Sriman Dutta

Hi!

I was trying to analyse the RC series circuit for a sinusoidal AC input. I fed the input with an AC voltage V1sin(wt), and found the differential equation for the voltage across the capacitance Vo. 

RC dVo/dt +Vo = V1sin(wt)

I assumed the capacitor was chargeless, so Vo = 0 at t=0. 

Next I tried to solve for Vo using two methods. I used Laplace transform and brute-force solving the differential equation itself. 

However I arrived at two different results, considering the same initial condition. 

Please see if I'm missing something. Or do give an explanation. I know it's very trivial, but this is getting on my nerves. 

PS: I didn't know the right category for this question so posted under Engineering. 

I am so sorry. It was a mistake. Please take back the post. 

On 12/17/2020 at 7:10 PM, MigL said:

Ahh, but Kirchoff, and Weins, just had a set of rules.
And Stefan-Boltzmann, as well as Raleigh-Jeans, didn't work, and headed for infinity at high frequencies ( UV catastrophe ).
Planck was the first to accurately describe Black Body radiation … in 1900  .

I wonder...
The fact that we all know QM to some degree, leads us to recommend textbooks which are fairly advanced.
But to a noob, a lot of material is left out ( or taken for granted as common knowledge ), leading to the confusion that prevails among people new to the subject , or the general population.
While the historical approach includes knowledge which is later discarded, a fully modern view leaves a lot of gaps.
Maybe a textbook which starts from basic principles, and gives a theoretical ( not historical ) foundation, before tackling advanced material is the best choice.

I agree. I personally followed Griffiths Quantum Mechanics and McGraw Hill's Demystifying Quantum Mechanics as introductory textbooks, later supplemented by JJ Sakurai's Modern Quantum Mechanics as I got interested in the concepts of angular momentum. 

10 hours ago, studiot said:

Modern ?

Black body 1860  :  Compton 1923  :  Bohr  1913  :  Davisson 1921  a century to a century and a half ago.

Here is something more modern, at least into the second half of the 20th century.

It is set in terms of more modern maths than Schrodinger,, Hilbert or Heisenberg had.

 

I actually meant to say that modern QM begins with Hilbert space definition.

However in introductory college lessons, they first teach you the phenomenon that raised doubt in classical physics. As an example, I told about my first sem lectures in QM. 

on a side note, can you tell me the name of the book? It looks intriguing. 

9 hours ago, Markus Hanke said:

I would argue here that it is in fact the other way around - the form that solutions of Maxwell’s equations can take depends on the geometry of the underlying spacetime. You can see this most clearly when you write down Maxwell’s equations in their most general form, using differential forms; you will find the Hodge dual appearing in them, which explicitly depends on the metric. Unless you use a metric that has the right form (i.e. metric signature), the invariance of c isn’t guaranteed.

I am also not sure whether the invariance of c alone is enough to derive the full set of Lorentz transformations - I don’t think it is. I think you need to postulate the invariance of the spacetime interval, which is a stronger condition; it implies the invariance of c, but maybe not the other way around. I’m actually not completely sure about this.

Yaa. There were three main postulates: the constant speed of light for all observers, the same laws of physics for all inertial observers and the homogeneity of space. 

7 hours ago, SergUpstart said:

In general, the main thing in QM is the uncertainty principle formulated by Heisenberg. It means the rejection of complete determinism in the laws of physics. And this is correct, because if there were only determinism in the laws of physics, that the fate of each person would be predetermined even at the time of BB.

Well, Heisenberg's uncertainty principle is a special case of a nore general result in mathematics, which is the uncertainty principle for two Fourier conjugate variables. If g(t) be a function of t, and it's Fourier transform be G(f) in the conjugate domain f, then the uncertainty principle is equally valid for them. 

