Thank you! But why is it that ‚the velocity you are trying to define has nothing to do with any velocity that could has been ascribed to the particle prior the first position measurement‘? Is that only a problem arising due to measurement ‚inperfection‘? This would be unsatisfying.
Thank you, this is helping me a lot! But I don’t understand another thing: If I measure the energy of an electron, how do I know in which state (e.g. a superposition of eigenstates) it was before? For example: Could it be that a single electron in hydrogen is in a superposition of 1s and 2s...
Hello everyone! I have two questions which had bothered me for quite some time. I am sorry if they are rather trivial.
The first is about the general solution of the hydrogen atom schrödinger-equation: We learned in our quantum mechanics class that the general solution of every quantum system...
Thank you! But would this mean, that the following is true:
So let’s say, that we have two masses m1, m2. Mass m1 is accelerated toward m2 with a magnitude of
$$a1=G\frac{m2}{r^2} $$
And then also m2 would be accelerated by
$$a2=G\frac{m1}{r^2} $$
So the acceleration of the approaching of both...
Hello everyone.
Probably this question is trivial, but nevertheless I am confused about Newtons law of motion:
$$F=G\frac{m_1m_2}{r^2}$$
Now, some sources say, that F is the force between the two masses m1 and m2. Other sources say, that F is the force that m1 exhibits on m2. But isn’t this a...
Yes, that’s absolutely right. I have a mistake in my calculation above. So here is what I think:
Let’s talk about only one sort particle:
H=U+pV=ST-pV+N##\mu##+pV=ST+N##\mu##
So if H is really ST+N##\mu## than
d(H-ST-N##\mu##)=0
This is true because of Gibbs dulem...
I have also ignored it in the definition of H=U-pV. U has the chemical potential in it. If I make the calculation with the chemical potential from the beginning on I get: d(H-TS)=Vdp-SdT-N##\mu## which is zero.
Thanks!
In my thermodynamics lecture notes, I can not find a indication that this only applies if T and P are constant. It is derived in a similar way like here: https://ps.uci.edu/~cyu/p115B/LectureNotes/Lecture6.pdf
Edit:
Could following be true: If H actually is TS than it follows...
Thanks for your awnser!
I know that the integration is in general not that trivial, but in this case it is. This is due to eulers homogeneous function theorem. See the Wikipedia article (https://en.wikipedia.org/wiki/Internal_energy) in section ‘internal energy of multi component systems‘.
Hello everyone!
I have a course in thermodynamics this year, and there is a question about enthalpy that I cannot answer: given the definition of enthalpy H=U+PV and the integral form of the internal energy U=TS-PV we conclude that H=TS.
We normally say that enthalpy equals the heat exchanged in...
Thank you very much, this is helping me a lot! Unfortunately I am very inexperienced in this topic.
At one hand, the step size 10^6 would be optimal from a theoretical standpoint (the Lorenz section would (if plotted in the right way) represent a u-sequence). On the other hand result of stepsize...
Hello everyone!
I was studying chaotic systems and therefore made some computer simulations in python. I simulated the driven damped anhatmonic oscillator.
The problem I am facing is with solving the differential equation for t=0s-200s. I used numpy.linspace(0,200,timesteps) for generate a time...
I really thank you for your helping hand!
I had a bit of trouble but I think I worked it out:
My posted equation is a first order differential equation. But if I use for example lagrange, I get a second order equation for r. I reduced it by considering energy conservation to the first order...
Thank you for the help!
In the summer I read a book about nonlinear dynamics and there the term phase space was used in this context. So sorry if i was on the wrong page!
But nevertheless I think, I have found the problem in my calculation: There is a square root in my r equation, so no point...
And exactly this is my problem. If I try my derivation with another potential (which is not time dependent) I get a solution of the same kind which is not time dependent either. I don’t understand how this is fitting together.
Thank you for the answer!
I have calculated following system of differential equations for the kepler problem (I am not sure if all prefactors are right, I don’t have my notes with me):
If I put in the same coordinates r and theta I always get the same velocity. What am I misunderstanding?
When it is on the same spatial location (and has same mass, momentum and energy) it has to follow the same trajectories. This is because the velocity vector at this location points in the same direction. So if a trajectorie in a two body problem crosses itself at a later time and goes in a...
Thanks for your answer!
But nevertheless: I could plot it like a vector field in a plane. And there should be no intersection, because that would violate uniqueness.
Or to say it another way: If the trajectories don’t cross in 4D, then I cannot ‘reduce‘ it to a 2D vector fiel with constant...
I think that’s a definition ( at least I read it differently from different sources). Some would define a 2D phase space with a vector on every point, some would define a 4D phase space with 4 axes and no vectors. The result should be the same?
I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem:
Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential...
Thanks for the answer!
So I thought that there is only one Liapunov function for a system. Like if i have a conservative system I can use a energy function to do the same things (and the Energy function is unique?). A liapunov function is a generalised energy function (at least so I was told)...
Yes that’s right, but on the edge of the trapping region all arrows should point inward (or at least be tangential to the boundary), so dr/dt should be less or equal to zero?
I was dealing with nonlinear systems of differential equations like the Lorenz equations (https://en.wikipedia.org/wiki/Lorenz_system). Now there is a trapping region of this system defined by the ellipsoid ρx^2+σy^2+σ(z-2ρ)^2<R.
I wondered how this region is found and I found out that a...
I am a bit confused about how kets in dirac notation are working.
I read on wikipedia, that kets are linear, so |a*Φ>=a*|Φ>.
Also I read (https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_04.pdf) that this is not true for the position state ket (...
I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here:
R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function.
I know that it is able to rotate a curve, vector...
So I don’t get a right result if I don’t consider momentum?
That’s a good question. That’s probably not possible. Nevertheless it is a bit confusing for me, that I can not get the right awnser without respecting conservation of momentum.
But shouldn’t it be, that energy for its own is conserved? Is there something that connects energy and momentum deeply? I thought both conservation laws arise from different properties of our universe (noether theorem).
Lets neglect conservation of momentum and assume that all frames of reference are inertial. Now imagine three objects: the sun, the Earth and an asteroid. In the inertial frame of the sun, Earth and asteroid are flying towards each other ( velocitys v and -v).
Now imagine you are standing at...
Thanks! I asked me this question also. But I came to the conclusion that this has to be. Because if the ball passes the wall before t_max/2, I could lower the velocitiy. So the peak would be closer to the origin and the height also decreases. But it could also be that I am wrong.
Every trajectory follows a parabola if we neglect air resistance. So we can calculate the maximum distance in x direction s_max. Also we can determine the time it takes to hit the ground again t_max. If the ground is everywhere the same height, I can assume that at t_max/2 the height (s_y) is at...
I appreciate your help very much! Thank you for spending so much time.
I have read your posts 3 times to be shure I am understanding everything. Now I have one (probably dumb) question.
You said:
I don’t fully get why in a local system I need only r \hat{r}. My guess: the local system is...
Thanks to all of you for your help!
Here is my solution. Could be possible that I wrote down something wrong because I erased a lot and may have forgotten to erase something.
Thank you. This examples help me a lot!
Thanks for the awnser! Ok now it’s clear to me that there must not be a phi...
Ok. Slowly I am understanding. I have one last question I am not shure about.
In the picture you can see the actual exercise. Is it clear from this context, that it is possible to reduce the number of basis vectors? Would the solution to c be any different if i include the therm with phi?
Thank you very much!:)
So I am right that r=p*p+z*z is not complete? Why is it legitim to use this equation in this case?
It is clear for me that the vector is not constant. I did this exercise also! Thanks! Nevertheless it is not clear to me why a not constant base should mean that I only...
I thought there has to be a term like ##\phi * \hat\phi## . Cause otherwise the vector is not fully described in there dimensions.
If I am getting this right: r could only be in a plane with constant phi but not ‚everywhere‘ ?
I am sorry if my questions are obvious! But it confuses me a lot.
Thank you!
But how is this consistent with the mathematical concept of basis vectors. Of course it is true if I look at the graph, but it’s not obvious for me in mathematical sense.
I am starting to learn classical physics for my own. One exercise was, to calculate the vector r (see picture: 1.47 b). The vector r is r=z*z+p*p.
I don’t understand this solution. My problem is: in a vector space with n dimensions there are n basis vectors. In the case of cylindrical...