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Jilang
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Does anyone have an intuitive understanding of why probabilities in QM are the amplitudes multiplied by their compex conjugates?
Jilang said:Does anyone have an intuitive understanding of why probabilities in QM are the amplitudes multiplied by their compex conjugates?
jedishrfu said:I thought it was to give you a real value that could be measurable.
jedishrfu said:I thought it was to give you a real value that could be measurable. The wiki article may explain it better:
http://en.wikipedia.org/wiki/Born_rule
or maybe not but at least there's some references to follow up.
Feynman says this is the fundamental difference between classical and quantum mechanics. Quantum mechanics predicts probability amplitudes which are complex numbers, and the addition of complex numbers in which their relative phase plays a role is the root cause of all interference phenomena.Jilang said:Does anyone have an intuitive understanding of why probabilities in QM are the amplitudes multiplied by their compex conjugates?
Bill_K said:Feynman says this is the fundamental difference between classical and quantum mechanics. Quantum mechanics predicts probability amplitudes which are complex numbers, and the addition of complex numbers in which their relative phase plays a role is the root cause of all interference phenomena.
audioloop said:and more interesting, superposition is limited on space tieme
Jilang said:Er ?? . Please expand.
Jilang said:Does anyone have an intuitive understanding of why probabilities in QM are the amplitudes multiplied by their compex conjugates?
bhobba said:We see there are basically two reasonable ways of modelling physical systems using probability models - standard probability theory and QM.
That's likely the deepest reason of all.
Thanks
Bill
Devils said:Since the study of probability has been around for hundreds of years, were the properties of QM discovered in the past century or earlier? ie why wasn't QM math worked out in the 1800s.
Devils said:Are there macroscopic or other examples of QM that can be seen intuitively?
bhobba said:I think that the 'intuitiveness' of QM can only be seen via mathematics.
Thanks
Bill
Jilang said:The recurrent theme appears to be assuming continuity of transition as an axiom, but I am still unsure as to why the continuity is important. Is it to do with reversibility?
Thanks for describing this so nicely. I think you can derive the EM wave equation from the Maxwell equations. Are there any equations analogous to them in QM for deriving the Shroedinger equation?Mandragonia said:There is also a historical reason why QM became developed in this direction.
Early in the twentieth century it became apparent that electrons behave more like waves than like particles. Their behaviour is not well described by classical mechanics. Thus there was a need among physicists to develop a theory that accounted for the waviness of particles such as the electron. Now nobody really had a clue how to do this... So what they did, is borrow ideas and techniques from the -by far- most successful wave-theory at that time: Classical Electromagnetism. This theory had developed over many years, and culminated in the 4 Maxwell equations.
The complex wave function in the Schroedinger equation was adapted from the (complex) electric and magnetic fields of EM. The probability density is analogous to the intensity of the electromagnetic field (the absolute value of the electric field squared plus the absolute value of the magnetic field squared). In doing so, the QM theory incorporated the wave-effects of diffraction and interference that were observed for electrons and light, and which had been successfully described for the latter by the Maxwell equations.
Jilang said:Thanks for describing this so nicely. I think you can derive the EM wave equation from the Maxwell equations. Are there any equations analogous to them in QM for deriving the Shroedinger equation?
bhobba said:The correct way to derive Schrodinger's equation...
So at rock bottom, classically, or in QM, symmetry is the real reason behind the dynamics.
By now, hopefully, you will have glimpsed, but only glimpsed, one of the deepest and most profound insights physics has revealed - symmetry is what really governs the world. Its not talked about much in the pop-sci press - it should be - it's startling and shocking - but you only appreciate it by studying the real deal.
I invite you, and anyone else, to take that journey.