In QM x and p are conjugate variables and gence there exists an uncertainty principle between them. But what relates the two is that p=hk, h is Planck's constant divided by 2*pi and k is the wavenumber( also called the wavevector in 3d commonly). Hence as you see the more fundamental thing is p=hk and the uncertainty principle directly follows from that.

5 hours ago, joigus said:

 

I think the word "deduction" points to a very interesting feature of how new principles come about, and why it's anything but easy to get at them. The immediate temptation we all have is that we must deduce the new underlying principle --it is perhaps a consequence of our overridingly-deductive education--, when actually what time and again proves to be the essential step is an inductive reasoning. Something along the lines of,

What simple assumption must I adopt so that all these facts can be an immediate consequence of it?

It's what Markus, in other post, called the overarching principle.

It's what Einstein was a master at.

That's pretty much interesting. Of course physics is an inductive study. Mathematics is more of a deductive study where results are deducted from some logical axioms or intuitively satisfying axioms.

Though the actual test of physics is obviously experiments, the great enthusiasm in theoretical developments in the last century has perhaps popularized the tendency to derive everything, even the most fundamental aspects of nature. Physicists are trying to understand now why the universal constants have those specific values. They could have taken any possible value but out of all random numbers, Nature assigned them those numbers. Is it completely arbitrar or has deeper meaning- that's the question.

8 hours ago, studiot said:

There has been a great deal of mathematical development in axiomatic QM since the workers you mention.

You should look up the work of your countryman, V.S.Varadarajan, who died last year.

He proved one of the 10 Mathematical axioms.

https://en.wikipedia.org/wiki/Veeravalli_S._Varadarajan

Yes. Most Modern QM books begin with study of Hilbert space and linear algebra theorems and then into topics of operators, eigenvalue equations and wave mechanics. Although this is a more methodical study, the actual historic proess of development was different. I remember in the first sem QM lectures, there was introduction to experimental observations which proved the failure of classical physics. Observations took discrete values. Black body radiation, Compton effect, Bohr's model and Davisson-Germer experiment were the stepping stones to modern QM. 

18 hours ago, joigus said:

Historically, they started as axioms, as said by Swansont. Those Einstein-DeBroglie axioms helped Schrödinger guess his equation, but he took a further step, because he involved the potential energy, which plays no role in the DeBroglie, Einstein, Bohr, etc. set of old quantum rules. Heisenberg used a more algebraic approach (matrix mechanics). Dirac proved that Schrödinger and Heisenberg's formulations are equivalent. But it was all guesswork.

But in the modern formulation, you can deduce them by using the postulates. In particular, the canonical commutation relations.

 

[X,Px]=iℏI

 

as well as the correspondence principle.

Even today quantization of fields rests on the correspondence principle, which relies heavily on guesswork, because there is no unique way in general to postulate a quantum operator for a classical observable.

Okay. So pretty much the fundamentals are a guesswork. And yet the entire formulation that relies heavily on those axioms works out so well. 

Hence there isn't any independent deduction of the Planck-Einstein relations, as I infer. 

This is unlike other branches. In relativity the main postulate that speed of light is constant for all observers has a well-reasoned deduction feom Mawell equations solutions on the form of plane waves. The rest part of relativity like length contraction, time dilation, Lorentz transformation can be deduced mathematically from that postulate. 

12 minutes ago, swansont said:

Planck's and deBroglie's work would be in their papers on these topics. 

In addition, Schrödinger developed the wave equation.

AFAIK, these basics were proposed, not derived. Subsequent work was derived, but not these building blocks.

 

Then these are the basic or the most fundamental equations ?

I might say then that they are axioms. 

Hi,

Planck while explaining the black-body radiation postulated that photon energy is quantised, that is, E=hf, f is frequency. 

Similarly to explain the matter waves, de Brogile proposed that p=h/y, y is wavelength. 

Using this two relations, the whole theory of QM has been developed. 

Is there any derivation of these results ? Or are they accepted to be fundamental relations of nature? 

Thanks !

