary, indivisible particles as being “pointlike”. While we cannot dive into mathematical details here, the basic principles shall be sketched. Quantum theory is, so to say, the mathematical formulation of particle–wave duality. Only with quantum theory atomic structure can be understood. Nevertheless, to understand how the colours which surround us come about, one needs some basic knowledge on the smallest parts of matter. If the light of the sun is spread out into different colours by a simple glass prism, the narrow absorption lines cannot be seen. Neon, which gives red colour in a gas discharge, is a colourless gas. The aurora borealis (northern light) is very rare at our latitudes, and to appreciate the colours of cosmic objects, powerful telescopes are necessary. Neon signs (or other gas discharge tubes) as used for advertising, sodium or mercury vapour lamps show atomic emission the colours of fireworks are due to it. Light emitted or absorbed by single atoms contributes only very little to the colours of our surroundings. The same file with adapted formatting can be found here. Atomic spectra The html formatting and custom instructions have been disabled on this server.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |