The Schrödinger Equation

The time-dependent Schrödinger equation is the postulated law of motion for a quantum state: it fixes the rate of change of ψ in time, through a factor of i, from the curvature of ψ in space plus the potential energy at that point. Use the supplied topic model Y_out = exp(-5.7884*q^2*s*cosd(th)), where Y_out is AMPLITUDE |ψ|, 1 AT t = 0 (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ; sind/cosd/tand use degrees. th is Phase θ of the coefficient C, measured in °. Keep q (Wavenumber k) = 1 10^10 m^-1; s (Elapsed time t) = 0.2 fs fixed. For this exercise use only 9.5625 ° ≤ th ≤ 80.4375 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.49658895, with a bounded reading uncertainty ±0.026592. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0070875 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

A wavefunction is admissible when it is single-valued, continuous, and square-integrable, and normalising it means dividing by √(∫|ψ|²dx) so that the probability of finding the particle somewhere is exactly one. Use the supplied topic model Y_out = pi/max(k,0.01), where Y_out is FRINGE SPACING π / k (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. k is Wavenumber k, measured in nm⁻¹. Keep r (Reflected amplitude r = |B|/|A|) = 0.6 fixed. For this exercise use only 2.6375 nm⁻¹ ≤ k ≤ 7.3625 nm⁻¹ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.56915488 nm, with a bounded reading uncertainty ±0.021368 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725 nm⁻¹, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Separation of variables writes a solution as a function of position alone multiplied by a function of time alone, which reduces the time-dependent Schrödinger equation to the eigenvalue equation Ĥψ = Eψ — but only when the potential has no explicit time dependence. Use the supplied topic model Y_out = cosd(th), where Y_out is REAL PART Re φ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. th is Phase θ = Eₙt/ħ, measured in °. Keep n (Level n in ψₙ) = 2 fixed. For this exercise use only 189.45 ° ≤ th ≤ 321.75 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.0026703506, with a bounded reading uncertainty ±0.1174. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.01323 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

A stationary state is a single energy eigenfunction carrying the phase e^(−iEt/ħ); its probability density and all its expectation values are frozen, so only a superposition of different energies can make anything move. Use the supplied topic model Y_out = 0.5*(0.5+2*sqrt(p*(1-p))*cos(2*pi*t*(n^2-1)/2.7496)*((cos(pi*(n-1))-1)/(pi*(n-1))^2-(cos(pi*(n+1))-1)/(pi*(n+1))^2)), where Y_out is MEAN POSITION ⟨x⟩ (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. t is Time t since preparation, measured in fs. Keep p (Weight |c₂|² on the upper level) = 0.5; n (Upper level n) = 2 fixed. For this exercise use only 0.10625 fs ≤ t ≤ 0.42125 fs on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.29117734 nm, with a bounded reading uncertainty ±0.0069002 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000315 fs, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.