WAVE MECHANICS: Star drive by John Gard

WAVE MECHANICS

The forces of rotation apply to everything including electromagnets. From the forces of rotation came wave mechanics.

Many have a false image of wave mechanics. This comes from a human trait of trying to explain one thing in terms of another. For example; when one mentions wave mechanics an image of waves on water appears. Since humans are 70 percent visual we assume the wave on the beach which we see is the wave itself. The image is of a two dimensional wave. In fact, the wave we perceive is really the result of the waves interaction with the shore and is not the wave itself.

A wave is really the movement of a force or forces. I believe that instead of wave mechanics it should be called dynamic forces. After all, the word wave really first stood for that mechanical condition which does occur to water at the shore and there is nothing mechanical about a radio wave.

Wave come in two basic types. Those waves which are linear and those which are not linear. A linear wave is a wave which follows the pure mathematical representation of a point moving in a true circle. Square waves and many other waves which follow movement factors other than a circle are nonlinear type waves. In reality, there is no such thing as a pure linear wave, but, we will not dwell on that. Waves can be in a two dimensional or three dimensional environment and translational or rotational. The following picture has examples of several wave types in a two dimensional environment.

The only linear wave by the above definition of linear wave is the sinusoidal wave. Even the continuous wave is not a true linear wave. True, it's a linear line but not a wave because it does not under go angular acceleration which is required to be representative of a wave. First thing to develop in wave mechanics is some mathematical time domain function to describe the wave. As we are going to deal with linear waves, the velocity and t is the time at any given moment what's position point with respect to time is angular velocity times time.

Types of t waves are those t waves that exhibit a continuous angular velocity and those which have their own function of t. The basic theory of the communications comes in the terms of AM radio band and FM radio band. Amplitude modulation is AM and FM is frequency modulation.

Since we are trying to find what is happening to the wave at some distance from the source and because the wave is moving through a media at some velocity, we must apply a propagation constant to the wave with respect to its position:

kp is the propagation constant and x is the position. This is commonly called a modulated wave. Note: k is used by many people to define many types of propagation constants and other things, it is hard to keep things straight and should not be confused with Boltzmann's constant either.

Because the wave exists we must give it some amplitude A and the wave must be represented as a function in time. The most obvious function which works is the sin function. Thus:

The function F(x) here is a representation of the waves intensity at a point in time and A is the maximum amplitude. In a contained system as a rope, ocean, or wave guide; what goes in, must come out, or do work.

This is fine for a translational wave ( which is a wave moving in a contained media like a rope or wave guide ) but what is happening with a spherical wave which is traveling outward from the source in all directions?

First thing, the distance x becomes r for the radii of the sphere and the distance of the point from the source. Next, the amplitude changes. That is it decreases with 4 times distance squared from the source. The reason for the amplitude decreasing is because the same energy must be dispersed over the entire area of the sphere. The system does not add energy at a distance from the source. It only accounts for what has been applied at the source some distance from the source. This becomes intensity per unit area.

This of course has its problems. Such as; what happens at the distance r = 0? To answer this simply, there is no such thing as r = 0. There is always a finite point which the wave begins therefore, there must be some distance in which r has a starting value. At this point the intensity is at its maximum per unit area. The above formula assumes that the source is continually emitting, but, what happens if the forcing function goes to 0 after a length of time or just after one half cycle? What happens when the system encounters some object? In the event the forcing function ceases, the wave stops generating but that which has been generated continues to travel outward. When the wave hits something it either absorbed by it, transmits through it, reflects off it, or a bit of all. The development of wave mechanics and quantum theory attempts to answer these questions using standard math.

To visualize the way the math is developed, I'll first start with a single wave drawing. Visualize a ball rolling along the ground, only fix on a single point on the surface of the ball along the axis of rotation. Imagine a line through the center of the ball parallel with the ground. The point will follow a sinusoidal mathematical relation ship. The angular velocity is the rate the ball is going around in circles and the forward motion is the velocity. We can use the usual x-y axis arrangement but occasionally there is a sever problem with the square root of minus 1. Therefore, instead of a y axis it will be called i or sometimes j.

