Can a particle have 0 energy?

[gib65]

I was thinking about how they say the quantum theory of the atom saved atomism from collapse by suggesting a way electrons can stay in orbit around their nucleus without falling into it. If energy can only come in discrete amounts, then electrons can only exist in discrete energy levels in their orbits around nuclei. But, in my understanding, this doesn't resolve the problem of electrons falling into the nucleus unless it means that the lowest energy level must still contain some energy. If you can have an energy level with 0 energy, that level would equate with the electron falling into the nucleus. Therefore, particles (or at least electrons) cannot have 0 energy. Is this right?

[swansont]

An electron in the ground state of Hydrogen has -13.6 eV of energy. In part because the way one measures potential energy (in this case electrostatic PE) is arbitrary. It's convenient to set zero energy to be at infinite separation, but we really only care about changes in PE, so in that sense it's all bookkeeping.

 

So "zero energy" is not entirely a well-defined term.

[Tom Mattson]

But' date=' in my understanding, this doesn't resolve the problem of electrons falling into the nucleus unless it means that the lowest energy level must still contain [i']some[/i] energy.

 

The requirement of discrete energy levels alone is not sufficient to guarantee that the atom won't collapse. The specific form of the Coulomb potential must be just so in order for that to be ensured. If the Coulomb potential were, say, an inverse square potential then it would you would get discrete energy levels, but the eigenvalues of the Hamiltonian would not be bounded from below, and you'd still get atomic collapse. But since the Coulomb potential varies as 1/r, there is a finite lower bound on the spectrum of energy eigenvalues.

 

If you can have an energy level with 0 energy, that level would equate with the electron falling into the nucleus.

 

Of course you're free to put the zero of energy anywhere you like, but convention you are choosing here is not common. It's far more common (in nonrelativistic QM) to associate zero energy with a free electron (that is, an electron that is infinitely far from the nucleus). Under the normal convention electron energy is taken to go to neg. infinity as the electron collapses to the nucleus. But as I said the atomic Hamiltonian is bounded from below, so this does not happen.

 

Therefore, particles (or at least electrons) cannot have 0 energy. Is this right?

 

No, it is not.

[ydoaPs]

don't particles have to have energy? e=mc^2, ZPE, and such?

[Tom Mattson]

While it's true that a massive particle is guaranteed to have an energy associated with its mass (E=mc²), it is also true that one is free to adjust a potential energy function by any number one wishes. That is because it is only potential energy gradients that are physically meaningful. So for instance say a particle of mass m is subject to a force that can be derived from a potential V(x), such that V(x₀=0. And say further that the particle is at rest at x=x₀. So it's total energy is it's rest mass energy, right?

 

Not necessarily.

 

You can just as easily use the potential V^*(x)=V(x)-mc². After all, it has the same gradient (and therefore gives rise to the same force) as the original V(x). So now, using V^*(x), we say that the particle has "zero energy".

 

That's why merely stating the energy of a system without also specifying a datum is physically ill-defined.

[brad89]

I think no because by eliminating all of the energy of matter, you eliminate time for that particle. As time and space are interconnected, you would destroy the matter, which violates the law of conservation of mass. It is impossible to reach absolute zero, but I think it is really all around us (that is probably getting annoying about now, I have posted that way too many times.)

[gib65]

As I am familiar with.

[brad89]

Probably, it is just about on every other thread on this site!

[Tom Mattson]

I think no because by eliminating all of the energy of matter, you eliminate time for that particle. As time and space are interconnected, you would destroy the matter, which violates the law of conservation of mass. It is impossible to reach absolute zero, but I think it is really all around us (that is probably getting annoying about now, I have posted that way too many times.)

 

Not one bit of this makes any sense.

[biggs]

hey i heard about this somewhere, if you take away all the energy from a particle (ie 0 kelvin) the electrons will fall to the center of the nucli and the particle will turn into energy(or some other thing will happen), cold fusion

 

is this true? or could it be true (since it's probably never been done before)?

