Why Isn't Montgomery Modular Exponentiation Considered For Use In Quantum Factoring?

How is "force" defined in quantum mechanics?

• Or perhaps simply, in a modern sense? I have always been curious what causes the repulsive force between matter. I soon came across Pauli repulsion, which explains the answer... But apparently it is not considered a "true force." Someone else posted an answer on Quora asking why this is so, and I delved through that. I think none of the answers quite hit on what was bugging me. (In fact, some of them seemed to agree that there is an inconsistency in terminology.) I think I was particularly confused at first because I have a classical definition of force in my mind. F =m a. A force is some influence that changes the motion of some mass. By this definition, Pauli repulsion IS a force. Friction, buoyancy, and the floor under my feet are all a testament to this. Yet in modern physics there are only 4 [apparently] "true forces." (I had expected the answer to my original question would involve electromagnetism or the nuclear forces.) So... how is "force" defined in quantum mechanics? What makes those "true forces" forces while Pauli repulsion is not? Is there any way of reconciling the idea that a particle's motion can be altered by something that isn't a force? (Or is something else meant entirely by "true force?") Does quantum mechanics have a means of addressing contact "forces" (like friction)? Perhaps most of the answer is that it's an issue of terminology, which is fine. I'm just curious what the terms ARE in that case?

• Answer:

  In nonrelativistic quantum mechanics forces are added into the Hamiltonian of the system in terms of potenials. F=maF=maF=ma can be written in the form −dU(x)dx=ma−dU(x)dx=ma-\frac{dU(x)}{dx}=ma. In this second form the force is expressed as a potential energy function of space. The hamiltonian of a system is H=T+UH=T+UH=T+U where TTT is the kinetic energy and UUU is the potential energy. Schrödinger equation, the equation that determines the time evolution of the state ψψ\psi of a quantum system (when there is no measurement) is −iℏ∂ψ(x,t)∂t=Hψ(x,t)−iℏ∂ψ(x,t)∂t=Hψ(x,t)-i\hbar \frac{\partial \psi(x,t)}{\partial t}=H\psi(x,t) For a free particle U=0U=0U=0, and for a particle under the force expressed by the Hook's law F=−kxF=−kxF=-kx, U(x)=kx2/2U(x)=kx2/2U(x)=kx^2/2. The probabilistic distributions of any physical quantity, such as position and momentum, are calculated using this ψψ\psi. This is how forces are included into the dynamics. Pauli exclusion principle is not a force in this sense. It just says that two identical fermions cannot be in the same quantum state. Therefore, in a bounded system, such as an atom, two electrons cannot have the same energy, angular momentum, and spin quantum numbers. But the reason is not that there is a force which generates a repulsion between electrons. In the formulation, they simply can't be in the same state, because otherwise their wavefunctions will cancel them out. The physical force that is responsible for friction and touch is the electromagnetic force. Electrons at the surface of "touching" object repel each other.