I have seen in many places that the equation $E=mc^2$ is useful for describing subatomic particles as well, and, this is the basis of Nuclear reaction. However, to my understanding, this equation is from the field of relativity and relativity and NRQM are not directly compatible.
So, I suppose there is different explanation of this in NRQM and in Relativity. Hence, my question, how does the explanation of $E=mc^2$ differ in relativity and QM?
However, to my understanding, this equation is from the field of relativity, and relativity and Quantum mechanics are not directly compatible.
Exactly! That's the reason, one relies on the Quantum Field Theory when the relativistic effect gets important. The equation $$E=mc^2$$ has the same meaning in both quantum mechanics and relativity. Quantum mechanics simply ignores this equation and assumes that the particle number is conserved. One can consider the applicability of this assumption by assuming that if you trap a particle in the size of box $L$ then $$\Delta p\geq \frac{\hbar}{L}\rightarrow \Delta E\geq \frac{\hbar c}{L}$$
When this uncertainty in the energy exceeds $\Delta E=2mc^2$, then the relativity comes in (Einstein's relation) and particle and anti-particle can pop out of nothing. At this scale, one has to consider the relativistic effects.
It's not quite right that relativity and non-relativistic QM are not compatible. Non-relativistic QM is an approximation of relativistic quantum field theory. For instance if you take the appropriate slow speed limit of the relativistic Dirac Equation or Klein-Gordon Equation, you get the Schrodinger equation.
As another example, if you want to calculate the de Broglie wavelength of a particle, the correct way to do it uses the relativistic definition momentum. For slow moving particles, the classical $p=mv$ is a very good approximation. But for fast particles, like an electron with $1$ MeV of kinetic energy, using the relativistic momentum is necessary.
Non-relativistic QM assumes that mass and particle number are conserved. This makes it trickier to model things like nuclear decay, where one particle splits into two and some rest energy is converted to kinetic energy. A non-relativistic QM model of this would have to start with two particles. Each particle would have fixed mass during the decay, and the nuclear binding energy would account for the mass defect and kinetic energy.
For instance, to model the alpha decay of Uranium-238, you could start a bound state of Thorium-234 and an alpha particle. This composite system would be your model of the Uranium nucleus. The two would be bound together in a finite potential well of some kind. The nuclear decay would be a quantum tunneling event where the alpha particle tunnels through the potential barrier and escapes. Given the observed decay rate of U-238, you could fit some shape parameter of your model potential, so that the tunneling probability gives the correct decay rate.