[swansont]

hey i heard about this somewhere' date=' if you take away all the energy from a particle (ie 0 kelvin) the electrons will fall to the center of the nucli and the particle will turn into energy(or some other thing will happen), cold fusion

 

is this true? or could it be true (since it's probably never been done before)?[/quote']

 

 

No.

 

Absolute zero is center-of-mass motion (kinetic temperature) and electron excitation distribution. At 0 K all COM motion would cease and all of the electrons would be in their ground state.

[5614]

um, what happened to QM's Zero point energy???

 

Classically at absolute zero a particle has no energy, but a consequency of QM's HUP (uncertainty) and also QM's Pauli exclusion (not all electrons can be in lowest energy level) means that there is a minimum energy an atom can have, referred to as zero point energy (ZPE).

[Tom Mattson]

The Zero Point Energy is just the minimum energy that a bound quantum mechanical particle can have. But that ZPE can be set to any real-numbered value, including zero.

[5614]

I know all of that, except for the last part.... how can a particle have zero energy? Seeing as we can't reach absolute zero anyway? Or is it some kinda of potential energy?

[Tom Mattson]

See post #5. You can always transform a potential to make the total energy of a particle zero at any arbitrary point, by adding an appropriate constant. Remember it's not the potentials that are physically meaningful, it's the forces. And F=-∇V. So if I replace V(x) in post #5 with V^*(x), I find that the exact same force is derivable from the new potential, because the gradient of a constant is zero.

 

All this leads up to the idea of a gauge transformation. Have you heard of those?

[5614]

hmm, you see if you look at wiki or hyperphysics they say that ZPE is minimum energy for a quantum harmonic oscillator, is what we're running into here the fact that you are not talking about a harmonic oscillator?

 

So could the energy of an electron (as opposed to the diatomic harmonic oscillator) have a ZPE of 0?

 

If so how does that fit in with HUP because it would be totaly stationary, or would measuring it's position give it energy?

 

(I just briefly looked up about gauge transformation, I have never heard of them before)

[swansont]

But the ZPE of a harmonic oscillator is infinite, since it's 1/2 hbar*omega for an infinite number of frequencies. But since changes in energy are what's important, we ignore that.

[Tom Mattson]

hmm' date=' you see if you look at wiki or hyperphysics they say that ZPE is minimum energy for a quantum harmonic oscillator, is what we're running into here the fact that you are not talking about a harmonic oscillator?

[/quote']

 

The harmonic oscillator is a special case. Any bound system has a ZPE.

 

So could the energy of an electron (as opposed to the diatomic harmonic oscillator) have a ZPE of 0?

 

Yes. As I said you can always transform the potential to "move the zero" to any point.

 

If so how does that fit in with HUP because it would be totaly stationary, or would measuring it's position give it energy?

 

What makes you think that a particle with zero energy would be totally stationary? Take a particle of mass m. Let's its KE have the value mc², and its PE have the value 2mc². What's its total energy? Zero. Is it stationary? No.

 

(I just briefly looked up about gauge transformation, I have never heard of them before)

 

Could you tell me what physics courses you've taken? It might help.

[Tom Mattson]

But the ZPE of a harmonic oscillator is infinite, since it's 1/2 hbar*omega for an infinite number of frequencies. But since changes in energy are what's important, we ignore that.

 

Huh? I thought that the ZPE was the minimum energy a particle can have. The spectrum for the SHO is [math]E_n=(n+1/2)\hbar \omega[/math]. The ZPE corresponds to n=0, which is a finite energy.

[5614]

I thought that the ZPE was the minimum energy a particle can have

Me too... but then I thought that ZPE was, well, >0

 

What I think swansont means is this:

from here: http://www.scienceforums.net/forums/showthread.php?t=4415

If you solve the QM "particle in a box" for the universe (solve it for length L and then let L get very big) you get an answer that looks like (n +1/2)hbar*w for each of the infinite number of modes. n is the occupation number' date=' or number of photons in that mode, and w is the frequency.