18 hours ago, CuriosOne said:

Are You 100% Sure On This??

Understood but:

Counting by base 10= 10, 20, 30 

Counting by base 2 = 2, 4, 6

Is this correct?

You seem not to understand what is meant by bases.

I will be trying to explain it in simpler terms.

Any quantitative thing is a number. We use 10 characters 1,2,3,4,5,6,7,8,9,0 to represent them. Count how many characters we use? It's ten characters. Therefore our standard base of calculation is ten. 

Now imagine a civilisation living in a far off galaxy (don't ask questions like where are they bla bla, I'm just trying to explain). They evolved just like us. However unlike us, they are familiar to calculate numbers in base 4. They use the characters @,#,$ and & to represent all kinds of numbers. So they have base 4.

That's basis. It has nothing to do with calculus or trigonometry. 

Trigonometry doesn.t require a special base. Why would it? tan 45 =1 in our base and character set. It will be @ in that far off civilisation's base and character set. That doesn't mean the two things are different. 

If you find it hard to understand, for a moment forget everything about number system and things taught. Try to see what I'm trying to say. 

I see.

Thanks a lot for the information!

On 11/29/2020 at 5:58 PM, joigus said:

The rigorous theorem requires heavy-duty maths: field operators, states and vacuum, and representations of the Lorentz group.

A relatively easy-to-follow discussion is provided by John Baez:

https://math.ucr.edu/home/baez/spin_stat.html

In a nutshell, what the spin-statistics tells you can be stated as this: The splitting of particle classes into bosons and fermions, as characterized by their exchange properties (statistics):

\[\varphi_{1}\left(x\right)\varphi_{2}\left(x'\right)=\varphi_{2}\left(x\right)\varphi_{1}\left(x'\right)\] (bosons)

or,

\[ \varphi_{1}\left(x\right)\varphi_{2}\left(x'\right)=-\varphi_{2}\left(x\right)\varphi_{1}\left(x'\right) \] (fermions)

Can be obtained also from their properties under the Lorentz group (the rotation factor of it) by rotating the whole universe around the midpoint between them. If the vacuum is Lorentz invariant, we won't be doing anything to it, and particles will just be exchanged if the rotation is of angle \( \pi \).

The name spin-statistics is because "spin" has to do with properties under rotations, and "statistics" has to do with properties under exchange. "Bosons under rotations are also bosons under exchange; conversely for fermions" is the content of the theorem.

I hope that helped.

I see.

Is the approach to the mathematical proof requires group theory?

Also imagine if there are two particles are in positions 1 and 2, then their exchange in positions is equivalent to rotation of the world around them keeping them constant. But how can that change the wavefunction ? Like I can view the two particles are seen one side, and from other side, their wavefunction values are different (a minus sign pops up!) ?

Hi everyone, 

While reading Quantum Mechanics from Griffiths, I came upon a point where the author writes that the relationship between spin of a particle and its characteristic statistical behavior is explained by relativity. 

I'm in complete darkness regarding this. Can someone please explain how this is explained? And also it will be great if someone can cite some source or mathematically explain the phenomenon. 

Thanks in advance!

10 hours ago, MSC said:

We all hear a lot of doom and gloom when it comes to man-made climate change. 

What I hear less of, is what science, technologies and policies are being developed that might have the potential to halt it. 

So here is a scenario; imagine we have access to the military budget of every nation and we could use it to fight climate change. What should we do with that money in this scenario? Assuming all nations agreed to cessation of all military conflicts until the climate is no longer under threat from us, for the time being. 

Obviously this is a highly unlikely scenario, but for the sake of argument I want to know what could be done with a massive re-prioritisation of resources in favour of fighting climate change. 

This is not my AOE outside of the ethics of it, so forgive my ignorance. Appreciate anyone who takes the time to respond.

From a realist's point of view, you cannot altogether discard military. You cannot absolutely disarm and channel all your military funds to other sectors. Because there are threats. Because there is fundamentalism and violence. A recent knife-attack and inhumane beheading of a teacher in France remains the solid evidence that there exists a vast populace whose sentiment is fragile enough to be hurt by caricrature of religious figures. In essence they are arrogant, orthodox and deeply fundamentalist, lashing and attacking all the elements of modernity, viewing them vile and obstructing the progress of humanity. Thus, you need an army, you need civil security against such fundamentalist groups. Imagine if a single person carried out an attrocity as shocking as this in the heart of France, what could an entire association of orthodox fundamentalists do if France or any nation was completely devoid of military? :")

However, I agree to the fact that there is a pressing issue of climate change. If we neglect it, it can only worsen and threaten our own existence. It needs to be addressed on a war-footing basis. I am optimistic that our present and upcoming technology has the potential to drastically reduce cardon emissions. We have highly durable and efficient solar panels, wind turbines, biodegredable plastic, recycling of waste, and so on. 

In this regard, I remember an old quote: " We do not inherit the earth from our ancestors, we borrow it from our children." 

13 hours ago, motlan said:

 

all functions of physics on a graph extends in both directions however they cannot go in both directions at the same time.  Usually the variables of the physics equation equate to the positive direction of the line or curve and terminate due to the finite value of the variables.  upon termination, it will extend in the negative direction of the function from the point of origin.  the key principle of classical mechanics is opposite direction equal magnitude for all equations.  so a positive value plus the same value negative value equals zero. it is assymetric in the math and physics and you can clearly see it in the graph.  case in point my reverse order math.  E=MCsquared + zero, move the MC squared over the equal sign and you get E - MC squared = zero.  the  same function can also be expressed as Zero + E=MC squared.  Move it across the equal sign and you get -E+MCsquared=zero.  Notice the energy and mass variables transition from positive to negative referring to opposite direction equal magnitude.  My simple manipulation of the same equation refers to equal to zero on both ends.  This is clearly stated in the function when you graph any equation.  It will reverse the effect in the opposite direction.  Time as time frames like all equations must obey this math.

What did you do to the equation?

Your entire passage sounded me to like: here's positive, negative, both direction, negative direction, E=mc squared, E=0+mcsquared, blah blah...

Wish to learn before you theorise? Follow the textbooks. 

1 minute ago, swansont said:

That's not a scenario for tunneling. That's a particle in a box.

Tunneling has a barrier of length L, with V = V₀, and V=0 elsewhere.

A commonality here is that the wave function extends into the barrier in both of these cases, but V=V₀ everywhere but the box leaves no opportunity to tunnel

yes, to be strictly speaking, this is not formally tunnelling. A better example of it can be the delta potential.

However, even in this finite potential well, the wavefunction behaves like that. I am actually trying to draw the analogy here between the probability of finding the particle on the other side of potential barrier and the probability of finding in the regions outside (-a,a) where E<V0, yet still it manages to get there.

4 hours ago, Maximilian2 said:

But in this case the exponential decay probability inside the step would be destroyed, because the same ammount of particles that cross the barrier go to infinity. (I forgot to mention explicity that I'm considering the case where E < V0, with E the energy of the wave-particle).

Assumin          V=0 when -a<x<a   

                          V=V0 elsewhere        (V0>0)

There are two cases:

Case 1: E < V0 (bound state)

In this case, the wavefunction has certain discrete energy levels, the number of which depends on the strength (aka shallowness) of the potential. The discrete states can be obtained after some numerical calculus, as there is no direct analytical method.

Also the wavefunction is not zero outside (-a,a). This is the striking feature of quantum tunneling. If you integrate the square modulus of the wavefunction in (a,infinity) or (-infinity, -a), that effectively gives you the probability to find the particle in that region. And surprisingly it's non-zero, which means there is some probability for the particle to exist outside the potential(finite) barrier in spite of insufficient energy.

Case 2: E> V0 (scattering state)

In this condition, the particle exists like a free guy, but with just a potential acting outside (-a,a). Here if you assume that you are directing the particle from one side, then it can be shown that there exists some probability that the particle is reflected back. Obviously, the transmission probability is more, and it increases with increasing E.

41 minutes ago, Capiert said:

 wrt the initial_speed vi;

 & relativistic if you (only)

 swap (=reverse) its perspective

 from wrt earth's minimum_speed 0

 to wrt earth's maximum_speed c.

 

Reversing the

 Kinetic_energy (perspective)

 KE=m*v*va (wrt Earth’s_speed), *(-1)' gives

 -KE'=m'*(-)*v'*va' (wrt light’s_speed) which

 is already relativistic.

 

Please notice

 c=v+u, u=c-v, u=-v', (prime_symbol ‘ is wrt light’s_speed)

 the (negative) speed_difference (wrt light’s_speed), is

 -v'=-(vf'-vi')=vi'-vf'

 & the average_(accelerated)_speed (wrt light’s_speed), is 

 va'=(vi'+vf')/2

 for initial_speed vi' (=-0'=v_min, wrt light’s_speed, or v_max=c wrt Earth’s_speed)

 & final_speed vf' (wrt light’s_speed).

 -KE'=m'*(vi'-vf')*((vi'+vf')/2), combine brackets

 -KE'=m'*((vi'^2)-(vf'^2))/2, let vi’=-0’=c

 -KE'=m'*((vi'^2)-(vf'^2))/2, bring c^2 out from the brackets (c^2)/(c^2)=1/1=1

 -KE'=m'*(c^2)*(1-(vf'^2)/(c^2))/2, let gamma’^2=(1-(vf'^2)/(c^2))

 -KE'=m'*(c^2)*gamma’*gamma’/2.

 

That equation

 has (=contains)

 "half" of (DePretto's 1903, vis_viva),

 Energy
 E=m*(c^2)

 & 2 (Fitzgerald_Lorentz, relativistic_contraction similar) coefficients (named)

 gamma_primed

 gamma’=(1-(vf’^2)/(c^2))^0.5

 where the rest_mass m=m'

 is (the same, =identical)
 constant(_variable)
 for either perspective.

 

I.e. Conservation of mass com

 (wrt Earth’s_speed) m=m' (wrt light’s_speed).

 

Speeds are the variables (instead).

 

2020_11_23_1010_KE_is_already_relative__Preliminary__2020 11 23 1419 PS Wi_(stripped).pdf 24.61 kB · 2 downloads 2020_11_23_1010_KE_is_already_relative__Preliminary__2020 11 23 1356 PS Wi__with old _Excel_ formula text_(mix).pdf 27.01 kB · 0 downloads

Suggestion 1: Know what you are talking

Suggestion 2: Know physics

3 hours ago, motlan said:

all functions on a graph extends in both directions, in the positive direction and negative direction.  the variables of the equation extends in the positive direction first followed by the negative direction to equal to zero.  for example the sine graph goes in the positive direction and negative direction.  In the positive direction the path of this circular function goes clockwise, in the negative direction it goes counterclockwise.  sine 90 degrees is 1.  sine negative 90 degrees is -1.  time is a coordinate in relativity which must be treated as positive direction terminated followed by negative direction back to the point of origin to equal to zero (equilibrium).  In the physics of an air conditioner to generate cold air it must release hot air that expels outside.  however eventually the cold air and hot air on earth eventually unite and neutralize on earth, this is the reverse order to equal to zero (equilibrium) the arrow of entropy eventually reverses.  Time is the same way.  

Ok my math is feeble but when did functions are plotted first in the positive side and then in the negative side 

On 11/22/2020 at 5:16 AM, motlan said:

Another reason time will terminate

All electromagnetic energy terminates with distance.  Electricity going through a cable diminishes and terminates with distance.  Radio waves terminate at a certain distance.  A sound wave through Doppler Effect stretches with distance from the source and eventually loses noise.  Light from blue shift to red shift loses energy in an elongated wave length and eventually terminate in travel.  That is why the more distant stars are dimmer in light than the closer stars.  The light from the more distant stars lose intensity with a greater distance to span.  Even light from further stars do not make it to our skies because at that distance the light energy terminates.  Star light in the skies can be millions of light years away, that is an incredible distance away. 

The universe began at the singularity with only one single time frame.  At the moment of the big bang, space and time in the form of time frames extended into the future.  It was electromagnetic energy that extended the time frames way into the future.  Like I mentioned, all kinetic electromagnetic energy weakens and then terminates with distance.  The energy that expanded the time frame weakens as it stretches away from the first time frame before the big bang.  At some point in time that energy shall terminate possibly trillions of years into the future.  Like the kinetic energy of a pendulum, the time frame shall cease to extend (equal to zero) followed by reverse order of time (time frames).  At this point it is the mother of all opposite reactions.  Electromagnetic energy travel terminates, the speed of light is finite, an object in free fall reaches terminal velocity and perception from consciousness terminates due to finite.  All variables and everything in this universe are finite and will resort to reverse order. 

Well well we have pseudo-metaphysics here. XD

Energy does not fade away. Distant stars which are still not visible is because either their light is absorbed or objected in their path, or they haven't yet reached us( the rays are travelling and might reach earth tomorrow, who knows).

Doppler effect or red-shifting has nothing to do with energy termination.These two concepts are related to shifting of frequency(or wavelength) with relative motion. They do not prove that energy is terminated.

And what is kinetic electromagnetic energy? XD Heard it for the some time.

And time is treated as a coordinate in relativity. The direction of time is always towards future. Well there are a couple of definitions of unidirectional time in different senses. I read these in Hawking's popular book "A Brief History of Time". 

Firstly, there is the notion of thermodynamic time. Every process in this universe occurs so that its entropy is increased. This increase of entropy causes an arrow of time, i.e., time always points to that direction in which the entropy increases.

Second you got the cosmological time. Time moves in that direction such that the universe expands.

And thirdly, psychological time- the sense of time in our mind. We "feel" that time moves in one direction. This intuitive sense comes from the cause-and-effec experiences over the years of evolution.

On 11/19/2020 at 4:51 PM, ScienceNostalgia101 said:

This is an occasional activity from my childhood. Whenever my backyard would be filled with an abundance of snow I'd dive off the patio into the snow. I am curious now how much of a risk I was assuming when I did that. I didn't jump from a very high height, but I am curious now how the ability to be safely slowed to a stop by the snow; without hitting the ground underneath and  without accelerating fast enough to injure myself; relates to factors like the wetness of the snow, the snow pack, and the snow depth.

I guess there are too many factors at play: compression, friction, velocity of impact, heat generated due to impact causing a little melting, depth of snow, temperature. Deducing a quantitative mathematical model is cumbersome at least. 

11 hours ago, John Cuthber said:

Something puzzles me.

Trump barely won the 2016 election and he did it even though he lost the popular vote.
He has had 4 years to sort out any issues he saw with the electoral process, and didn't so I guess he accepts that the results in 2016 were pretty much a reflection of reality.

He won that on essentially a single issue campaign; "Build a wall". He didn't build the wall.

He's the ultimate "You had one job" meme.

Obviously, that's not going to go down well with the people who voted for him.

In the meantime, he also oversaw the unnecessary deaths of more US citizens that the Vietnam and Korean Wars put together.
Overall, 4 years ago he had less than 50% support, but fluked a win.
Since then he has screwed up virtually everything he has tried to do.

Why would anyone think he was now more popular?

