There is a beautiful discussion of this subject in the book entitled 'Principles of Magnetic Resonance' by C. P. Slichter in pp.18-22/32. This choice is closely related to the selection of  rotating reference system in the analysis of kinematics of magnetic dipole and applied static magnetic field driven by the linear alternating  field, which may be decomposed two counter acting rotating fields.

Dispersion and absorption signals are always related, and the mathematical connection is known as the'' Kramers-Kronig Relationship' in the literature. (SEE: Eq. 24 ibid:)  Similar equations are hold for the dielectric constants or the electrical susceptibility.

Very simplest reason is that frequencies, such as those in NMR signals, can be explained equally well as either sine or cosine functions.  The question is whether the signal at a point zero has a discrete value of zero or one.  When two discrete frequencies are present, one could propose likewise, each could have a value of  zero or each could have a value of one, except that is not the only solution:  one could have a value of zero, the other could have a value of one.  Complex numbers are in essence the accounting system to keep track of whether two frequencies are both cosine functions, both sine functions are something in between.

Thanks for a very intuitive answer Dr. Walter. In NMR we are detecting the signal from both x and y plane from time zero. So in reality we should be able to write the signal with a trigonometric function. According to your answer the uncertainity of the phase is taken care by the complex number? How does complex numbers take care of this? What has "i" or the imaginary part do with this? 

The format in NMR is always frequency  v = cos xt +/- i sin xt.   where t is in unit of  time and x is in units  1/t .  Intensity is v^2 function,  so   is also a function of cos ^2 xt  +/-  sin^2 xt.

All these arguments are have validity as long as the system is linear that means the connection between the action (external driver) and the response of the system is linear. 

Tarik,

This is true and not necessarily true. Two simple examples:

Every roll of toilet paper has a finites number of sheets and a finite length and width. A linear system, so one can mathematically roll the toilet paper onto a cylinder.  The amount of squares added increases the diameter to the roll increasing the volume in a very predictable way.  Except increasing the length and volume concurrent require a correction:  the length at the bottom of the sheet (inner diameter) and the length of the top of the sheet will always be different.  What happens actually is that the diameter increase is always a function of the TOP of the sheet, not the average length of the sheet. If you do not do this, the number of sheets calculated on any roll of toilet paper will be incorrect. A linear function following a curved space requires knowing the relative phase of in this case the cylinder.  Never is the diameter precisely the same between:  distance x  and distance + sin y  and distance x - sin y.

Not a harmonic oscillator model.

When curvature is in 3 D instead of in 2D, such as the thickness of a balloon as a function of air pressure, the volume inside the balloon and the volume including the outside of the balloon is never the same.  Calculating the rate of expansion of air in a balloon is not identical to calculating the rate of expansion of the surface area surrounding the air.  The  cos y +/- i sin y component is most probably an essential mathematical requirement for every 2 D construct that is to describe any 3D process.

In any XYZ coordinate system, the coordinates are  left-handed or right handed.  In a harmonic oscillator model,  -X and +X  could just as easily be  +X and -X and nothing would be different. In polar coordinates with a similar harmonic oscillator,  cos y + sin x   and  cos y - sin x also would be distinguishable.  The numbers  i, i^2  and i^3  are critical in any accounting of the magnitude of frequencies that are not [fully] independent of each other.  The sign of how vibrational modes add up depends upon how the sine functions add up or do not add up [as predicted]. .    .

Dear Walter,  I should admit that I could not understand your remarks above.  Complex algebra definitely useful 2D space (Mathematical Theory of  Elasticity By Muskhelishvili) or  for  Minkowski space-time [ i c time+3D] as demonstrated by him 1907 for the special relativity.

The length of roll of  toilet paper may be calculated by the following formula which may be useful for some one in RG ;

Lk  (Theta)  =  2 Pi   SUMn=kn=1  { ro  + (n -1 )  t }   +  (ro  + k t) Theta,  

 Length at the end of  kth    full  turn  plus some angle Theta if any.  Sum term is valid when  k> 0  other wise it is zero. Where  t  is the thickness of the paper,  ro is the radius of the cylinder over which paper is rolling.  This formula may be further compacted by 

Lk (Theta) = 2 Pi   k{ ro + 2-1 (k -1) t } +(ro + k t) Theta 

Best Regards.

Tarik,  The answer is about a increasing thickness of a roll is a correct engineering solution to the problem, except the answer can/will always be off by half the thickness of the paper: is   r  the distance to the middle or the outside of the piece of paper? .  This is both a theoretical and a very practical problem.  We have two ply toilet tissue, and the question is: is the length of the inside ply shorter than the outside ply?  Of course it is.  the thing is that it is equally true if only a single sheet of paper.

The inside half is compressed, the outside half is stretched.  On average, nothing is happening, except clearly both are happening systematically, not randomly.

One can live life perfectly well with  pi =  3.14  or  3.1415   or  .   .      .  except one can choose to add as much precision as desired.  My point is that ignoring this exactness is obviously truncating data.

When computers can do complicated calculations exactly, why not include the same degree of exactness in the equations?  

One can do a very practical experiment with two ply toilet paper: remove the several sheets of the outer ply and then begin removing two layers at a time so the top layer is now the bottom layer.  The two layers no longer align because the bottom layer is always compressed and shorter than the upper longer layer.  .       .

It is not that the simpler model is wrong, just that it is incomplete and fails to explain what is clearly observable and experimentally verifiable. 

Models need to be internally self-consistent, so if r is the length to the outside of the sheet of paper around a coil,  then rolling paper around a cylinder compresses the paper as a function or theta, and the smaller the cylinder, the greater the degree of compression. 

To be even more exact, does compression in the r direction also result in compression in XY plane of the sheet itself?  

In DFT calculations, all frequencies / vibrational modes are assigned as positive, in part because a negative frequency has no physical meaning.  The symbol  i has no physical meaning either, except it is essential to keep tract of the sign of the sign function in vibrational modes.   Nothing is lost by always including it in equations whether or not it adds something to the answer.  Truncating it from equations in some circumstances precludes having experimental results correlate positively with theoretical results. .   .

Thanks for the conversations!

Walter.

Dear Walter you go too much in details by putting unnecessary emphasis on the secondary features like most German scientist does. Buy I like them for that.  In reality cross section of the rolling toilet paper describes a spiral in 2D space, and the  equation of which can be easily  obtained for the inner or outher surface in terms of   r, Theta assuming that it is incompressible. When it comes to the case of elastic material, you have to know applied hook stress i.e., traction) acting on the  paper while it is exposed during the rolling process. Good mechanical engineering problem where the radius of curvature enters into scenario to compute the radial stress, which varies continuously with theta angle.   That is enough for me dear Walter. But I would say that  one has the following connection. 

Pin -  Pout = Hook Stress / Local mean radius of curvature,  Best Regards

Tarik,

Cool!  Thanks for having that be clear.  The mathematics figured out at the macroscopic level can be precisely as valid at molecular level, though this too needs to be fully thought through.    Suppose the rolling toilet tissue problem at the molecular level is a roll of graphene  which gets increasingly thicker.  Seems to me the forces on this planar molecular structure would have to mimic exactly the same radius of curvature problem.  The molecular forces below the plane and above the plane would be [predictably] unequal , and the same mathematical equations would apply.  .    .    Would be cool if someone were to do this experiment and prove the math is correct including  Pin -  Pout

concurring that shape follows the same math at the molecular level [except when it does not].

This of course is a very different problem than a tree ring increase in radius over time.

I did publish what I thought was a rather interesting article on applying shape mathematics to an alpha-helix.  The helical shape of course cannot depend upon which site one starts / observes / measures / ends the shape.  Because every peptide sequence contains  N-C*-C sequence, the shape cannot be different if one starts the helix at an N or a C* or the carbonyl C.  The molecular problem in this case is that 11 backbone atoms equals one loop of a helix,  which means there are three different stoichiometries that will exist for every helical shape, i.e. one less N or one less C*, or one less carbonyl C. If the shape is uniform and the three chemical compositions are different, how does the molecular structure and shape mesh?

