Conjugate Pairs Practice Questions Answers: Fill & Download for Free

Indeed you cannot. And its not just wave functions ...It comes from the fact that +i and -i are mathematically indistinguishable. This post is about what that strange statement means, why its so, and some of the implications (including the question you asked).The complex conjugate of a + bi is a - bi. That's the definition.The complex conjugate of the wave equation f(a + bi) = g(a + bi) + h(a + bi)i where g and h are real valued is g(a + bi) - h(a + bi)i.Massive spoiler alert: If f is a wave equation, and f(a + bi) = g(a + bi) + h(a + bi)i, then f(a - bi) = g(a + bi) - h(a + bi)i. Take a number, find its conjugate, pump it through a wave equation, then take the conjugate of the answer. That gives you the same answer as simply finding the value of the function at that point. That is a rather odd and useful fact, and you may well wonder why is it so?You already know that for any polynomial in Reals, if a + bi is a root, then so is a - bi. If you changed every reference to i to being -i, here is what you get:If a + b(-i) is a root, then so is a - b(-i). Which simplifies to if a - bi is a root, so is a + bi. Which is the same as the statement for +i.So which is the real (whoops, proper) interpretation of the square root of -1, +i or -i? How can you tell them apart? How can you distinguish them?You can't.That the y axis is labled with "positive" imaginary numbers going upwards (+1i, +2i, +3i ..) is just a convention. Draw the y axis around the other way, pointing downwards, and the only affect is that you form a mirror image. Clockwise rotations become anticlockwise rotations. You can't tell if you are the real thing or the mirror image, because both make identical "predictions". They are indistinguishable. Its like arguing whether left is the opposite of right, or whether its the other way round.So we have no way of ever distinguishing +i from -i in mathematics let alone physics. It is completely arbitrary which one you pick as the real (whoops) i.But in physics, there is an interesting twist, related to the rotation I mentioned above. Wave equations (all equations in physics, really) are analytic functions. This is connected to the idea that it shouldn't matter how you lay out axis (eg which direction you choose as north) because space has no preferred direction. The functions where the value of the derivative of a function is independent of the choice of axis are these analytic functions.But the complex conjugate function isn't analytical. If you rotated the Real and Imaginary axis about (0,0), the graph of conjugate functionf(a + bi) = a - bidoesn't simply rotate. It becomes a reflection about the x axis, a very different thing.So this requirement that physics equations be analytical makes this stuff about i, in my opinion, even weirder. Not only are conjugate pairs indistinguishable in mathematics, no underlying physics law can separate conjugate pairs.Seriously hard to wrap your head around. Practical implication: not a lot. Two solutions which say the same thing for the price of one calculation.

Question : How do we balance our desire for worldly success with our desire for spiritual liberation?The answer to this question solely depends on definitions that one chooses to assign to each of the two terms, and these definitions largely dependent on an individual’s outlook of life.Likewise, this attempt to address the posed question too, depends mostly on my current perspective of life - which might be dynamic and may or may not be in agreement with views of other readers.DefinitionsWorldly Success : Utilizing, sharpening and developing skills while working towards raising the overall quality of life in the ecosystem that we’re a part of.Spiritual advancement: Efforts made in controlling thoughts to direct the mind inward with the sole aim of realizing ultimate truth.I personally don’t see these two goals of human endevour conflicting each other. On the contrary, they form a conjugate pair - mutually supplementing each others’ growth.One of the best ways to achieve worldly success is by following a set of simple virtues (samanya dharma) that are innate but concealed within each being. Virtues such as truth, benevolence and non-violence. Which essentially are a set of practices that enable an individual to effectively discharge his/her duties without causing harm to others. Maintaining these standards in daily transactions not only leads to worldly success , but also helps an individual in maintaining better control of mind by paving way for purity and clarity of thoughts - which is a step forward in the direction of spiritual progress.

You’re probably looking to get a simple answer to this question, one that will help you out with learning organic chemistry for some limited-time goal, like graduation. You’re probably not a chem major. The answer, in that sense, for hydrocarbons is to look for double or triple bonds which have some clear alignment or overlap in three dimensions. So if you think about simple molecules like butadiene(image from Butadiene - Wikipedia)and borrow a molecular model kit or some kind of online equivalent, which will produce models that look like the one shown above, you can see that the two double bonds, between the 1,2 carbons and the 3,4 carbons, are in a plane and will likely be conjugated, and therefore there will be some degree of delocalization. On the other hand, if you look at a molecule like biphenyl(image from Biphenyl - Wikipedia)you can see that the bond between the two benzene rings is twisted off-plane. In this case you can expect that the double bonds are not delocalized across the single bond between the rings. There’s delocalization, true, but it is localized into the two separate rings.One other aspect that is commonly cited is the existence of resonance structures- a bond is delocalized when you can draw equivalent (or even in some cases non-equivalent) resonance structures, structures where all the atoms of the molecule are (roughly) in the same place but the bonds drawn have been rearranged. The classical example of this is benzene(image from Benzene - Wikipedia)The Kekulé structures above are the classical model of delocalization: because you have a planar system of double bonds on adjacent carbons, you can draw the double bonds either between carbons 1–2, 3–4, 5–6 or 2–3, 4–5, 6–1 and the two structures are chemically equivalent. If you measure the bond lengths, assuming that single bonds are longer than double bonds, you get an apparent equal bond length. (It is absolutely arguable, as an aside, whether or not the measurement is truly genuine- the molecule could be vibrating between two isomers rapidly and what we are measuring is apparently the average.)The resonance argument can be extended to organic molecules with heteroatoms, too- the classical example of this is the keto enol rearrangement(image from Keto–enol tautomerism - Wikipedia)The lone pair of the oxygen is conjugated with the adjacent double bond, and there is a partial delocalized character to the double bond.That’s how, at a first approximation and using the methods generally given to undergraduate students of organic chemistry, an undergraduate student who isn’t a chemistry major is expected to identify a localized and delocalized bond- through planarity and conjugation of adjacent double bonds.But, that’s not the whole story. Not even a little bit.In order to identify true localized and delocalized bonds in molecules, you have to step into the quantum world. In a quantum chemistry picture, electrons are not little planetary particles zipping around nuclei, and they are definitely not pairing up in convenient little groups like you might see in a Lewis diagram.Modern chemists probably wouldn’t go through all of the backbreaking labor of hand-calculating wavefunctions. There’s reasonable desktop tools that allow you to punch in a given geometry of atoms, and beep beep boop within seconds you get a so-called “bond order” between all of the atoms for a particular set of quantum mechanical tools. But if you just use the tool and rely on the numbers, you don’t actually understand what’s going on in the calculations, and it’s pointless to say “this bond is delocalized because the computer said so” because the truth is way stranger than either binary condition, “delocalized” or “localized”. The truth is a mixture of both.To understand why I say that, you need to understand one of the extreme simplifications that was done to try to understand the nature of bonding in hydrocarbons, you need to work through the math of HMO- The Hückel method.I’m not going to walk you through that- there are dozens of websites which you can go to to get the nitty gritty details of that. But I think it’s important to illustrate exactly what the method computes for things like butadiene, the molecule I showed you first. The website, “Examples of pi atomic charge and bond order calculations:”, says the following:What this result is saying is that the bond order, the degree of double bond ness between the four carbons in butadiene, is 1.89 for the 1–2 and 3–4 bonds, and 1.45 between the 2–3 bond. OK, so add it up- that means the quantum mechanical calculation is saying that there are 5.23 bonds in butadiene. Now, you’re going to look at the molecule, and count bonds, and get one, two, three single bonds, and four, five with the double bonds, so WTF did the 0.23 of an extra bond come from? The problem here is that generally speaking, all electrons are delocalized. Each molecular orbital stretches, more or less, over the entire molecule, and each electron that fills each orbital is essentially shared equally between all of the atoms. The problem here is one of indeterminancy- you can’t actually argue that an electron “exists” at any one fixed point, and because all electrons are the same, you really can’t tell if two electrons switched positions while you weren’t watching, no matter how closely you try to watch. Your concept of what a bond is comes from those plastic molecular model kits where *pop* a bond looks like a real physical thing, but it aint so. It’s just a way we talk about the degree of apparent overlap between adjacent atoms’ atomic orbitals adding up into a bigger molecular orbital.And boy oh boy, there are a lot of misleading facts out there. For example, let’s try to answer what the bond order is between each of the pairs of carbon in benzene using the same technique. The correct calculation shows 1 2/3 of a bond, such as found at https://ocw.mit.edu/courses/chemistry/5-61-physical-chemistry-fall-2007/lecture-notes/lecture31.pdf. But many websites, and perhaps your undergrad lecturer, will say something like 1 1/2, because it’s a nice i-didn’t-do-the-math-and-wanted-it-to-add-up-nicely type explanation that you will see at websites like How can I calculate the bond order of benzene? | Socratic. They do that, because a sophomore chemist will look at the Kekule drawings each and add up 1,2,3,4,5,6 single bonds and 7,8,9 with the double bonds added in, divide by six, and expect 1.5 to be the right answer. The difference is, the extra stabilization provided by delocalization is increasing the apparent bond order and lowering the total energy, the heat of formation, of the molecule. And the Kekule structures, or the dotted half bonds, actually have zero physical relevance- they’re a nice picture, nothing more, a simplification of a much more complicated quantum picture that can’t be drawn with neat little lines.Huckel has its limits, though. While it gets the lack of stabilization in cyclobutadiene from an energy perspective, it also happens to predict a bond order of 1.5 between all of the atoms, and so gives you a nice square triplet ground state, in general disagreement with attempts at measuring the bonds in cyclobutadiene, which tend to the rectangular and non-conjugated picture. And hopefully, you won’t take any single calculation as gospel and suggest that the 1 2/3 of a bond I assert as gospel above is actually the final answer in terms of bonding for benzene. All methods of calculation resort to some level of assumption and interpretation.Identification of localized and delocalized bonds is a kind of exercise that undergraduate instructors put poor suckers who aren’t chem majors through because that’s how undergraduate chemistry is taught, and unfortunately chem majors usually don’t know they’re sticking with it at the point that they take the class and get exposed to that dreck too. It’s an abomination, in my opinion, and needs to die in a hot fire. There’s some rudimentary hand-wavey reasons why it might be useful to a limited extent, kind of like teaching little kids about Santa Claus so that they are encouraged not to misbehave. But I don’t think it fits with modern quantum mechanics, and how bond order is actually understood by most modern practicing organic chemists today.