 

That means that even if there are no photons, there is an energy of hbar*w/2, and since there are an infinite number of terms, that sum is infinite. Physicists have long recognized that this infinity is meaningless and that the relevant energy quantity is the difference between two states.[/quote']

 

But anyway, I thought ZPE was simply the energy a particle retains at absolute zero, or the lowest amount of energy a particle could have.

 

So looking here:

 

Let's its KE have the value mc2, and its PE have the value 2mc2. What's its total energy? Zero. Is it stationary? No.

So we have KE = 2PE where KE>0 (assume we are not talking photons), but how does this make its total energy 0? I think my prob is more with the potential energy, if it has KE (>0) and PE in a ratio of 1:2 how is the total e=0 ?

[swansont]

Me too... but then I thought that ZPE was' date=' well, >0

 

What I think swansont means is this:

from here: http://www.scienceforums.net/forums/showthread.php?t=4415

[/quote']

 

Yes, that was the context of my comment.

 

But anyway' date=' I thought ZPE was simply the energy a particle retains at absolute zero, or the lowest amount of energy a particle could have.

[/quote']

 

I guess it would be helpful to define the system that's being discussed. When I hear ZPE I think of the vacuum energy by default, so I kinda locked in on that. In a bound system that was described in the OP, the lowest energy state is just the ground state. For a particle in a bound syste that can be treated as a harmonic oscillator, the energy is 1/2 hbar*omega.

[5614]

Well I just really read about Casimir effect, and, well... wow! Physics rocks! But I did know about vacum energy before, I just knew it existed and not much more...

 

So vacuum has a minimum energy.

 

Particle in a box or quantum harmonic oscillator has energy, well, I want to see the response to the end of post #20 first.

 

For an atom can e=0 be true? Because Pauli exclusion, you can't have all the electrons in the lowest energy level (so assume atom has >1 electron) can an atom have e=0, I don't think so. But for a single electron atom, could e=0?

[Tom Mattson]

So we have KE = 2PE where KE>0 (assume we are not talking photons)' date=' but how does this make its total energy 0? I think my prob is more with the potential energy, if it has KE (>0) and PE in a ratio of 1:2 how is the total e=0 ?[/quote']

 

Ugh, I made a mistake. The PE should have been -2mc². So we have for total energy:

 

Rest Mass Energy + KE + PE = mc² + mc² - 2mc² = 0

 

The total energy is zero, but the KE is nonzero (which means that the particle is not stationary).

[5614]

OK, so what particle are we talking about here (atom/electron/photon/vacuum)? And how do we achieve this negative potential?

 

But otherwise, fair enough... I get it.

 

And then (as above):

For an atom can e=0 be true? Because Pauli exclusion, you can't have all the electrons in the lowest energy level (so assume atom has >1 electron) can an atom have e=0, I don't think so. But for a single electron atom, could e=0?

[Tom Mattson]

OK' date=' so what particle are we talking about here (atom/electron/photon/vacuum)?

[/quote']

 

Any particle with mass will fit the bill.

 

And how do we achieve this negative potential?

 

Just let the particle be sujected to any potential, and do a transformation on it like I described in Post #5. The two potentials (V and V^*) give rise to precisely the same dynamics.

 

And then (as above):

For an atom can e=0 be true? Because Pauli exclusion, you can't have all the electrons in the lowest energy level (so assume atom has >1 electron) can an atom have e=0, I don't think so.

 

Yes, an atom can be said to have zero total energy. As I keep saying, you can add any constant you like to the potential function and it does not make one lick of difference to the physics of the atom. All you have to do is define that constant to be such that the total energy is zero.

 

But for a single electron atom, could e=0?

 

Yes.