# Energy of free particles not quantized ?

#### DarthVolta

Joined Jan 27, 2015
521
For ages I read about how when you measure things on the atomic scale and they seem quantized, in multiples of Plank's constant , like E=hv. I don't know much QM on the math level, I need a lot more classical mechaincs 1st, and more time LOL.

And only recently I heard a physicist say that free particles DO NOT have to have quantized energy ?

Before I imagained that if you accelerated a free electron in an E field, that overall, it's energy/velocity/momentum, did seem to be quantized. Even if the measurement of such a particle seems to be only as accurate as the Heisenberg relations (at this point in history), does modern physics say that free particles can have any real valued speed/energy/etc ? Like choosing from the real numbers, vs the integers ?

I "know" how in bounded systems they end up with quantized states, and most particles 'emitted/absorbed' might start out with a quantized energy level/momentum/etc

I suppose I'll die before learning what this place is really about. But I want to know !

#### Papabravo

Joined Feb 24, 2006
14,857
Quantization occurs because of constraints in space or time. A free particle has no such constraints. Bound particles do.
Q: Is a conduction electron in a copper wire free or bound?

#### nsaspook

Joined Aug 27, 2009
7,869
Interesting question. I would imagine in a space with zero potential energy a free particle with no forces acting on it is equivalent to a classical particle that can have any possible classical energy state.

#### Delta prime

Joined Nov 15, 2019
610
Potential energy is the energy possesed by a particle by virtue of its position in a field.Now we calculate potential energy relative to a standard energy at a point.This point is to be considered the zero of potential energy, though the effect of field is present its still neglected as it would be small compared to any other point in the field.Usually 0 of potential energy will be placed at infinity,& you never discuss infinity with a mathamatision because you'll never here the end of it.
Zero point I like to think it means there is only a single particle in the whole universe and it contradics Newtons 2nd law as mass must be defined as a ratio of 2 quantities.So there can actually never be any particle having zero potential energy. Then again.. never say never.

#### cmartinez

Joined Jan 17, 2007
7,219
So there can actually never be any particle having zero potential energy.
Which translates into a philosophical assertion: something that exists cannot manifest non-existence ... or at least that's the way I see it ...

#### Delta prime

Joined Nov 15, 2019
610
Which translates into a philosophical assertion
To to be self aware of one's existence
self-consciousness partly definitive of the very concept of a person, a person being "a thinking intelligent Being, that has reason and reflection, and can consider it self as it's self. Now that's philosophy baby!

#### bogosort

Joined Sep 24, 2011
571
And only recently I heard a physicist say that free particles DO NOT have to have quantized energy ?

Before I imagained that if you accelerated a free electron in an E field, that overall, it's energy/velocity/momentum, did seem to be quantized. Even if the measurement of such a particle seems to be only as accurate as the Heisenberg relations (at this point in history), does modern physics say that free particles can have any real valued speed/energy/etc ? Like choosing from the real numbers, vs the integers ?
Indeed, the definition of a "free particle" is that it's total energy is simply its kinetic energy; that is, free particles are not subjected to any potentials. And without a potential, the solutions to its Schrodinger equation are unconstrained sine waves, which -- when plugged back into the particle's Hamiltonian -- result in unconstrained (unquantized) energy values.

But such solutions are unphysical in the quantum formalism -- they are non-normalizable and so they cannot represent eigenstates in the square-integrable Hilbert space of states.

So, how should we interpret this? Well, if we look to QFTs, the answer becomes clear: there are no such things as "free particles". Every particle is a non-zero energy state of a quantum field, and so -- by definition -- there is always a potential. Hence, energy is always quantized.

#### DarthVolta

Joined Jan 27, 2015
521
Thanks, I've seen these names before, so I informally follow. So if some proton was created in a quantized system, and travels across the universe, getting accelerated by electrons here and there, their charge is quantized, and so would the number of them be an integer, so IDK what it would mean to accelrate the proton to a non-quantized amount. Also I guess the Plank-scale of measurment would also come back around to saying a given particle/wave seems quanitized.

Now that I know matrices, I want to try the 1st chapter of a QM book I have, it does only use lin.alg, calculus and diff. eqn's.
so I can do most of it. Years ago when I tried, I'd get frustrated when they'd say stuff about a full set of vectors in an H-space.
I don't know Classical mechanics/Field theory, the only real time I used and could sort of follow Lagrange/Hamiltion eqns was in some QFT....but I didn't know what I was doing, I was just following the calculus

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#### ColonelJ

Joined May 22, 2020
1
I think as far as the Time-Space plane is supposed to be a continous domain, we have nothing in a thorough meaning of quantized or discrete level.
but we can take advantage of quantization or discreteness in special physical conditions. like what happened in analog ==> digital migration for electronics, it's gonna happen for classic ==> quantum in physics.
there's also some difference between fundamental particles like Fermions and Bosons in behaviour in think, that you can read more in Solid State physic books and articles.