They Say It Cannot Be Done

Monster-Mat · August 7, 2003

Is there anything faster then the speed of light? Can anything travel along laser light? Could some day we could have space vehicles traveling by laser light at unheard of speeds?

Sorry, no signal and no object travels faster than light in a vacuum. If any could, effects could happen before their causes. For example, a super-light-speed spacecraft could arrive before it started. Impossibilities would abound.

Suppose I could lope along at light speed, "c", and notice a light wave traveling beside me, of course, at speed "c", too. That light wave will look like it's standing still, relative to me. Just like the car in the next lane looks stationary when it's going at my speed.

But this is impossible, according to Einstein's Theory of Special Relativity. The speed of light in a vacuum must be the same ("c") for every observer in a uniformly moving reference frame. It can't be zero for me. Therefore, I can't lope along at "c" nor, therefore, exceed "c".

What happens as an object approaches light speed? Suppose I'm the pilot of a spacecraft cruising at near light speeds. I decide to beat my headlight. Can I?

I push in the throttle and my velocity increases, slightly. So far, so good: give it the "gas" and it goes faster. However, as my velocity increases, so does my motion energy (called kinetic energy). Energy is equivalent to mass as Einstein discovered (E = mc²) and therefore also has inertia. As my energy increases, so does its inertia. It gets harder and harder to accelerate the spacecraft to greater speeds.

I shove in the thruster control all the way; the engine yields an enormous energy pulse-its best. But my speed remains much the same, near light speed-and the headlight shines merrily ahead of me.

Inertia increases without limit as the ship's velocity approaches "c". Thus, it takes an infinite force and an infinite amount of energy to accelerate the spacecraft to "c" and that is impossible. Once again, since I can't attain "c", I sure can't pass the headlight.

I'm curious, though, as I watch the headlight skip ahead: What happens to the enormous thrust of energy? It no longer appreciably increases my speed. It can't. My inertia's too great. The energy squeezes into my mass! An outside observer sees me and my craft swell as I shove in the thruster. Soon we appear twice as large-good grief! As I hit 99.9997% of "c", my craft and I loom 410 times bigger than when we started. "E" becomes "m" as Einstein predicted.

This really happens in the CERN or Fermilab particle accelerators. Protons accelerated to 99.9997% of "c", appear enlarged by a factor of 410 times.

But, all is not lost. Our space vehicles may, someday, travel at unheard of speeds. Theoretically, they can travel as close to "c" as they please (and we can engineer the required propulsion energy). Remember, too, that spacemen traveling at near light speeds age slowly.

Fidgits · August 7, 2003

Of course... if you want to get as close as you can to the speed of light.. you'll have to use Solar winds... which travel at the speed of light away from a star (i.e. sun)... figure a way to harness them and voila... not even the IS430 could keep up...

Monster-Mat · August 7, 2003

that is indeed true

Monster-Mat · August 7, 2003

Scientists at the University of Arizona in Tucson hope to harness sunshine to point and stabilize future space telescopes.

Sunshine exerts a weak force on spacecraft. This has given space scientists headaches for years, gently turning spacecraft off target or off-orbit. But there has long been the idea of harnessing solar pressure with huge, gossamer solar sails to push spacecraft like high-tech clipper ships.

"While most people think of solar sails for pushing with the pressure of light to accelerate and move spacecraft, our thought is to use that force to point the telescope and keep it in position," says Roger P. Angel, founder and director of the UA Mirror Lab.

Currently, space telescopes are rotated, pointed and steadied by motors, gyroscopes and thrusters. As space telescopes become lighter and lighter, the vibrations and oscillations created by these devices can blur images. They also require using finite fuel supplies, whereas sunshine is an inexhaustible source of steering force as well as energy.

Angel and UA researchers Blain Olbert and Paul Calvert want to take advantage of the solar shield that already is needed for space telescopes. This shield sits between the telescope and the sun to keep the cryogenically cooled telescope from heating up.

The UA scientists visualize a shield shaped like a pyramid, with its sloping surfaces covered with hinged, reflecting tiles. Normally the tiles lie flat against the surfaces. When the tiles on one side of the pyramid are raised, however, the pressure balance is upset and the sunshade is pushed to one side. By raising tiles like ailerons on the right pyramid faces with electrical energy collected from solar cells, the solar pressure could be used to hold the spacecraft stable or to change its orientation or angular momentum.

Key to all of this is developing suitable lightweight tiles that bend when a voltage is applied to them.

UA engineers are exploring the feasibility of building these tiles under a $100,000 grant from NASA. Professor Paul Calvert and graduate student Blain Olbert, both of the Materials Science and Engineering Department, are working on the project. Olbert also is a staff engineer at UA¹s Steward Observatory.

They are constructing tiles from piezoelectric polymer films that bend when a voltage is applied to them.

"We are looking for polymer films that already are produced commercially and want to see if any of them are suitable for this application," Olbert says.

The tiles will lie on the solar shield surface like shingles on a roof. One edge of each tile is glued to the shield. This glued edge acts as a hinge on which the tile rotates. Each glue joint and tile must survive hundreds of thousands of cycles during the spacecraft¹s 10-year life span without detaching or delaminating.

"We are evaluating commercially available films and adhesives and the technologies for sticking them together," Olbert says. "Then there¹s the whole problem of lead technology. How do we attach the electrical leads to each tile that are needed to energize it? And we also have to think about redundancy because there are micro-meteorites out there that are constantly punching tiny holes through the tiles."

In order to evaluate the films and adhesives, Olbert and others are building test tiles in a Steward Observatory laboratory. This is a difficult, tedious and time-consuming task because the films are like thin plastic wrap. Static charges make them stick to everything, and once they¹re creased, they¹re ruined.

The adhesives also have to be carefully squeezed out from between the films as they¹re sandwiched together, which means low-viscosity adhesives are a must.

Currently Olbert¹s work is funded under Phase I of NASA¹s Gossamer Spacecraft Initiative. NASA envisions gossamer spacecraft as large, ultra-light vehicles that can reconfigure themselves or evolve in response to changing mission conditions.

Phase I is basically for evaluation of ideas. If NASA thinks the UA work shows promise, the project could be funded for Phase II, which calls for manufacturing and testing complete solar shield roof panels.

"We are in the weeding-out phase right now," Olbert says. "There are lots of ideas that need to be turned into working solutions."

UKPulse · August 7, 2003

Ahhh, but the phenomenon of quantum tunnelling demonstrates that particles can cross a quantum barrier at speeds in excess of the speed of light....

Brettster · August 7, 2003

lets ask Prot

he knows the amazing power held within a beam of light

vince-com · August 7, 2003

Mat get an umbrella, the german sun is definitly warming you too much today :)

Fidgits · August 7, 2003

Ahhh, but the phenomenon of quantum tunnelling demonstrates that particles can cross a quantum barrier at speeds in excess of the speed of light....

Yes.. but the theory also states that this can only be achieved on a sub-atomic level... :P

UKPulse · August 8, 2003

Yes.. but the theory also states that this can only be achieved on a sub-atomic level... :P

Ahh yes, but it proves the exception when they declared 'nothing' could exceed the speed of light. Clever these lil' sub-atomic wassits - like certain sub-atomic particles being able to pass straight thru the earth! :)

For my part, I like the old time dilation effect - I could do with some of that! Hang on a mo.... get younger (relatively), have less money, pay higher insurance premiums.... doh! :duh:

Loz · August 8, 2003

I am very impressed by our collective knowledge.

I read Mat's first peice and was itching to reply mentioning partical accelerators. then he beats me too it. the rest was all learning for me., THanks guys. I wish I was able to access so much information and channel it into a rewarding endevour.

Loz

Fargo · August 8, 2003

Ahhh, but the phenomenon of quantum tunnelling demonstrates that particles can cross a quantum barrier at speeds in excess of the speed of light....

Ahh yes, Quantum Theary also says that time travel is possible as the partical exists in two states at the same time..and Teleportation can be achieved

Matthew_McNally · August 8, 2003

Ahhh, but the phenomenon of quantum tunnelling demonstrates that particles can cross a quantum barrier at speeds in excess of the speed of light....

Ahh yes, Quantum Theary also says that time travel is possible as the partical exists in two states at the same time..and Teleportation can be achieved

but I've always believed that Heisenberg's Uncertainity Principle rules out teleportation?

we can only know either a particle's postion or momentum with absolute precision - therefore ruling out teleportation?

Monster-Mat · August 8, 2003

What is the Quantum Theory though, lets take a look shall we

Quantum theory was not created "out of the blue". It's mathematical framework and ideas grow out of a long history of classical mechanics.

A number of certain experiments around the turn of the century created the need to replace the classical theory of matter with a new quantum theory.

The first attempt's (Bohr) had remarkable yet limited success. This became known as the Old Quantum Theory.

Schrödinger and Heisenberg developed the modern theory, which was extended by Dirac to include relativistic effects.

An important aspect of understanding of one's position is to appreciate where one came from. This is also true of modern science. Though admittedly more complex, it is possible to follow a thread through the last four centuries up to the birth of modern quantum mechanics. Rene Descartes is often credited as the Father of Modern Mathematics. While a mercenary soldier, he experienced a dream one night that triggered in his mind the idea of modern algebra. (Some nightmare!) It is helpful to follow a short tour from Descartes through to Newton, Bernoulli, Euler, Hamilton, Maxwell, Einstein, Schrodinger, Heisenberg, and Dirac. It is revealing to see how the mathematical ideas of one generation opened new avenues for the next.

As mathematical physics reached a plateau one century ago, new vistas were being suggested as new experiments produced unexpected results. Spectroscopy and heat capacities, black body radiation and the photoelectric effect, all presented simple results that stood in stark contrast to accepted theories of nature.

With Planck's bold suggestion of the quantum nature of light and Einstein's equally impressive explanation of the photoelectric effect in quantum terms, Bohr, who was working with Rutherford on the nuclear scattering experiments, took another step and developed a mathematical framework to explain the spectroscopy of the hydrogen atom. He invoked the ideas of standing waves to justify the quantal stability of certain levels in the atom and this simple process provided such startlingly good results that the world was hooked on the quantum. This came to be known as the "Old Quantum Theory" since its comparatively excellent results were nevertheless limited in their scope. For another decade, extensions to this quantum theory proved unsatisfying

It was Prince Louis de Broglie who challenged the scientific world with his Ph.D. thesis wherein he suggested that if light could behave as particles, then matter could also behave as waves. He wrote down his famous de Broglie relation which was taken up by many people - often to try to prove that it was wrong. One of these was Erwin Schrödinger who soon recognized the remarkable nature of the idea and wrote down his now famous wave equation for matter. Werner Heisenberg developed a completely different pathway but ultimately identical in results following his matrix mechanics methods. When Dirac wrote down the relativistic wave equation, modern quantum theory had arrived.

so now we fully under stand where the Theory came from chaps, lets look at the Mathematical Basics for the Theory

Probability

The occurence of quantum events is interpreted entirely in terms of probability. We usually understand probability reasonably well where discrete events are involved. For instance,when a single die is thrown, we readily accept that there is a one-in-six chance of getting one of the possible numbers. There are six possible outcomes, each of which is equally likely (assuming a perfect cube for a die), and each one has a 1/6 probability of occuring.

Another way of defining probability is to throw the die many times, keeping track of the total number of throws (T), and the throws which produced a given number (say four) (F). As the total number of throws increased, the ratio F/T would approach 1/6.

What about multiple events? Take three dice. Throw the first one. The probability of getting a 1 is 1/6. Throw the second one. The probability of getting a 1 is also 1/6, and so on with the third. However, throw all three together. The probability of getting a 1 AND ALSO getting another 1 AND ALSO getting a third 1 at the same time is 1/6 x 1/6 x 1/6 = 1/216. This is fundamental about probability, where the outcome of multiple events is characterized by the product of the probability of each individual event.

Sometimes you have to count carefully. With these three die, you can ask the chance of rolling a 7. Think carefully and you will find 15 distinct ways to get the number 7 (three 5-1-1, six 4-2-1, three 3-3-1, and three 3-2-2). The chance of getting a 7 is 15/216, or 5/72 or 0.06944 or 6.944%. By contrast the chance of getting a 3 (only 1 way, 1-1-1) is only 1/216 or 0.0046296 or 0.463%. By careful counting of the desired outcomes and the total attempts, we can easily determine probability in these discrete systems.

With these three dice, the numbers from 3 to 18 are possible to obtain. We can graph the probability against the number and obtain the following. Note how in a single throw of three dice, the probability of getting 10 is 0.125, or 12.5%, whereas the probability of getting the number 4 is 0.01389 or 1.389%. The probability of getting a 10 OR an 11, is 12.5 + 12.5 =25%. The probability of getting a 9 OR 10 OR 11 OR 12 is about 48.1%. To find the probability of a selection of altrnate outcomes, we add the probability of each event. If we throw the set of three dice twice the chance of getting a 10 the first time AND THEN getting an 11 the second time is 0.125 * 0.125 = 0.015625 or 1.56%. The probability of successive events occuring is the product of each individual event.

In Quantum Theory, we must deal with probabilities of a variable which varies continuously, for instance, the x-coordinate. In such a case it does not make any sense to ask, what is the probability of x = 0.250000... occuring, since there are an infinite number of possible points on the line and the probability of getting EXACTLY 0.25000... is infinitesimally small. Instead, we speak about the probability of x lying within a small interval on the axis, say between 0.25 and 0.30. When the interval is decreased to an infinitesimal length, the interval between x and x + dx, the probability as a function of position x is described by a function D(x) which is called the probability density. Then the probability that the variable will have a value between x and x + dx, is D(x)dx.

Imagine the probability density function to have some complicated shape, To answer the question of the probability of finding the value between A (say 1.5) and B (say 2.5), we add the probability at each point between these two endpoints, exactly as we did in the discrete case above. Except for a continuous variable, the summation is replaced with an integral - we are looking for the area under the curve between these two points. The region under the curve represents the probability of finding the variable between the values 1.5 and 2.5. We have the relation

Probability is a real, non-negative number, so the function D(x) must be real and positive everywhere. In quantum theory, the wavefunction can be complex and negative, so that it cannot be a probability. However, the SQUARE of the wavefunction is real-positive and the square of the wavefunction is the probability density. You may note that for the curves drawn above, I have scaled the probability so that if you add all of them up (or integrate over the entire space), you get the number 1. This can be interpreted as being the chance of getting ANY result out of all possible results is 1 - namely something must happen. With such scaled values, the probability density is said to be NORMALIZED. A wavefunction, when squared and integrated over all space gives the result 1, is also said to be normalized.

Complex Numbers

Since wavefunctions can be complex, a good understanding of complex numbers is valuable. In the real number system, the square root operation is undefined for negative numbers. The complex number system, however, validates that operation and defines the sqaure root of -1 to be a new number called "i". Some simple manipulations with i produce:

An arbitrary complex number z can be written as z= x + iy, where x and y are two real numbers. In this case, x is called the real part of z and y is the imaginary part of z.

One way to represent a complex number is as a point in the complex plane, where the horizontal axis is known as the real axis and the vertical axis is the imaginary axis. By comparison with circles in the real number plane, we can define the same point in terms of a radius and an angle. The radius is the distance from the origin and is called the modulus or absolute value of z, denoted by |z|. The angle that this radius direction makes with the positive x-axis is called the phase or argument of z.

Since, z = x + iy and since it can be shown that (by comparing the Taylor's series expansion's of cos x, sin x, and exp x) we can write

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part of the complex number.

Some further relations involving complex numbers are

On this basis, it is important to note that the square of a complex wavefunction is actually the product of that wavefunction with its complex conjugate.

One final point about complex numbers is useful, for with them we can define roots of the number 1. In the real number system, the only root of 1 is 1 itself (1 x 1 x 1 x 1 = 1). In the complex number system, the result is slightly more rich. We start by recognizing that 0 raised to any base (including e) gives the result 1. For the complex number with phase of 0 and modulus of 1, we have the number 1. But as the phase rotates the number through the plane, it comes back on itself after 2p radians. Hence, all multiples of 2p phases also move back through 1 (k can be any integer, positive or negative).

Now consider a number w, which is defined by

where n is a positive integer. If we take the product of w with itself n times, wn, it is readily seen that we obtain 1. In other words, w is the n-th root of 1. By choosing successive values of k = 0, 1, 2, ... , n-1, we obtain n different n-th roots of 1. All other values for k give numbers which vary by multiples of 2p and hence are not different numbers. For instance, the three cube roots of 1 are

Differential Equations

The Schröodinger Equation is a differential equation. We look here at ordinary differential equations, which are equations in only 1 variable. Equations with more than one variable or called partial differential equations. The time-independent Schröodinger equation is ordinary while the time-dependent equation is partial. An ordinary differential equation is a relation involving an independent variable (such as x) and the dependent variable y(x) and its derivatives. The order of the equation is the order of the highest derivative in the equation.

The above equation is of fourth order. A special kind of differential equation is a linear differential equation. All derivatives of y and y itself appear to the first power and all coefficients of each derivative are functions of x only (not of y). For instance

If a differential equation cannot be put into this form, it is a nonlinear equation. If the functional form on the right hand side is 0, the equation is said to be homogeneous. If it is not zero (whether it is a constant or some function of x), then the equation is said to be inhomogeneous. The one-dimensional time-independent Schröodinger equation is a Linear Homogeneous Differential Equation of Second Order.

Solving differential equations often involves as much art as skill. For a few simple equations, there are techniques to solve them directly. But in most cases, differential equations are solved by inspection and trial-and-error. In general, any linear homogeneous differential equation of second order can be put into the following form by dividing through by the coefficient of the second order derivative.

In general, a differential equation of nth order has n arbitrary constants in the solution. In this case, we will find two independent functions y1 and y2 which define a general solution to this second order equation.

where c1 and c2 are these arbitrary constants. By independent functions, we mean two functions which are not simply a multiple of each other (2x and 3x are not independent functions, but 2x and 2x2 are). Any value for these will satisfy the differential equation. However, every problem must have additional information or constraints in order to specify the value of these two constants. The additional information may involve the position and velocity of the particle at t=0, the beginning, or perhaps it may involve a vibrating string which is held at both ends. Since it is held at both ends, we know that the amplitude at those two locations must be 0. These additional constraints are called boundary conditions. The general solution, along with two additional boundary conditions (for the equation has two arbitrary constants since it is an equation of second order) constitutes the solution to a given problem for the Schröodinger equation.

Operator Algebra

Though a differential equation solution carries the essence of a given problem in quantum theory, we can develop a more general approach through the use of Operator Algebra, which will provide a more rigorous framework and more widely applicable results.

The 1-dimensional Schrödinger Equation can be rewritten as follows

The term "H" in this equation is called an operator. The operator concept is straightforward. It is simply a rule which transforms one function into another. You have used this concept all your mathematical life. The operator "5" is simply the operation of multiplication by the number 5. The operator "cos" takes the cosine of the argument. The operator "D" takes the derivative of the function.

As you can see, there is nothing too cryptic at this point. (Look at some examples.) But you will soon realize that this framework provides us with some powerful tools for analyzing problems. Here are some rules for working with operators. These rules define an algebra for operators.

Operators obey the associative law of multiplication, just like for numbers.

However, they do not NECESSARILY obey the commutative law of multiplication, unlike numbers. This is an extremely important point. Some operators commute with each other,while others do not. We will find that this has very important consequences for quantum mechanical operators. To investigate this property we define a new entity called the commutator of two operators.

If AB = BA, then [A,B]=0 and we say that the two operators commute with each other. (Look here for some examples.)

There is a special class of operators called linear operators. For an operator to be linear, it must obey these two properties:

The important point here is that operators which occur in Quantum Mechanics are linear.

Eigenvalue Equations

As described in the text about Operator Algebra, by definition, an operator "operates" upon a function and produces another function. For every operator, there is a collection of functions which, when operated on by the operator produces THE SAME function, modified only by a constant factor. Such a function is called an eigenfunction of that operator and the multiplicative constant factor is called the eigenvalue of that eigenfunction. In equation form, this looks like

This is an eigenvalue equation where f(x) is an eigenfunction of A and a is the eigenvalue of f(x). If it is given that we know the operator, what are the eigenfunctions and their associated eigenvalues? This is the general problem that is to be solved with eigenvalue equations. Here are some examples.

You can immediately see that there is a whole collection of eigenfunctions of the differential operator, e4x, e5x, e6x, etc. and each one has a different eigenvalue. By contrast,

Quantum mechanical problems can be cast as eigenvalue equations. As we find out, any observable property of our system (energy, momentum, angular momentum, dipole moment, etc.) has its own operator. The eigenfunctions are the wavefunctions of the system and the eigenvalues are the values of the property - the number that we measure. The Hamiltonian is the energy operator for the system and the Schrodinger equation is an eigenvalue equation

Vectors

In studying mechanical systems in general and ones with angular momentum in particular, it is valuable to have a good handle on vectors. Any physical property which requires the specification of both magnitude and direction requires a vector (for instance, velocity, momentum, or angular momentum). This is in contrast to scalars which only need the specification of their magnitude (for instance, temperature or pressure).

To work with vectors, a vector space is set up. A very useful such space is often formed as a Cartesian coordinate system in 3 dimensional space. This is useful since it correlates to the space in which we live. However, higher dimensional spaces are also useful - even infinite dimensional spaces - in quantum theory. It is important to appreciate that the usage of the term "space" refers to the domain in which a family of vectors operates. Our usual understanding of space ("The Final Frontier" variety) is only 1 example of an infinitude of spaces (the vector space variety). With spaces of a different dimension (NOT the science fiction "Dimension X" variety), it only describes the number of basis vectors needed to describe the space. Our universe needs three vectors to describe a position in space whereas our universe needs an infinitude of vectors (though we usually get away with 60 or 70) to describe a molecule. But the algebra we need is the same for a 3 dimensional space or for a 70 dimensional space, so we usually teach in 3 dimensions for that is easiest on our minds.

Here is such a space. The unit vectors i, j, and k are chosen so that through the addition of multiples of themselves with each other, the three of them can describe all vectors possible in the space. An arbitrary vector V, is described by specifying the amounts of i, j, and k which, when summed together, make V.The components of V, Vx, Vy, and Vz, are these multiples. To specify V, it is sufficient to specify its three components (Vx, Vy, Vz). Hence, a three dimensional vector is an ordered set of three numbers. A seventy dimensional vector is an ordered set of 70 numbers.

Two vectors, A and B, are equal if and only if each of their components are equal: Ax = Bx; Ay = By; Az = Bz. It is interesting to observe that 1 vector equation (A = B) is equivalent to three scalar equations. This brevity is a nice aspect to vector algebra.

When we add vectors, we add each of their components separately. By this it is clear that in order to add two vectors, they must have the same dimension - otherwise the operation is undefined. When we visualize this in space, we imagine moving the start point of one vector to the endpoint of the other vector. The sum vector is the resultant.

In this two dimensional case, we have mathematically A + B = (4,7) + (4,1) = (8,8). We should also remember that multiplication of a vector by scalar is multiplication of each component by the same scalar, namely cA = (cAx, cAy, cAz)

The length of a vector is called its magnitude and is usually denoted by |A|. The directionality of the vector is lost for this quantity so it is a scalar.

Matrices

A matrix is an array of numbers. They can be combined together according to various sets of rules - those of addition and multiplication. The numbers in the array are called elements and specified by indicating its row and column number,

Addition

Two matrices can be added only if both have the same number of columns and the same number of rows. if this criterion is met, then

Multiplication

Two matrices can be multiplied together only if the first matrix has the same number of columns as the second matrix has rows. Hence, square matrices of the same size can always be multiplied together, but rectangular ones need to be carefully matched. If two matrices are not of the correct shape, the operation of multiplication is not defined for them.

It is important to realize that the operation of multiplication is not commutative for matrices, as it is for numbers. This means that the product AB does not equal the product BA in general. It may happen in some instances, but it is not guaranteed, except for special instances - these will be important special cases in quantum theory.

Determinants

The determinant of a matrix is a number, but is represented and calculated as indicated above. These are important in solving many matrix problems.

Diagonal Matrix

A diagonal matrix is one whose elements are all zero except for those on the major diagonal, that is where the row number equals the column number, r=c. For example

The Unit Matrix is a special case of a diagonal matrix, where all the diagonal elements are 1.

Transpose of a Matrix

The transpose of a matrix is formed by exchanging the rows and columns: row 1 becomes column 1,

Inverse of a Matrix

The inverse of a matrix is defined so that M-1M = 1. It is a little tricky to construct mathematically. NOT ALL MATRICES HAVE AN INVERSE. To form the inverse of a matrix:

Form the determinant of M, as |M|.

Form the transpose of M, as Mt.

Form another matrix Mt', from the transposed matrix, where Mt'rc is formed from the signed cofactor for element rc, that is the determinant formed from the transposed matrix where row r and column c has been removed.

Finally, form the inverse M-1 as Mt'/|M|.

Complex Conjugate of a Matrix

This is formed by replacing every element of the matrix M with its complex conjugate M*.

Adjoint of a Matrix

The adjoint of a matrix is the complex conjugate of its transpose, and is written as M+.

A Unitary Matrix

A matrix is unitary if its inverse is equal to its adjoint, M-1 = M+.

A Hermitian Matrix

An Hermitian matrix is also known as a self-adjoint matrix. This means, of course, that M = M+.

Matrix Solutions to Equations

Matrices can be used to succinctly describe and to solve many mathematical problems, and are of particular value in dealing with sets of simultaneous equations. Solving simultaneous equations is an important problem in general, and the eigenvalue equation is usefully solved in this manner in particular. There are two steps to such a solution: first one finds the eigenvalues and then, with those eigenvalues, we can find the eigenvectors.

and if you wish me to continue, we then go into such Mathamatics as:

Coordinate Systems

Cartesian

Spherical Polar

Special Functions

Trigonometric

Sine: Sin

Cosine: Cos

Tangent: Tan

Cotangent: Cot

Secant: Sec

Cosecant: Csc

Hyperbolic

Hyperbolic Sine: Sinh

Hyperbolic Cosine: Cosh

Hyperbolic Tangent: Tanh

Hyperbolic Cotangent: Coth

Hyperbolic Secant:Sech

Hyperbolic Cosecant: Csch

Exponential

Logarithmic

From Differential Equations

Bessel Functions

Hermite Polynomials

Legendre Polynomials

Laguerre Polynomials

thanks

Zee007 · August 8, 2003

What the f................

Bazza-old · August 8, 2003

Determinants
The determinant of a matrix is a number, but is represented and calculated as indicated above. These are important in solving many matrix problems.

well thanks mat, im sure this will help me solve some of my matrix problems

Fargo · August 8, 2003

Determinants
The determinant of a matrix is a number, but is represented and calculated as indicated above. These are important in solving many matrix problems.

well thanks mat, im sure this will help me solve some of my matrix problems

PMSL

UKPulse · August 8, 2003

Way to go - my man! :)

How about some Quantum electrodynamics or some Gravitational theory? Personally I like fractals myself - the concept of the mandlebrot set and the fact that a finite area can be enclosed by a boundary of infinite length, now that's fun!

Oooops... shall I get my coat?

Monster-Mat · August 8, 2003

QUANTUM ELECTRODYNAMICS

Also known as QED, quantum field theory that describes the properties of electromagnetic radiation and its interaction with electrically charged matter in the framework of quantum theory . QED deals with processes involving the creation of elementary particles from electromagnetic energy, and with the reverse processes in which a particle and its antiparticle annihilate each other and produce energy.

The fundamental equations of QED apply to the emission and absorption of light by atoms and the basic interactions of light with electrons and other elementary particles. Charged particles interact by emitting and absorbing photons , the particles of light that transmit electromagnetic forces. For this reason, QED is also known as the quantum theory of light.

QED is based on the elements of quantum mechanics laid down by such physicists as P. A. M. Dirac , W. Heisenberg , and W. Pauli during the 1920s, when photons were first postulated. In 1928 Dirac discovered an equation describing the motion of electrons that incorporated both the requirements of quantum theory and the theory of special relativity . During the 1930s, however, it became clear that QED as it was then postulated gave the wrong answers for some relatively elementary problems. For example, although QED correctly described the magnetic properties of the electron and its antiparticle, the positron, it proved difficult to calculate specific physical quantities such as the mass and charge of the particles. It was not until the late 1940s, when experiments conducted during World War II that had used microwave techniques stimulated further work, that these difficulties were resolved. Proceeding independently, Freeman J. Dyson, Richard P. Feynman and Julian S. Schwinger in the United States and Shinichiro Tomonaga in Japan refined and fully developed QED. They showed that two charged particles can interact in a series of processes of increasing complexity, and that each of these processes can be represented graphically through a diagramming technique developed by Feynman.

Not only do these diagrams provide an intuitive picture of the process but they show how to precisely calculate the variables involved. The mathematical structures of QED later were adapted to the study of the strong interactions between quarks, which is called quantum chromodynamics .

Monster-Mat · August 8, 2003

Subatomic and Gravitational Theory

The UMass subatomic and gravitational theory group works on fundamental physics at scales ranging from the quantum gravitational Planck scale -- 10-35 m, to subatomic scales - 10-15 m, to the very large -- 1025 m, the size of the universe! At the subatomic level the ongoing projects involve efforts in both nuclear and particle physics regimes, often in support of the ongoing UMass experimental efforts. The work generally involves the use of nonperturbative techniques in order to study what is thought to be the correct picture of strong interactions -- quantum chromodynamics or also known as QCD -- using hadronic and other probes. The nuclear physics work has featured chiral perturbative methods in order to study electromagnetic interactions of the nucleon, while the particle physics component has also utilized chiral techniques, but in this case as a probe of weak interaction amplitudes. Additional studies involve heavy quark physics , an area that will be at the forefront of particle physics over the next decade. The gravitational work is concentrated on the classical and quantum physics of black holes and higher dimensional black branes, and on gravitational aspects of string theory.

Mandelbrot fractals

A fractal is a mathematical object with a rough or fragmented geometric shape. They are said to be "self similar and independent of scale" according to Ermel Stepp. They can be subsdivided into parts, and each smaller part is a image of the original whole. Thus, zooming in on a fractal would lead you to see the same image repeated over and over. This is why is is said to be "self similar" and "independent of scale". Most fractals are generated from a mathematical equation where the results are iterated, that is the results from the equation are fed back into the equation, and this process is continued until the number grows larger and reaches a certain boundary.

A "rigorous" mathematical definition of fractals was stated by Benoit Mandlebrot (famous for his Mandlebrot set, rediscoving fractals, and naming these mathematical objects) as "a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension". Stepp cites that this definition is not totally satisfactory for it does exclude some sets that are considered fractals. There are many types of fractals--such as Sierpinski's triangle, the Kock snowflake, the Peano curve, the Mandlebrot Set, and the Lorenz attractor. Fractals have also been shown to describe real world objects that don't follow normal Euclidean geometry. Such examples are mountains, coastlines, and clouds.

Fractals are usually colorful, with colors assigned to values produced by the equation of the fractal. An algorithm is used to assign these colors. Most fractals generated by computer for it is almost impossible to create them by hand. In fact, it was the computer that LED Benoit Mandlebrot to rediscover them. The first to discover fractals was Gaston Julia but the inexistence of computer hindered the study of them. When the computer was available as a resource for Mandlebrot, he began the first in depth study of them. The set he discovered and is famous for is the Mandlebrot set. The black area of many fractals is considered to be the Mandlebrot set.

below is an example of Mandelbrot set

insider · August 8, 2003

matt u need to get out a bit more lol :duh:

Monster-Mat · August 8, 2003

i know........

but its hot outside...37 degrees

and that means i will have to go to work.but ......but....i dont want to

AltEzZaBoB · August 8, 2003

so what about quantum chromodynamics?

Monster-Mat · August 9, 2003

you really wanna know BoB

Monster-Mat · September 15, 2009

quantum chromodynamics (QCD), quantum field theory that describes the properties of the strong interactions between quarks and between protons and neutrons in the framework of quantum theory.

Quarks possess a distinctive property called color that governs their binding together to form other elementary particles.

Analogous to electric charge in charged particles, color is of three varieties, arbitrarily designated as red, blue, and yellow, and—analogous to positive and negative charges—three anti color varieties.

Just as positively and negatively charged particles form electrically neutral atoms, colored quarks form particles with no net color.

Quarks interact by emitting and absorbing massless particles called gluon's , each of which carries a color-anti-color pair. Eight kinds of gluon's are required to transmit the strong force between quarks, e.g., a blue quark might interact with a yellow quark by exchanging a blue-anti yellow gluon.

The concept of color was proposed by American physicist Oscar Greenberg and independently by Japanese physicist Yoichiro Nambu in 1964. The theory was confirmed in 1979 when quarks were shown to emit gluon's during studies of high-energy particle collisions at the German national laboratory in Hamburg. QCD is nearly identical in mathematical structure to quantum electrodynamics (QED) and to the unified theory of weak and electromagnetic interactions advanced by American physicist Steven Weinberg and Pakistani physicist Abdus Salam .