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Great. Seems weird, and frustrating frankly. You suggested money was the only difference between prostitution and rape. Zapatos rightly said this was a dangerous position to espouse since a rapist could simply throw money at their victim and claim he was a customer. You then asked him which cost more. It was a weird question which didn’t make sense, but Zap politely shared he didn’t have experience in this space and asked you which YOU thought cost more. He was trying to progress the conversation forward. In response, you claimed money plays no part. Again, seemed contrary to your previous posts so he next asked why you were asking him which costs more, and you said “I’m not asking.” Seriously, stop playing games or don’t post if that’s not possible for you.

I'm with an advanced civilisation from another planetary system. Perhaps the "human resemblance" evolutionary process, is simply the most favoured to reach advanced intelligence and associated abilities. What I mean is that while an Octupus is obviously "intelligent" it is still confined somewhat by its evolutionary path. I pick a civilisation from another planet simly because of the "near infinite" extent and content of our universe, and the stuff of life being everywhere we look.

Due to the extrem extent of space, I voted another planetary system. In my opinion life is quite common in the universe, considering that every sun has a habitable zone, during their entire lifespan. I think there might be billions of intelligent life’s out there with different levels of evolution. type 0 = 99.999999999…9% type 1 = 00.0000000000…4% type 2 = 00.0000000000…3% type 3 = 00.0000000000…2% type 4 = 00.0000000000…1% I think distance (Space) with time can be conquered, by the help of Science, so an aged, matured, advanced intelligent life form could be able to pay us a visit.

Thank you, excellent answer. +1 Now we are in business. It is not necessary to be super good at differentiation and integration, just to be able to recognise what they are and what the dx and ydx means in ∫ydx One step on from this is to understand the basics of differential eqautions. Consider [math]\frac{{dy}}{{dx}} = some\,function\,of\,x = f\left( x \right)[/math] Which simply says that the derivative is some (given) function of x called f(x). This is all differential equations really are, and the solution or integration of the equation is to find the function of x that yields the given one. That is to find a function of x that when differentiated gives f(x). So rearranging dy=f(x)dx Taking the integral of both sides ∫dy=∫f(x)dx y=F(x)+C Note that variable y includes some arbitrary constant C. F(x) on its own its known as the Primitive of f(x) or the antiderivative. This is the format which you will find in swansont's excellent link to hyperphysics that he gave in his first response. This free site provides an exceptionally clear presentation of Physics and its connections to other sciences and is very well respected. http://hyperphysics.phy-astr.gsu.edu/hbase/index.html Swansont's link has several itegrals in it, which we can now explain. They are of the type [math]\int {ydx} [/math] Where y and x are different variables. So we need to change one of them into the other in order to evaluate the integral. That is express y in terms of x or dx in terms of y and dy. This format ydx is used when y is a function of x such that y =f(x) that provides a value for y at every point of interest of x. In the case of classical (wave) mechanics our variable y is sometimes called y but often the greek letter phi [math]\Phi [/math] is used and is the classical wave function. In the case of quantum mechanics this variable is psi [math]\Psi [/math] and is called the wave function. Now I have said that the wave function, or its square, is not a probability so the obvious question arises What is it ? More in a moment. But whilst we have both classical and quantum side by side, let us quickly look at your question about hyperfine splitting swansont mentioned. Splitting occurs when there is an additional disturbing or influencing force or field acting on the particle. That is additional to the base force or field that creates the (quantum) energy levels for the particle. For us the particle is the electron and the base field is the (electrostatic) field of the nucleus in which the electron moves. When an additional influence occurs (which may be self generated as with the angular momentum) or may be an external field such as the one in magnetic resonance, The original levels each split into two, one of higher and one of lower energy than the original. These can be further split to form the hyperfine splitting, but that is not really of interest here in my opinion. Splitting more generally is important in other areas such as chemical bonding where the original level splits into a higher level called an anti bonding level (orbital) and a lower level called a bonding orbital. Here the disturbing influence is the field of the second atom involved in the bonding. Splitting also ocuurs in classical situations where we modulate a classical wave with a second wave to obtain new waves of higher and lower frequencies (and therefore energies). OK back to your question and the second part. What is the wave function and what is its connection to probability ? Well a good way to study this is to do a dimensional analysis. A probability is a pure number and has no dimensions. The dimensions of the wavefunction are rather odd because they are not 'fixed', but depend upon the dimensions of the space you are working in. Again swansont has noted this very briefly. In one dimension (x) the wavefunction has dimensions [math]\sqrt {\frac{1}{{length}}} \;ie\;{L^{ - \frac{1}{2}}}[/math] In two dimensions (x,y) the wavefunction has dimensions [math]\sqrt {\frac{1}{{area}}} \;ie\;{L^{ - \frac{2}{2}}}[/math] In three dimensions (x,y,z) the wavefunction has dimensions [math]\sqrt {\frac{1}{{volume}}} \;ie\;{L^{ - \frac{3}{2}}}[/math] So neither the wavefunction nor its square are pure numbers. So they cannot by themselves be probabilities. OK so what do we do about this ? Well the method of dimensions says that if we multiply by two quantities together the dimension of the product is the product of the dimensions. That is dimension (A . B) = dim(A) . dim(B) So if we square the wavefunction to get rid of the square root and multiply by the a length or an area or a volume as appropriate we will obtain a pure number. and the differentials dx, dA and dV have the appropriate dimensions. That is how and why we form one of the integrals [math]\int {{\Psi ^2}} dx[/math] [math]\int {{\Psi ^2}} dA[/math] [math]\int {{\Psi ^2}} dV[/math] Now we have a pure number all that is left to do is to scale it into the appropriate range for a probability that is a number between 0 and 1. This process is called normalisation. This is why the probability must always be associated with a region of linear, aerial or volumetric space. Sorry it was so long and rambling but I hope it helps. F