Why in quantum mechanics we continue using integral if the energy is quantized? I mean, the assumption of continuous now is invalid
On the other hand, to find the electric field in in classical electromagnetism given potential and boundary conditions (which would be the equivalent to solving the Schrödinger equation in a potential) have to make the sum of all orthonormal functions, while mechanical quantum each function is a different value for energy. Do you have any connection with the foregoing that I just asked? What text I can better clarify my doubts?
>Believe that someone is going to answer that
Can you put this in the form of a frog meme?
>>7995680
>Why in quantum mechanics we continue using integral if the energy is quantized? I mean, the assumption of continuous now is invalid
Because of what is called the dual nature of elementary particle behavior. They exhibit both
discreet and wave properties. Waves tend to be "continuous".
>>7995737
Too simple to be true.
>>7995680
Because you're interesting the wave, which is continous. The allowed wavelengths of that wave in finite space are quantized
>>7995680
What is the reason for energy being quanitzed?
This should be made clear by an understanding of the Bohr model of the hydrogen atom, or even particle in a box mechanics.
like a lot of things in QM that don't make intuitive sense, IT JUST WERKS
>>7995680
>Why in quantum mechanics we continue using integral if the energy is quantized?
We don't. You can only replace summations with integrals in the cases where delta E is small. For example, for the hydrogen atom all energy levels above 10 (more or less) are so close in energy that energy states collapse into a continuous function. You need to remember these are all just approximations and at a certain point the difference between the integral and summation is negligible. This is coming from someone only in thermodynamics so maybe I'm wrong but we have gone over a lot of QM in that class.
Bump because no one answered this part of the question
>On the other hand, to find the electric field in in classical electromagnetism given potential and boundary conditions (which would be the equivalent to solving the Schrödinger equation in a potential) have to make the sum of all orthonormal functions, while mechanical quantum each function is a different value for energy. Do you have any connection with the foregoing that I just asked? What text I can better clarify my doubts?
>>7996487
Bump because OP needs to elaborate on what is being asked.
>>7995680
>Why in quantum mechanics we continue using integral if the energy is quantized?
Surely you've solved the free particle schrödinger equation at some point. Is the energy quantized?
Supposing you have a system whose energy is truly quantized, if the difference between different energy states is "small" you replace summations with integrals because it's easier to work with them, that's often the case in statistical physics (but not always) as >>7995827 already pointed it out.
I'm not quite sure what you meant with your last question, could you rephrase that?
OP here
In the book (Classical Electrodynamics - Jackson, JD) we can see that to find the potential function given boundary conditions, we have to solve using the same method that we applied when we solve the Schrödinger equation, but in quantum mechanics the value "n "that depends solutions ends up being the value that energy is quantized, however in this problem the value of" n "disappears with gimmick, in more complicated cases by the method of summation of Fourier series.
What accounts for this difference? Simply heuristic is derived from the fact that in quantum mechanics is quantized energy or is there something different about both problems? Because in both cases it is working with waves.
>>7996572
I don't think n in these types of problems is the same as the principle quantum number. It's just supposed to represent the integers n=1,2,3,4,5. But I'm >>7995827 and I've only dabbled in PDEs so I really don't know how they apply to quantum mechanics. Maybe n is the principle quantum number. The pic you uploaded seems to explain pretty clearly why it only uses the first term, so I'm afraid I still don't understand exactly what you're asking.
>>7996604
Okay. But why that number "n" is not the same in quantum mechanics if you have the same mathematical origin? Apply the first term is an approximation that as you already said is explained in the book. Then, explain that you could find a complete solution using the method of sum of Fourier series, something never done in quantum mechanics.
>>7996572
I dont't get what QM has to do with anything here. What you just posted is about finding an electrostatic potential. This has nothing to do with solving the schrödinger equation, neither conceptually nor methodically. To me it seems like you just read the word "boundary conditions" and made a few assumptions here and there.
Quick question: I assume you didn't have any lectures on QM if you're hearing now classical electrodynamics, did you? If that's the case, I would just wait until you hear a proper lecture on QM, that should be much mor helpful than anything we could post here.
