# Can a particle have momentum without energy?

Can a particle have linear momentum if the total energy of the particle is zero? Even if a particle has a certain velocity, can its potential energy cancel out the kinetic energy as to add to zero ?

• "Total energy" is not an absolute quantity in most contexts. You can set the energy level of zero to wherever you like without changing the physical results. Note that momentum is not an absolute quantity either, since it depends on the frame of reference. – Mark Adler Dec 15 '13 at 1:33

As weird as it sounds, the answer is "yes."

Take, for instance, a satellite in gravitational orbit around some heavy body. It's energy is given by $$H=\frac{p^2}{2m}-\frac{GMm}{r}$$ Clearly, there are solutions to this equation which have $0$ energy (look at a slowly moving particle that's really far away), but those solutions necessarily involve a non-zero momentum.

This answer may seem artificial because it also allows for negative energies (oh! horror of horrors), but mechanically, this gives all of the correct equations of motions. The details are very enlightening, too: positive energy orbits correspond to hyperbolae (unbound orbits, scattering orbits), negative energies correspond to ellipses (bound orbits like those described by Kepler's Laws or, colloquially, just "orbits"), and zero energies correspond to parabolae. Parabolic orbits are non-periodic, but never escape the effective gravity of the heavy mass (unlike hyperbolic orbits). These shapes, of course, all collapse into lines when the particle has zero angular momentum.

Edit: As David mentions, in relativity a free particle with zero energy simply does not make sense because it would have to be both massless and without momentum. Massless particles are massless in all reference frames, so the particle would have to be momentum-less in all reference frames as well (which sounds like a pretty boring particle to me). But if you include interaction potentials in your definition of a particle's energy, positive, negative, and zero energies are possible once again.

• Why artificial? Why horror? Potential energy is often negative, by definition. Total energy (including potential energy) then does not need to be positive. – Ján Lalinský Dec 14 '13 at 23:23
• @JánLalinský: Oh, no reason. Just a little bit of light-heartedness at the expense of low-level physics pedagogy. Many introductory-level physics courses always define energy as being a positive scalar to facilitate students' first steps into energy conservation. I was merely addressing the likely perspective from which this question was being asked and making a little joke at its expense. As I said, even if this way of thinking about energy seems unusual to the neophyte it gives the correct equations of motion - which is all that matters. – Geoffrey Dec 14 '13 at 23:34
• Interesting. Can you post a reference (textbook)? I'd like to read an example. Some people tend to think of energy as of some kind of stuff, like potato mash with positively semi-definite density. Perhaps those textbooks are partially responsible... – Ján Lalinský Dec 14 '13 at 23:38
• I found this with a Google search. Ch. 9, pg.148 is a good place to start. He mentions that PE (although not total energy) could be negative but then says "to avoid negative numbers... always choose the lowest level as the zero potential mark." Moreover, he defines energy as "the capacity to do work" earlier in the chapter, which subtly implies that energy can only be positive (how can you have a negative capacity to do something?). Even Feynman's famous "toy blocks" analogy reinforces the idea of energy as positive (what's a negative block?). – Geoffrey Dec 15 '13 at 0:14
• Thanks.I agree that choice of the min. possible energy as zero energy is useful for beginning students. The capacity to do work is a nice idea too. But there are cases when other aspects lead one to use concept of energy which can be negative - like the gravitational potential energy you mentioned above or electrostatic potential energy of positive and negative particle. One should not create fear of negative energies in students, they are very useful. For example, electromagnetic energy of hydrogen atom is negative - hence the hydrogen atom has lower mass than simple sum of masses of e and p. – Ján Lalinský Dec 15 '13 at 0:49

As I cannot post any comments I have to post this as an answer although the essential points were already given:

In classical mechanics energy itself was no meaning. Only energy differences have a physical interpretation. Thus in the classical case energy is only defined up to an arbitrary constant. So any fixed state's energy can be set to zero (but of course not all state's at the same time).

As correctly stated by David, in special relativity, this constant has to be chosen in a particular way (As energy there is directly connected to mass. Mass of course also has a meaning without looking at mass differences).

So the answer really depends on the context here.

No. According to the energy-momentum relation, the magnitudes of the energy and momentum of a particle of rest mass $m$ are related through the equation $E^2=p^2+m^2$. Obviously $0\leq p^2\leq p^2+m^2=E^2$, so if the energy is zero we have $0\leq p^2\leq0$, or $p=0$.

• I'm not sure how correct this answer is. The formula you wrote is only valid for a free particle. For a scalar particle under an EM field the energy is $E=\sqrt{(\textbf{p}-\textbf{Q})^2+m^2}+U$, so it could be negative. – jinawee Dec 14 '13 at 22:23

yes a body can have momentum even its total mechanical energy is zero as all energy are relative quantity so if a body is not bound to any system then momentum can be exist if observer is present on that body itself which is not bound to the conservative force