Would like your comments, if you are so inclined.

 Kindest regards,

Walter.

Dear Walter all my life (T am now 81 years old) went by solving mathematical problems and putting the physical events into the mathematical format such as a hobby!!   

In the derivation of  Lk (theta) problem I always use inner surface of the sheet which was the natural way to start the problem.  In the second problem above the mean radius is the arithmetic mean:  rm  = (rin + rout)/2. 

Actually  we have the following connection for curved interfacial layers along which hydrostatic pressure  Pr varies:

dPr = - Q dr/r  where the normal   force acting on the cross section of sheet is  T= Pr -Q .Then     Pin  -  Pout   = Integralinout  [  Q/r   dr]

Tarik,

Thanks for integrating 81 years of experience and mathematics into elegant formula!

So if one applies your formula P in  -  P out   = Integral [  Q/r   dr]  to graphene, then one can have virtually any value of Q only by "dialing in" a thickness of "number of turns" of  graphene.  Do not have the engineering parameters to know which range of r in and r out would product the useful result. .     .     The model of nanoparticles is that of tree rings:  every tree ring has a specific discrete curvature.  One is then stuck with tubes of only specific dimensions.  It seems to me, in your P in - P out,  you can have /engineer virtually any Q value you want, just by cutting the graphene the right length (and figuring out how to start and stop the roll of graphene .  .     .

I was born in 1948 so still have some years of catching up to do. .   .

Walter.

There is one simplification for above mention exact expression which is when the thickness of the sheet is much less then its radius of curvature. Then one writes

[integral  dr Q/r ] about equal to   [ 1/rm  integral  dr Q]  where  Gama= integral  dr Q  may be called interfacial tension.  Hence

P in - P out =  Gama / rmean 

Thorough   treatment    of interfacial layer is presented in the last paragraph of p-3921 and  first paragraph of p-3922  in the following paper.  Where  the real meanings of  Q  and  Pr  in terms of  3D interfacial layer stress tensor is presented.  Also  the reversible work associated with anisotropic interfaces is advocated in that paper by introducing the concept of  interfacial tension tensor and surface layer 2D deformation tensor.

Best Regards

Tarık

Tarik,

Love exact solutions!   The question that remains in clear focus from your discussion is: beginning from the example of graphene,  a film that happens to be only one molecule thick, the same formula can be used to discern molecular properties!   The molecular forces can be divided into inter-sheet [i.e. inter-molecular] components, and a sheet to sheet component [intra-molecular] forces.  The interfacial tensor corresponds to the sheet to sheet resistance/attraction.  

Would be really remarkable if the mathematics that work at the size of a kernel of wheat are precisely as exact at the thickness of one molecule.  Of course, there is no chemical structural model for whether the math works for smaller than the size molecular distances.   No chemical structures against which to test the model.

This is what intrigued me about the alpha-helix:  the engineering model for a helical shape was so more accurate and more precise than the biological model of a helix.  Ergo the need to update the biological model of the helix. .    .

Do not know if anyone else would be interested in knowing if molecular level bending forces of graphene on the angstrom scale would be self-consistent with bending of sheets of graphene at the nanoscale ,  micron scale, maybe even at the mm scale.

Do not know if this would be an experimentally complicated or simple thing to do. Could be fun, though. .    .   What do you think?

Walter.

Dear Dr.Walter,

Your example of tissue roll was very intuitive. Can you provide some similar analogy on why complex number would be needed to explain some physical observation? I really hope i grow up to understand your conversation with Tarik.

Vijay,

It is not really complicated to understand, leave it to mathematicians to prove.

The square root of 144 is simply 12, except that answer is correct and wrong at the same time.  Both 12 and -12 give exactly the same answer.   The incomplete answer is to say the positive number is the real answer and -12 is not a "real" number.  Of course -12 is as real as 12.  If you happened to choose label a right handed Cartesian axis and someone else choose to label the axis as a left handed axis,  that alone would switch the positions of -12 to 12 and 12 to -12 eg. along the X axis.   Exactly the same "physical' data point, just labeled differently.