Thanks
Bill
bhobba said:Your next question is probably - why complex numbers? Well that's the quantum mystery isn't it? Feynman actually sorted it out - if you don't have complex numbers then you do not get phase cancellation on all the possible paths a particle can take leaving only those of stationary action. In fact you can actually derive Schrodingers equation from the Hamiliton-Jacobi equation of Classical Mechanics if you do one simple thing - go to complex numbers:
http://arxiv.org/abs/1204.0653
You are welcome. And you are right. From the Maxwell equations (which are first order differential equations) one can derive the EM wave equation (which is a second order differential equation).Jilang said:Thanks for describing this so nicely. I think you can derive the EM wave equation from the Maxwell equations. Are there any equations analogous to them in QM for deriving the Shroedinger equation?
Jilang said:Thanks Jedi, I didn't get too far with Wiki as the maths got a bit tricky. But I just found this on this forum from 2009
http://en.wikiversity.org/wiki/Making_sense_of_quantum_mechanics/Principles_of_Quantum_Mechanics
Which I found very helpful. It suggests it is a joint probability between the particle and detector, a joint probability, which to a fair approximation would be the amplitude of the moving particle multiplied by the complex conjugate. This makes some sort of sense, but I don't know if it's generally accepted so would appreciate any comments on this.
San K said:Good link.
Sometimes it feels as if space-time, matter, energy, dark energy, dark matter etc...are all inter-convertible...
on a separate note:
what is "linear objects"? ...as in below:
"If we keep in mind that state vectors represent ordinary linear objects, there is nothing mysterious about quantum physics"
are linear objects (one or) two-dimensional objects?
Yes, I can't get it to work either.stevendaryl said:I do not understand that paper. Equation 4.2 says:
[itex]S = -i \hbar ln(\psi)[/itex]
Equation 4.6 says:
[itex](\frac{\partial S}{\partial x})^2 = -\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi[/itex]
But that doesn't seem correct. Let's take the ground state of the harmonic oscillator, which has the form: [itex]\psi = e^{-\lambda x^2 - i \omega t}[/itex]
For this [itex]\psi[/itex], we have:
[itex] S = -i \hbar (-\lambda x^2 - i \omega t)[/itex]
[itex](\frac{\partial S}{\partial x})^2 = - 4 \hbar^2 \lambda^2 x^2[/itex]
[itex]-\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi = -\hbar^2 (-2\lambda + 4\lambda^2 x^2)[/itex]
So it's not true that
[itex](\frac{\partial S}{\partial x})^2 = -\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi[/itex]
Mandragonia said:You are welcome. And you are right. From the Maxwell equations (which are first order differential equations) one can derive the EM wave equation...
You ask whether there is something similar to the EM wave equation hidden in the Schroedinger equation. No, not really. It turns out things are somewhat different in QM. That is because the left hand side (with the Hamiltonian) is second order in space. So their is a mismatch between space and time derivatives. In fact, the Schroedinger equation describes the time evolution of an electron wave as a kind of diffusion process. Electron waves have a tendency to spread out.
Jilang said:I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?
stevendaryl said:I do not understand that paper. Equation 4.2 says:
Jilang said:I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?
Jilang said:Symmetry is my weak spot so I have ordered the Ballentine. I will wrap it up for myself for Xmas, 25 quid well spent and I've started my Christmas shopping also-not bad for a days work!
Jilang said:I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?
Mandragonia said:1. Is it just coincidence?
Yes, I think so. The main focus of Erwin Schroedinger and his colleagues was on building a model that describes the atomic structure and its properties, in particular emission and absorption spectra. The behaviour of free electrons was (presumably) less important.
2. Is there some underlying process driving it?
The spreading of a wave packet is regarded a consequence of Heisenberg's uncertainty principle.
3. Why would it become more uncertain over time?
A free electron can be in sharply localized state at t=0, with small uncertainty in position dx and some uncertainty in velocity dv. If you wait for some time, the wave packet evolves. The uncertainty in position increases due to the additional uncertainty in the velocity. Finally it becomes of the order t*dv, which can be much larger than the initial uncertainty dx.
Jilang said:Thanks for this. Does the uncertainty in the velocity stay the same or does that increase too?