As the ball is moving, the point scribes a sinusoidal shape in space. In the case of a rolling ball, the angular velocity is a function the forward velocity, proportional to the radius. Two things are occurring with the point; the point is moving up and down, plus along the x axis. One full revolution is called the wave length lambda . Now, imagine the ball off the ground. In this state the angular velocity is not a function of forward velocity. However, the same sinusoidal wave shape is formed with a constant velocity. With respect to the center of the ball any point on the ball in time will follow:

Rasing e to j is Euler's identity and is far easier to deal with than trig functions. Because both positions of the point are of value the total function for our wave is:

A major development to wave theory came with DeBroglie's wave packets. Here again, I will not attempt to give a detail description of the development of wave mechanics. Much detail will be left out, however, a development of the theory will be presented. For the lack of a better name this wave packet when used to describe the affect on light radiation became photon. It was shown that energy seemed to have defined states, with an even jump in energy level E = hf; h is Plank's constant, and f is frequency. Then came from Rayleigh-Jeans blackbody radiation theory. As a body got hot it always emitted a discrete frequency (color) of light. It was demonstrated that a radiating body would make even jumps such that E = nhf where n=1, 2, 3... As light had energy, and it was moving, it was shown that a light photon had momentum of:

where k is the unit directional vector. To some degree, this was verified by the Compton effect. Compton showed that energy was conserved in an individual scattering process. This lead to the condition, where if energy can only be transferred in quantum amounts, the momentum can only be transferred in quantum amounts too.

A very general view of the pulse of oscillating energy can be seen to the right.

Unfortunately, those who attempt to explain this phenomenon visually use the rolling ball concept as I did. For some odd reason, the ball becomes the packet of energy. The problem is that the photon is not a packet in the typical fashion people view it. Most people view the photon as a particle. A photon is not a particle. A photon is a spherically expanding electromagnetic energy system.

To a viewer some distance from the source of the wave packet, the oscillating point is passing by the viewer. The point at where the wave packet has value is called the wave front. As the wave front passes by, the point's electromagnetic characteristics change in time. As the wave passes, its back end has 0 value. The wave packet only has value between the points of t1 and t2. It is between points t1 and t2 that the mathematical function looks like that of a rolling ball to the viewer at a point along the sphere of expansion during t1 < t < t2.

Typically, when an electron state change occurs, energy is given off. This is without regard to how the electron state was raised in the first place. The energy given off has the characteristics of an angular velocity wave traveling in a spherically outward direction. Light is an example of such a phenomenon. It actually is a packet of energy. Meaning it has a discrete beginning and ending. We wouldn't see any thing if all that was given off is one photon. In fact, what is given off from any light source, is grillions of photons. Many photons are given off at the same time, which would account for intensity but, most are given off one after another, which gives the impression of light for a length of time. Also, light is oscillatory in that both the electric and magnetic components of the wave are alternating with a constant angular velocity, this gives it it's color.

Light, and as with all other electromagnetic waves, travels at some speed. In space it is called the speed of light. The very common symbol for the speed of light is c. But, in glass, the speed is much different. In glass, the speed of light is a function of the type of glass, and the frequency of light. As the velocity of electromagnetic waves are very close the same for any given frequency, the frequency and angular velocity are in affect one and the same.

The full energy function for a range of angular velocities passing some point in time is given as:

One of the relationships this equation lead to was the modulation requirement. In radio, this states if the receiver is tuned to accept a band width of , the shortest pulse that can be received has a duration t= 1/. In effect this equation is:

Along the same line of thought, Heisenberg developed his uncertainty principle. Stated: if a measurement of position is made with accuracy x, and if a measurement of momentum is made simultaneously with accuracy p, then the product of the two errors can never be smaller than a number of order h/. The term h/ is inherently a vague quantity with a considerable latitude of possibilities.