>>7995680
>what are waves
Probabilities are continuous
>>7996658
>I dont't get what QM has to do with anything here. What you just posted is about finding an electrostatic potential. To me it seems like you just read the word "boundary conditions" and made a few assumptions here and there.
We had a thread some time ago where the OP (of that other thread) basically complained that the word "quantum" is a misnomer. His point being that momentum is never quantized, and in cases where it is (e.g. particle in a box), that is only because of some boundary conditions we impose on the system. He then compared it to a vibrating string in classical mechanics, where only specific frequencies are allowed due to bounary conditions, so it is also "quantized" in a sense.
Now to describe a vibrating string and solving the wave equation, in general you have to do a sum of all those frequencies, vibrating with only one of them would be a special case.
Now in QM, your state |Psi> is in general a superposition of many states with different quantum numbers, say "n". The thing OP is talking about is then the result of a measurement giving only one specific "n", or the collapse of the wave function after which |Psi> is a pure state made up of only the term corresponding to that quantum number "n" as a result of the measurement.
I think OP (here) might be confused about this, possibly because he hasn't taken a QM course yet.
>>7996658
Please, focus on the arguments to my question.
Precisely my question is why in QM the same method to solve PDE is used, however, when you get to this number "n" in the case of classic we proceed otherwise than obviously the case I'm showing is not QM.
In a case where you have to solve the time indepenndent Schrödinger equation in a potential well, you have a PDE and boundary conditions, the results are given by harmonic solutions depends on a number "n", the number "n" it comes after to relate energy levels. In case I'm showing them a while, you want to find potential function given on a boundary conditions, both equations are very similar indeed to be Laplace's equation.
Why must proceed differently?
Is it the fact conceptually different problems? They are mathematically equal?
>>7996692
Now there are two answers to my question, some say it's because the waves are continuous, and now you tell me that the probabilities are continuous.
They are equivalent those answers?
You really are waves or that are interpreting the probability distribution as waves?
>>7996729
waves represent probabilities and probabilities are expressed as waves
this picture is a single particle's wave probability
now imagine many particles in a system: their probability waves all interfere with each other to create weird-looking probabilities for each individual particle and a total probability wave for the entire system
>>7996722
This response is the most satisfactory. TY.
>>7996729
> You really are waves or that are interpreting the probability distribution as waves?
In QM it would be more like:
Interpreting the mapping of a wave function to a "continuous" set of probabilities.
That interpretation can then be used to give the probability of a quantum event.
So you could say the waves are not "real" -
they should not be confused with macroscopic Fourier manipulation of electromagnetic waves.
>>7995680
Because we integrate over spacetime, which are not quantized (in first or even second quantization, I believe). It is assumed that spacetime is continuous in QM and QFT -- it is the background on which quantum fields exist. Quantum theories of gravity (like LQG) are working to quantize spacetime as well, to make it consistent with QFT and GR. Obviously, this poses multiple challenges, and is why we don't have a working theory of quantum gravity yet (even disregarding experimental verification).
>>7995680
The energy is only quantized if the particle is bound
>>7997044
but why it can not be confused? They are mathematically similar problems.
>>7997127
Okey, this makes sense, because it is still permitted to integrate.
>>7997158
Yes, but the procedure is mathematically is identical to the case of electromagneticosmo classic, and this is because in both cases Laplace equation is solved by separation of variables.
>>7997652
>but why it can not be confused? They are mathematically similar problems.
Because for the case of the Schrodinger equation we are solving for energy, while in the case of the electromagnetism problem we are solving for the electric field.
I'm >>7996604 so I'm not certain, but this could be why n is the principle quantum number in one problem but not in the second, even though the same differential equation describes them both. Wave equations have many different applications. I'm just trying to find a justification for my own sake because I really don't understand the problem myself.
>>7997652
> but why it can not be confused? They are mathematically similar problems.
Nobody else here seems to see that similarity in a way that means anything important.
