So to start off, a disclaimer: I don't have any qualifications in physics whatsoever, but I have taken elementary physics courses up to quantum mechanics, so I have an amateur interest in the topic, being an Index/Railgun fan myself.
As the OP noted, the to aru wiki gives a very vague explanation about the Meltdowner ability and they claim it has to do with "stopping" electrons. From a quantum standpoint, this is nonsense, since by the Heisenberg's uncertainty principle, a "stopped" electron would simply be a standing wave that spans infinite space. Clearly this does not match up with the anime depiction of Mugino's ability. So I thought I would throw all that out the window and start with the assumption that Mugino has the ability to control electrons (more details on what that actually entails later).
There are several properties of the Meltdowner we need to explain:
heat: clearly it is very hot, given its name, and how it is seemingly able to ignite flammable materials
piercing/blocking: In addition, it seems to have the ability to "phase through" metal with a short delay besides actually melting it. She is also able to form a shield that instantaneously blocks projectiles (as opposed to simply melting them)
stationary: Mugino is able to hold her Meltdowner in a ball-shaped stationary form
beam: She is then able to fire off beams, while maintiaining the stationary form.
So here is my assumption: Mugino is able to control not just how electrons move, but also manipulate their quantum state. To explain her abilities, I will use the concepts of wave-particle duality of nature and quantum superposition.
According to the wave-particle duality, an electron is both a wave and a particle. From a quantum perspective, "wave-ness" and "particle-ness" are just properties of the electron. So if we observe an electron in a certain way, it may collapse to either a wave or a particle state, but without measurement, it just exists in a superposition of the two. In nature, an electron has very little mass (the defining characteristic of a particle) and quite a large wavelength. The Meltdowner is very clearly not a wave-type attack in the slightest; it is either a beam or a ball of electrons.
So what might be happening is that Mugino forces electrons to behave more like particles than waves to the extent that their wavelength is so long that it isn't even noticeable. On the other hand, the electrons become significantly more massive. The massive electrons give the Meltdowner its piercing/blocking properties quite trivially; you can block or break through stuff with more stuff.
The Meltdowner is hot because even though the electrons aren't moving, their kinetic energy remains the same. However, as a very large particle with very high mass and very little wavelength, this means each electron stores excessive energy, which it dissipates by vibrating in place. The green glow of the Meltdowner is not the electrons themselves, but the surrounding air being heated up.
Finally, the transition between stationary/beam forms. Heisenberg's uncertainty principle states that there is a trade-off between knowing momentum and knowing position. With control over the states of electrons, Mugino can essentially change the measurement. If she needs the electrons in stationary mode, she measures their position. If she needs them in beam mode, she measures their velocity.