Similar to a rotating body, as the angular velocity goes up, the energy stored in the system goes up. The angular momentum of a electromagnetic wave exhibits the same characteristic. As the frequency goes up the angular velocity goes up and the angular momentum goes up. Therefore, as the frequency of the wave increases, it requires more energy to propagate the wave.

Different from a rotating body, a generated electromagnetic wave exhibits secondary waves. The secondary waves are frequency multiples of the driving electromagnetic wave. The primary wave is called the fundamental and the secondary waves are called harmonics. Harmonic frequencies can be either odd or even. Odd being 1, 3, 5, 7 ... and even being 2, 4, 6, 8... In spherical waves, harmonics can have values which are not discrete numbers as 1,2,3... times the fundamental, values such as 1.5933 or 2.294 are possible. With waves, it is said that any thing which can be developed in a two dimensional system, can propagate in a three dimensional system, but not visa-versa.

All waves must propagate. Without the ability of the medium to propagate the wave, there can be no wave. On the other hand, if the medium is pron to a particular type of wave, then given any energy source, the wave will propagate. Most times, the force of the energy driving the wave is independent of the wave. A good example of this type of action is a flag waving. One can go to great length to keep a constant velocity wind blowing by a flag, but the flag will wave regardless of the continuous nature the wind. With sufficient ambient energy available and the whole flag pole will wave.

The differential of the wave function will have an affect the propagation of the wave. All the wave's harmonic functions must propagate. In many instances, the differential of the wave's harmonics will not allow the wave to propagate. When the proper energy and conditions of wave propagation are present the wave will amplify on its own. The term for this some-what automatic function is resonance. In electronics, it is easy to see resonance. Resonance is the term given to a media which is supporting a none vanishing wave. In a typical electronic circuit a resonant condition exists when the value of a capacitor's reactance equals that of an inductor's reactance. This condition can occur in either a series circuit in which the combination seems to be of low impedance to the electric forcing function, or, a parallel circuit in which the combination seems to be of high impedance to the electric forcing function. Capacitors are devices which store energy by electric stretching within a dielectric and have capacitance. Inductors are devices which store energy by a magnetic field and have inductance. The equation for the resonant frequency:

Given a source of energy and a medium of propagation which will resonate, the wave generated within the medium will absorb energy from the source. As the resonate wave continues to absorb energy from the source, the resonate wave will continue to gain intensity. This process will go on indefinitely. Most the time this does not occur because the medium reacts to the wave and changes it's propagation function. The term for this is damping factor and generally is called resistance. Typically, the damping factor is some function of aoe^x. It is also possible that the damping factor's affect is greater than the source energy. In the event that the damping factor is larger than the sustaining energy, the wave will die out.

In the above two pictures of wave points, notice the points which are nonvanishing. The largest or longest wave is at a point where is capable of sustaining a full wave. This is because a wave at 0 and have the same value. However, it is possible to reflect the wave at 0 and without changing momentum. At each nonvanishing point, all the higher harmonic components of the wave can pass through that point without a change momentum. At nodal points where the wave vanishes, note that at the next harmonic the nodes do not pass through that point.

Waves in a spherical environment can propagate in more than one direction:

When an electric wave is sent to an antenna and the antenna is of proper length, it will radiate efficiently. The antenna may radiate if the length is not at resonance, however, the losses become more prevalent as the radiated frequency is further from the resonant frequency of the antenna. Antenna's can be of many types and shapes. Most antenna's radiate linearly in a two dimensional form. This is a hard concept to understand because the radiated energy takes up three dimensional space. For example; the dipole, which the wave pattern is given here, is a length of conductor which an electrical angular velocity force was applied, driving a charge up and down the length of the conductor. This is a linear system with one degree of motion missing.

The larger circles in the dipole radiation pattern are multiples of the wavelength. As the wave expands into space, the wave follows the same pattern. The particular wave pattern shown in the radio antenna, is of fixed frequency and amplitude modulated. Thus, for a constant propagation velocity, the nodal wave lengths are the same distance apart. Waves can add together. A complex wave of many different frequencies can be present in a given media at one time. Waves can pass through each other. Waves typically are vibrating, however, waves while vibrating can be moving in a given direction too. Waves can be reflected back on themselves.