Trump is another epitome for the rising trend in aggressive ethno-nationalism in politics. His most popular slogan "Make America Great again" and bullying others, including activists, social workers, leftists, and even non-whites is something that will instantly portray him as a white supremacist. There was a surge in the white supremacist groups and hate groups in US in his regime. He also totally destroyed America's diplomatic relations. He often quotes false statements in press conferences, and many popagandas. He did nothing to help people in the covid pandemic. 

Indeed he was not deserved to be a President. 

On 11/21/2020 at 6:27 AM, joigus said:

Conjugate variables are certainly peculiar. Their properties cannot be simulated by any finite-dimensional space of states and thereby cannot be completely understood with discrete mathematics. They are the domain of transcencental mathematics. Unlike the famous \( J_x \), \( J_y \), \( J_z \) that people use in all the completeness theorems, they always pair in couples, one of which is conserved, the other is not.

Indeed. And the fascinating thing is how their operators' commutator give a constant ! Like take x and k, the k-observable will have a d/dx operator form in x-space and hence their commutator will yield a constant. Conversely, you have d/dk operator form for x-observable in k-space, and again their commutator is a constant. Tricky maths!

On 11/21/2020 at 3:22 PM, studiot said:

 

Yes QM is still very much a work in progress/unfinished business.  +1

Pretty much. And I guess I read somewhere that this immediate state collapse violates the mximum speed postulate in relativity. I might be wrong, but I guess that this immediate effect of any field (like Newtonian gravity) had the drawback of effects coming instantly, yet any information can at most travel at c. 

3 hours ago, Anchovyforestbane said:

This was the kind of explanation I was looking for, I thank you.  : )

You're welcome

1 hour ago, Anchovyforestbane said:

Is that so? Any book I've read matches them together as though one must necessarily imply the other.

The many world interpretation is not pecular to QM. It can be seen as an interpretation of probability itself (although I don't like it).

If you toss a coin, it can be either head or tail. Suppose you get head. There was an equal probability of getting tail before the start of the experiment. Thus, why one event is partially favoured when tossed randomly ? This might raise the thought that there exists another world( read universe), where you tossed the coin and got a tail.

Since QM is all about probabilistic nature of the world, people interpret it using this argument. Such theories naturally go to multi-verse concepts and scifi.

.

.

But I don't consider this uncertainty in QM to be a source of theorising multiverse. Rather it is a fundamental property of the Fourier transform! Take an example of a signal. If you squeeze its time period, it's frequency curve is flattened. The Fourier transform has a remarkable property. If you try to squeeze or localise a signal( or a wve or any function) in one domain, it will not be localised in its conjugate domain (not a good terminiology, but I am using it to illustrate the property). Conjugate domains or variables simply mean two domains or variables whose functions form  a Fourier transform pair. For example, take time and frequency. 

The uncertainty in QM comes from the fact that position x and its associated conjugate variable wavenumber k form a Fourier pair. By de Brogile's hypothesis, you have p=hk/2pi, or the momentum is directly proportional to k. This thus clearly forms another pair of conjugate variables, with just another constant in the exponentials. Indeed I actually kind of believe that the entire mysteries of QM can be dragged down to this fact that p and x are conjugate pairs.

If you want maths, just google about it and there's plenty of lecture notes. If you are a beginner, I would suggest get a good textbook or try an online course. 

Hope I cleared your doubts  .

The Copenhagen interpretation simply can be intuitively understood as another striking property of the Hermitian operator. 

In QM every observable has an associated operator. This operator is Hermitian because the observable value need to be real.

Now this Hermitian operator, say Q, has the property to return eigenvalues, say q, when operating on a function, called the eigenfunction or eigenstate. In a determinate state, the measurement of Q will always produce a certain eigenvalue q. Putting it other way might help. If measuring a physical quantity returns a certain value q, then we can be sure that it was in the state |q>, the associated eigenstate of q. Thus, you can draw the conclusion that "immediately" after the measurement,  the state collapsed to |q>. 

Thanks a lot. It all makes it clear

 Plus I myself also derived the results and found the connection. 

Thanks once again!