The same conceptual error is possible in the realm of sine and cosine functions. The absolute value of  sin x is positive and -sin x is negative, except it is purely a function of definition whether + sin x is to one's right or to one's left of an arbitrary zero.

The mathematics works exactly as identically if the +sin x were to one's left of zero as long as one is self-consistent using one's labeling, and as long as you know your following a different "handedness."  Everyone knows a left-handed person has to hold a pen or pencil differently to write something right to left.   This is another half-truth. A right-handed person has to hold a pen or pencil differently were he/she to write left to right.  Totally arbitrary whether one writes left to right or right to left, can write exactly the same things, except by convention, there is an agreement to write one way, and if one chooses to write the mirror image, that one states this explicitly.

If one in a painting draws a mirror,  can be done in a way that it would be impossible to discern which of the two is the mirror.

The complex number is an accounting system to preclude that happening in 3D. Symmetry in 2D does not necessarily mean symmetry in 3D because  i and i^3 have different signs.  Thus the "real" component in the 2D painting of mirrors if complex numbers were used would have one of the images with i above the plane and the mirror image with i^3  or -i on the opposite side. Would be unambiguous which is which. 

Vibrational modes are considered as positive [value] functions.  NMR  frequencies are examples of vibrational modes which can couple,  Two frequencies that couple result in a doublet of doublets  A1A2B1B2.   The positive model is that A1and B1 couple, and A2 and B2 couple.  Except this is another one of the half-truths things.

Also is possible A1 and B2 couple and concurrently A2 and B1 couple. The same frequencies in each set can add up differently because they pair differently.

In 2D NMR phase sensitive COSY experiments, coloring red above the plane and black below the plane, the two can be immediately distinguished from each other.  Why in some structures coupling goes one way and  in other structures coupling goes the other way, no one really knows.  And one could no know unless one used complex numbers to explain vibrational modes.

In the presence of complex numbers enables distinguishing things that would most probably be overlooked [and/or misinterpreted].  The normal way this works is only when someone suspects something is being misinterpreted does one invoke complex numbers as a way of distinguishing something else had actually been present.

The attached manuscript may also be useful to understand frequencies.

If a circular clock is one second per minute fast or one second per minute slow, 59 seconds clockwise can show up at the same time as 61 seconds counterclockwise; again unless one knows which direction the clocks are turning, one cannot know whether the clock is faster or slower.  Assuming all clocks flow in the same direction is nothing more than an assumption. .     .

In vibrational modes, a plus and minus is contained in a clockwise and counterclockwise context.  In a circular clock 59 seconds always shows up at the same frequency as 61 seconds counterclockwise.  So at any steady state measurement, collecting data a certain frequency, actually it is impossible to tell whether the frequency observed is a little faster or a little slower than the observation frequency:  a little faster and a little slower will show up at the same observed value.  This is why NMR spectra are collected in magnetic field gradient.  This is why in our lab we sometimes choose to collect Raman Spectra within an applied temperature gradient instead of steady state.

It is a Schrodinger's Cat thing. One cannot tell whether a cat is left-pawed or right-pawed unless one is in the box (in baseball they call one left-handed a “Southpaw”).  A gradient provides a [right] handedness /direction and detection is observing a coupling between the applied gradient with whatever cat happens to be in the box.  Coupling can be positive or negative and coupling cannot be observed unless there is a gradient. Left-handed pitchers and right-handed hitters couple differently than right-handed pitchers and right-handed hitters.  

The use of complex numbers in NMR in part is essential to correctly account for differences in direction of e.g.  J-coupling.  NMR spectra can be predicted exactly from fundamental principles if the relative direction of J-coupling is known.  Some frequencies unambiguously do not couple in the same [relative] direction as others. Unless one knows a specific pitcher is left-handed, and a specific pitcher is left-handed (or right-handed), this effect would not ever show up in any measurement. 

The dispersion part of the NMR signal is simply the sine portion of an NMR spectrum + 90 degrees from the absorption i.e. Cosine portion of the NMR signal.