A medium can be anything. A rope, steel bar, electronic circuit, water, the solar system, or the Earth, will easily act as a medium. Within the specific limitations of the medium, all fundamental frequencies can be generated. There are many types of waves as; acoustic, sound, impact, shock, radio, electromagnetic, water, fluid, strain, stress, elastic, and more with varying degrees of subtlety. A medium can be of any shape, however, the shape affects the wave function. A medium can be either a transmitter, absorber, or converter of waves. We can apply some philosophy to waves; if they can happen, they will happen.

In most industrialized areas, electrical power is delivered to the site using a 50 or 60 cycle per second wave. However, on very long runs, like across the country, large very low frequencies waves begin to appear. Power begins to run across country on its own, totally independent of usage. Systems of transformers and capacitors are installed specifically to dampen the affect of these very low frequency resonance.

Since harmonic oscillators are important to any oscillatory system, a short discussion on the subject is in order. Schrodinger worked on the mathematical development of harmonic oscillators. The first approximation for a potential curve is referred to as a potential well. At the bottom of the well, the potential is approximately equal to one half some force constant times (x-a)^2 where a is some distance away from the wall of the well and at a place where the potential is a minimum. An angular momentum function is given to the equation. The well is normalized about a position with edges of +/- a. With the use of Hermite Polynomials, Schwartz' inequality, Hermitean Operators, Hamilton's Operator, Wentzel-Kramers-Brillouin's approximation, out fell the famous Schrodinger wave equation.

Schrodinger's equation in simplified form:

E is the total energy of the system and V is the potential energy of the system. When one subtracts the potential energy from the total energy of the system what is left over is available as momentum energy. This is some what like dropping a ball. As the ball is falling it is gaining momentum and is loosing potential energy.

For particle wave Schroedinger's wave equation is:

where;

Schrodinger's equation in polar co-ordinates, r , & for a function only of r:

Further complexing the equations for function of r , & can be done by allowing:

Applying the total angular momentum as L^2 = Lx^2 + Ly^2 + Lz^2 will give values to points where the equation will resonate. These are called:

The eigenvalue of H or energy.
The eigenvalue of L^2 or total angular momentum squared.
The eigenvalue of any component of L.

Eigenvalues are typically momentum values and are numbers.

Restricting one component and solving simultaneously for angular momentum yield what is know as eigenfunctions of the Hermitean operator Lz. It has also been shown that such eigenfunctions of the Hermitean operator expand as a series.

To get the series, a little game playing with functions was done. Oc = Cc in this case where C is a constant, C is called an eigenvalue, or characteristic value of the operator O, and c is called the eigenfunction, or characteristic function, belonging to the eigenvalue C. O is an operator. This gets quite complex for various types of waves. It is not my intention to teach Quantum or Wave Mechanics. It is only my intention to show that such operations are mathematically sound. First of all, I am only interested in one form of wave. The form of wave that I am looking for is that which satisfy's the Bode-Titus relationship. This relationship seems to follow the normal geometric series. As eigenfunctions follow a general power series of the operator function given by:

Given an operator of a function of r and a constant An for all values of n allows:

Typically an eigenfunction is in the form of:

This has been the classical form of wave mechanics. Since these equations were originally develop by other people who are considered very sharp, and scrutinized by many, I can only assume that they are correct. From the ripples on a pond, to radio, and light waves, the mechanics of the waves are all the same. The term wave mechanics, is an expression of how a wave is generated, or started, and propagated. The easiest of all the waves to work with are electromagnetic. Einstein showed that mass itself is a form of contained wave. Even square waves have functions which follow the same equations to a good degree as sinusoidal waves. Waves tend to be three dimensional in spherical form, but for the most part, are interpreted in a two dimensional plain for ease of understanding. Without explanation, here are some more common wave equations.

Maxwell's electrodynamic wave equation: