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Degrees of freedom 
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#1
Nov2912, 02:39 AM

P: 58

HI PF, I wanted to know what are the degrees of freedom of an electron? How should it behave when it is left with its least energy in vacuum?



#2
Nov2912, 06:00 AM

PF Gold
P: 1,168

Hello dev70,
welcome to PF. The answer depends on the theory. In pointcharge classical electrodynamics, electron is a mass point and has 3 degrees of freedom. There are also approximative models of electron as charged rigid sphere  6 degrees of freedom. In nonrelativistic quantum theory without spin, just 3 degrees of freedom. In nonrelativistic quantum theory with spin, spin variable has two possible values, so it seems 3 coordinates +1 spin variable =4 degrees. But the spin is a different kind of variable. 


#3
Nov2912, 09:54 AM

P: 58

ok..3 translational degrees of freedom? if we take up a thought experiment like a particle in a box (let the particle here be an electron) and if the electron has the lowest possible energy i.e., zero point energy then how would the electron behave then?



#4
Nov2912, 11:05 AM

P: 184

Degrees of freedom
The concept of degrees of freedom only generalizes poorly to QT. In classical mechanics we have a configuration space whose dimensions are the degrees of freedom. This configuration space is completed by the velocities of the DoFs to form the phase space which holds the real state of the system.
In quantum theory the configuration space is already the state space, there is no extra tangent space to it. So you can't really speak of degrees of freedom anymore. In addition, the state space of a quantum particle has infinitely many dimensions, and so the best approximation for the number of degrees of freedom of an electron state must be infinitely many. As you can see, this makes only little sense. Finally, there's no relation to what you're asking about zero point energy. There is not even a lowest energy state for a free electron, because the energy is frame dependent. Furthermore, the theoretical lowest energy free particle state is not physical, because it is not normalizable. And since the energy spectrum is continuous, you can always prepare an electron with even lower energy (relative to a given frame) for any electron you're already seen. Now electrons are not really particles in the sense you might imagine. They're excitations in a quantum field. However, if you widen the scope of your question to quantum field theory, the answer would not be much different. The major difference will be that you cannot speak about a single electron anymore. 


#5
Nov2912, 11:43 AM

P: 133

If you have an electron in a box, then you have a single, welldefined groundstate wave function. The only "degree of freedom" you would have is an overall phase factor. The electron would not "behave" in any way, it would just sit in its ground state forever (similar to the ground state electron in an H atom).



#6
Nov2912, 01:59 PM

PF Gold
P: 1,168

In quantum theory, the answer depends on the interpretation one chooses. In addition to the point of view of Jazzdude, there is also a point of view according to which the particle jiggles back and forth in a chaotic motion, even at zero temperature. Say that the box is a cube wide side [itex]L[/itex]. The probability density for its position would then be assumed to be given by the square of the eigenfunction of the Hamiltonian corresponding to three quantum numbers 1,1,1: [tex] \psi_{111} (x,y,z) = \frac{1}{ \sqrt{V}} \sin \frac{\pi x}{L}\sin\frac{\pi y}{L} \sin\frac{\pi z}{L}. [/tex] 


#7
Nov2912, 02:29 PM

P: 184




#8
Nov2912, 04:33 PM

PF Gold
P: 1,168

It is not more complicated than how statistical physics works. We use probability density function of possible configurations to evaluate likelihood (probability) that the system has some definite configuration. In the model, the system (say, gas) is supposed to always have some, we just do not know which, so we assign probabilities.
In quantum theory, this approach is called statistical interpretation. The core of this approach is the Born rule for Schroedinger's wave function: the value of [itex]\psi(\mathbf r)^2\Delta V[/itex] gives the probability that the particle is at position [itex]\mathbf r[/itex]. Similar rule can be stated for wave function describing manyelectron atom. The particle can be thought to exist and have definite position and momentum  we just do not know which. See great article by L. Ballentine, Statistical Interpretation of Quantum Mechanics, esp. the end of the page 361. http://rmp.aps.org/abstract/RMP/v42/i4/p358_1 Similar point of view is adopted by Bohm's theory. 


#9
Nov3012, 01:36 AM

P: 58

ok..i got a brief idea about this. thank you all for your help. and lets me move on to vacuum fluctuations of the space around an charged particle say electron? do these electrons have enough energy to convert virtual particle to real one and thus radiate? I am a undergraduate student of grade 12. so, please help me out?



#10
Nov3012, 02:42 AM

P: 184




#11
Nov3012, 05:02 AM

P: 1,020

there is an intrinsic degree of freedom also which is the spin of electron.It can be naturally accounted by dirac eqn.wavefunction of electron is the usual thing(single particle) which is used to describe the electron.



#12
Nov3012, 08:34 AM

PF Gold
P: 1,168

However, I want to point out that such isolating wall does not exist and there is always interaction with the surroundings. So even the box itself will radiate to the outside and be able to exchange energy. 


#13
Nov3012, 08:36 AM

PF Gold
P: 1,168

Quick search lead to me this recent article, from which it seems these ideas work quite well: L.S.F. Olavo, L.C. Lapas, A. Figueiredo Foundations of quantum mechanics: The Langevin equations for QM http://www.sciencedirect.com/science...349161200019X# 


#14
Nov3012, 08:40 AM

P: 58




#15
Nov3012, 09:17 AM

P: 1,020




#16
Dec112, 01:18 AM

P: 58




#17
Dec112, 06:47 AM

P: 1,020

I think you might be talking about this wikipedia page
http://en.wikipedia.org/wiki/Quantum_fluctuationwhich says that energy is not conserved for small times due to vacuum fluctuation.It is wrong,energy is always conserved.It is only the off mass shell condition,which appears.vacuum fluctuation however has a finite probability but it does not say that virtual particle are converted into real particles. 


#18
Dec212, 06:00 AM

P: 58

i have read it, starting from Energytime uncertainty principle to quantum fluctuation. And i dont find any exact interpretation of the energytime uncertainty principle. Some say it means energy conservation can be violated for some time some say its not possible. I am stuck at that point. Whats the reality?



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