We have two pages of math text posted
without enough context to understand what you are really getting at.
OP are you rejecting the idea that we must invent this "not real" wave to explain physics since we already have the "real wave" ?
>>7997652
> Okey, this makes sense, because it is still permitted to integrate
Why do you think it might not be permitted to integrate ?
Consider the pseudoscience of economics for example. The units or real world effects are discrete quanta of things like dollars, pounds, or yen. Calculus is used all the time to predict things like the probability of a given selling price generating a given demand.
>>7997652
> Yes, but the procedure is mathematically is identical
You are using the word "identical" in an ambiguous way because in both cases
there is a difference in the final usage of the result and what it means in terms of answering a physics question.
The time-independent Schrödinger's equation with a given potential and the Helmholz equation in phasor form share, mathematically, some similarities. Hence we have things like photonic energy band gaps (periodic permitivity) just as we have energy band gaps in semiconductors (periodic atom potentials). We can trap light like we can trap electrons.
To solve an electrostatics problem (laplace/poisson-DEQ) with a boundry condition, you need to find the potential. One wa to do so is to use the separation of variables and then assume each subfunction can be expressed by a fourier series. These are not actual physical waves - its just a math tool.
In the case of a QM potential problem, aka. potential well, we already know the electrostatic potential and want to find the corresponding wave functions. The boundary condition (given by the potential) has to be satisfied by the wave functions. The result is only discrete waves do satisfy, hence the "quantisation".
>>7995680
The integral you are using [math]\int_{-\infty}^\infty |\psi|^2(r,t) d^3r [/math] represents probability over a region of space. Space is continuous in QM. Yes, space is continuous in QM so that means we can look at probability anywhere and it's exactly why we're using an integral.
It's pretty much impossible to really understand any QM until you've learned enough linear algebra to understand what eigenvalues and eigenvectors are. The reason I say this is because all functions can be expanded in a linear combination of the eigenbasis of some operator. If your state is represented by more than one of these eigenfunctions (from your eigenbasis) then you are in a superposition of states. Why? Because every time you make a measurement in QM your results are always ONLY eigenvalues. Since these observables correspond to orthogonal eigenvectors, you're never unsure about "what did this come from?" unless you have no idea what's going on. I'll say that more clearly. You ONLY, ALWAYS observe eigenvalues. Since observables are represented by hermitian operators, their corresponding eigenvectors are orthogonal and so they represent distinct states. But they need not be beforehand, which is what superposition is about and collapse of the wave function is before you observe it.
So what determines if your position, velocity, or energy is continuous or discrete? Since these are all literally eigenvalues of operators of possibly infinite dimension, there's no reason why it's gotta be one way or the other. In fact, free particles can have a continuous range of positions, velocities, and energies. No problem. That's because there's nothing trapping it. In fact, if your specific state has more energy than the potential energy of your system, there is a continuum of energies you can have and this is called a scattering state. If you have less total energy than the potential energy then you're in a bound state, THAT is where quantization happens by self-interference.
>>7995680
When you integrate in QM you are integrating over space, which is continuous.
>>7997880
I had only one question and wanted to see if someone gave me an answer I understood.
>>7997932
It was something that came to my mind.
>>7997938
I wanted to know if there was any explanation beyond physical argument, which does not seem so obvious and all overlook, but of course, people who know physics if you understand.
>>7997961
>To solve an electrostatics problem (laplace/poisson-DEQ) with a boundry condition, you need to find the potential. One wa to do so is to use the separation of variables and then assume each subfunction can be expressed by a fourier series. These are not actual physical waves - its just a math tool.
Really??
I thought there was no interpretive problems with classical electromagnetism. Although you have a point to say they are mathematical tools, I thought if those waves really existed.
>>7998000
Wow, this really is satisfying for me, thanks for the reply.
