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.

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.

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.

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. 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. 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. 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.

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.

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 !

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!

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!

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."

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.

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.

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.

Suggestion 1: Know what you are talking Suggestion 2: Know physics

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

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.

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.

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.

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! 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.

You're welcome

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!