In quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.
Dielectric surface loss from junction wiring represents a critical secondary limit for superconducting transmon coherence. We present a quasi-one-dimensional analytical framework to minimise this loss, enabling rapid optimisation without computationally expensive 3D simulations. We compare uniform strips (V1) against linear (V2–V3) and optimised hybrid tapers (V4–V5). We demonstrate that geometric tapering suppresses wiring participation by up to 99.6%, reducing it from 0.0756 ppm (V1) to 0.0003 ppm (V4). Crucially, however, a simple linear taper (T1 ≈ 127.306 μs) yields coherence virtually indistinguishable from theoretically optimal complex profiles (T1 ≈ 127.323 μs) in the current pad-dominated regime. We thus establish a definitive design rule: standard linear tapering is sufficient to eliminate wiring loss as a bottleneck, rendering fabrication-sensitive complex shapes unnecessary for next-generation, low-loss devices.
This record provides an index and dependency declaration for a one-way trilogy of works: (1) Minimal Ontological Foundation (MOF v3)(2) Phase–Flow Coherence (PFC v4)(3) Alpha Locking (v6) The trilogy establishes a finite-resolution rigidity chain, proceeding strictly in one direction. MOF v3 fixes the minimal ontological conditions for world-formation, from which irreversible internal update order (time), internal projection (measurement), and structural probability follow as necessary consequences. PFC v4 fixes the unique internally coherent representation of dynamics and probability under those conditions, eliminating all alternatives except Schrödinger-type bookkeeping with Born-rule weighting. Alpha Locking v6 completes the chain by showing that finite-resolution update structures cannot terminate arbitrarily. A unique minimal admissible depth is forced under purely discrete accept/reject predicates, leaving the fine-structure constant as a structural residue. No result depends on downstream assumptions.No fitting, tuning, or optimization is introduced at any stage. This overview introduces no new claims and contains no additional results.It serves solely to fix reading order, dependency, and scope.