>>7995713
This literally sums up the entire 4chan.
just saw this video yesterday
https://www.youtube.com/watch?v=LZie2QC5Jbc
this guy is the master of physics visualisation
https://en.wikipedia.org/wiki/Position_operator
https://en.wikipedia.org/wiki/Operator_%28physics%29#Wavefunction
-Basically you can represent a discrete basis (and a continuous basis) as an integral over all space of an operator acting on a given position, this will give the eigenvalue of that operator which is an observable (via Postulate 2/3)
http://web.mit.edu/dvp/www/Work/8.06/dvp-8.06-paper.pdf
^A more complete explanation
(at least I think this is what you are asking?)
>>8000947
That is not a good video at all. He just confuses things with the video.
A more intuitive explanation of quantum mechanics is in Dirac's principles of quantum mechanics.
>>8000947
That's pretty good.
So guys, what exactly causes a particle to end up at one position over another? Although some positions have a higher probability than others, can't it still end up at a less probable position?
>>8002176
It can but once you observe the particle the other wave functions collapse and the state is now projected onto that state
>>8002320
But why one position over another? Is it pure randomness? Is the universe fundamentally indeterministic?
>>8002331
Random in the sense that we can't predict the exact location, however we can, if we know the potentials the particle is under, find exactly the wave function and therefore the probability density, so it's not indeterministic in that sense
>>8002336
How is knowing the probability density an indication that it is deterministic?
Sounds like it's not completely random, but within certain parameters it is still random; within the parameters is it not indeterministic?
Hmmm.. I think I'm actually getting it. Sounds like it's a 'relative' thing. Relative to the rest of the world it's deterministic, but relative to anything within that bubble of disentanglement it is indeterministic?
I appreciate the responses, since I am just a layman
Wait.... So when something disentangled interacts with something else outside of it's own entanglement...
Isn't the interaction indeterministic?
I realize now I am not using the word entanglement correctly at all. IDK what word to use instead though
>>8002399
Okay, what are trying to describe?
>>8002503
I'm getting confused now on what exactly this probability density itself describes.
Considering a particle's probability density, it's just showing the probable positions it could have. But if the particle potentially interacts with another, then calculations for new wave function would be needed, considering all the potentials of both particles.
So are there two kinds of probability density? One arbitrary that doesn't involve potential interactions (like the most simple orbitals taught in high school) and others that do involve interactions?
What I was saying originally is how, considering the parameters of all the potentials of how two particles may interact... Okay I have no idea what I meant how it was a relative thing.
I'm just gonna ask this again. How is knowing the probability density an indication that it is deterministic?
>>8002549
I think what you are on about here is symmetry and asymmetry. If you consider two particles that are indistinguishable and evaluate there probability densities you may find that one is different two the other. This can't be the case as they are indistinguishable. You need to apply something called the permutation symmetry wave function on the system.
http://arxiv.org/pdf/quant-ph/0301020.pdf
As for the question of indeterminism, the particles location is not random as such but based on the probability density. If you define indeterminsm as not knowing exact the location of the particle then is indeterminate. However we can perfectly know the probability of it being there.
Engineering major taking an applied QM class right now, anyone got some good stuff on density matrix formalism and WKB approximations? I didn't take the pre-req for this class (kek) so I'm a bit behind
>>8003295
Thanks! I had not seen that thread.
You're famous OP
https://www.reddit.com/r/iamverysmart/comments/4elnlx/trying_to_impress_4chan/
>>8003560
lol
>>8003560
Kek.
>>8003560
In this discussion mentioned that / sci / is not an appropriate place to ask those questions. Where I could ask those questions seriously? I have many questions of physics that would like to clarify, but do not really know where to ask.
>>7995680
The reason why energy is quantized is because there is a discrete set of solutions to the wave equation, meaning that the allowed frequencies (and thus energies) are quantized. However, due to the fact that the solutions to the PDE are still continuous on the interval for which boundary conditions are specified, we can use integration to solve for the probability of a particle being in a particular range of positions (this is due to the probability distribution quality of the wavefunction)
>>8003560
now i remember why i hate reddit
>>8005242
/sci/ is a nice place for your questions.
You were reading their replies on reddit and reddit just hates 4chan, so it was pretty obvious they were going to call us shit.
