Saturday, August 4, 2012

Orthogonal: Riemannian Quantum Mechanics [Extra] - Greg Egan

"Units Throughout these notes, we've adopted a system where time and distance are measured in identical units. This is the equivalent of setting the speed of light, c, equal to 1 in our own universe (for example, by measuring distances in metres and using the time it takes for light to travel one metre as the corresponding unit of time). In the Riemannian universe, it amounts to choosing units for time such that Pythagoras's Theorem holds true, even when one side of the triangle involves an interval of time rather than space. In the novel Orthogonal, it is found empirically that this is the same as setting the speed of blue light equal to 1. In this section, we will go one step further and choose units for mass and energy such that the “reduced Planck's constant”, ℏ = h/(2π), is equal to 1. Mass and energy are then measured in units with the dimensions of inverse lengths or spatial frequencies — or, equally, inverse times or time frequencies. Our particular choice means that the Planck relationship between frequency ν and energy, E = h ν, becomes E = 2 π ν = ω, where ω is the angular frequency of the wave, and the relationship between spatial frequency κ and momentum is p = 2 π κ = k, where k is the angular spatial frequency. The maximum angular frequency ωm that appears in the Riemannian Scalar Wave equation is then simply equal to the rest mass of the associated particle. Relativistic Energy and Momentum Operators In the non-relativistic quantum mechanics we have discussed so far, we have simply applied the usual Schrödinger equation to the potential energy associated with the force between charged particles, on the basis that non-relativistic classical dynamics in the Riemannian universe is identical to Newtonian mechanics, so long as we treat kinetic energy as positive and choose the sign for the potential energy to be consistent with that. For Riemannian relativistic quantum mechanics, we will need to do things slightly differently. The structure of quantum mechanics in its usual formulation is closely linked to the Hamiltonian form of the corresponding classical mechanics, and in the Riemannian case the momentum conjugate to each coordinate in the Hamiltonian sense is the opposite of the relativistic momentum in the same direction."\ 5 out of 5

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