# Degrees of freedom

## Main Question or Discussion Point

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?

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Jano L.
Gold Member
Hello dev70,

welcome to PF. The answer depends on the theory. In point-charge 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 non-relativistic quantum theory without spin, just 3 degrees of freedom.

In non-relativistic 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.

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?

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.

If you have an electron in a box, then you have a single, well-defined ground-state 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).

Jano L.
Gold Member
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?
In classical theory, the lowest possible energy would be if electron stood still anywhere inside the box, with zero velocity.

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 $L$. 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:

$$\psi_{111} (x,y,z) = \frac{1}{ \sqrt{V}} \sin \frac{\pi x}{L}\sin\frac{\pi y}{L} \sin\frac{\pi z}{L}.$$

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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
IMHO, that's a very esoteric view. Where have you got this from?

Jano L.
Gold Member
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 $|\psi(\mathbf r)|^2\Delta V$ gives the probability that the particle is at position $\mathbf r$.

Similar rule can be stated for wave function describing many-electron 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.

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?

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 $|\psi(\mathbf r)|^2\Delta V$ gives the probability that the particle is at position $\mathbf r$.

Similar rule can be stated for wave function describing many-electron 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.
What you describe is definitely not Bohm. And I think you put a little too much into the statistical interpretation. The idea of random walks has been given up long ago, simply for the reason that it doesn't work out.

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.

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Jano L.
Gold Member
do these electrons have enough energy to convert virtual particle to real one and thus radiate?
In quantum theory of light, the expression "system radiates" means basically "system loses energy". If the whole system electron + box was in its lowest ground state, there is no way the system could lose more energy, so in theory the box would not radiate to the outside. However, you want to know what happens to the electron inside. Now the electron is not isolated and not in ground state (only the whole large system is). It can lose energy to the wall if it has some; or accept some from the wall. One can imagine this as a chaotic exchange of energy between the electron and the wall.

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.

Jano L.
Gold Member
And I think you put a little too much into the statistical interpretation.
Actually I think I am quite close to Ballentine's article. How would you describe proper understanding of statistical interpretation?

What you describe is definitely not Bohm
I agree, but I just said it is a similar point of view. By this, I mean that in Bohm's theory, there are particles as well and they have position and momentum. They do not jiggle in Bohm's theory, so in this respect my view is different.

The idea of random walks has been given up long ago, simply for the reason that it doesn't work out.
Perhaps. Can you give some reference to article which comes to this conclusion?
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/article/pii/S000349161200019X#

In quantum theory of light, the expression "system radiates" means basically "system loses energy". If the whole system electron + box was in its lowest ground state, there is no way the system could lose more energy, so in theory the box would not radiate to the outside. However, you want to know what happens to the electron inside. Now the electron is not isolated and not in ground state (only the whole large system is). It can lose energy to the wall if it has some; or accept some from the wall. One can imagine this as a chaotic exchange of energy between the electron and the wall.

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.
Does that mean electrons motion wont cause the vacuum fluctuations to be able to convert virtual particle to real ones?

Does that mean electrons motion wont cause the vacuum fluctuations to be able to convert virtual particle to real ones?
This is more or less rubbish.there are no virtual particles,it is possible to formulate quantum electrodynamics without virtual photon type thing.

This is more or less rubbish.there are no virtual particles,it is possible to formulate quantum electrodynamics without virtual photon type thing.
i guess, the Hawking's Radiation is somewhat based on the principle of vacuum fluctuations?

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.

i have read it, starting from Energy-time uncertainty principle to quantum fluctuation. And i dont find any exact interpretation of the energy-time 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?

energy is conserved at all moment,it is just that particles can have different momenta which does not satisfy on-shell condition.

Jano L.
Gold Member
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?
Depends on what you mean by energy. In quantum theory, there are states which are not eigenstates of the Hamiltonian and there are people who refuse to assume it has a meaning for such states. Then the question is dissolved, energy does not exist, hence no question about its conservation.

Of course, this is not very satisfactory, because from thermodynamics we have good reasons to believe that energy has meaning all the time and is conserved all the time.

The proposals for temporary violations of such conservation are not very credible, because that would require special equations of motion (for example, not reversible) and I think there are none in the most basic theory. Also, if there was temporary non-conservation, one would expect that the energy would fluctuate away (random walk) or perform some run-away. The fact that these things do not happen (it seems) are best evidence that the energy is conserved all the time.

thank you everyone for your help..