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In quantum mechanics a particle is represented by a wavefunction ψ(x) taking values in the complex numbers. We demand that all physical particles are in bound states where by definition. ψ(x) → 0 as |x| → ∞. ∫R3 ψ∗ψ d3x = I(ψ) < ∞.
As quantum mechanics is defined in a complex space, this does not pose any problem (we can have negative kinetic energies even if the total energy is positive) and we can solve for k′′ simply finding an imaginary number k′′ = iκ, κ = √2m ℏ2(VH − E) (with κ real).
$d/dx$ on ${\cal S}(\mathbb R)$ is unbounded. The shortest way to prove it is passing to Fourier transform.
Pure resonance occurs exactly when the natural internal frequency ω0 matches the natural external frequency ω, in which case all solutions of the differential equation are un-bounded. In Figure 20, this is illustrated for x′′(t) + 16x(t) = 8 cos 4t, which in (1) corresponds to ω = ω0 = 4 and F0 = 8. x. t.
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Mar 13, 2021 · Unbounded Solution refers to one solution being unbounded, in the sense that the y = x y = x is unbounded. Infinite Optimal Solutions refers to an infinite family of functions that are all optimal solutions, in the sense that x = π + 2nπ x = π + 2 n π is an infinity family of solutions to the problem min cos(x) min cos. .
Feb 11, 2023 · An unbounded solution space occurs when the objective function can be made arbitrarily large or small, and there is no optimal solution. A bounded solution space occurs when the optimal solution of the problem lies within a finite range.
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Jan 13, 2012 · There are physical reasons against considering unbounded observables, except for the Hamiltonian itself (which, as often remarked, is a case apart from other observables). The analysis of the measuring process by Wigner, and by Araki and Yanase, famously shows that unless an observable $Q$ commutes with the Hamiltonian, no measurement apparatus ...
