Qm One Dimensional Potentials — Examples

Work in the declared line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with mass parameter $m>0$, action constant $\mathrm{\hslash }>0$, and lengths, energies and currents read in the units of Dimensions, SI unit conventions and quantity types of the framework. Declare the incident energy $E>0$ and the barrier height $U:=2E$, so that $\kappa :=\sqrt{2m(U-E)}\mathrm{/}\mathrm{\hslash }=\sqrt{2mE}\mathrm{/}\mathrm{\hslash }=:k$; declare further the barrier width $a:=\ln (10)\mathrm{/}\kappa >0$, so that $\kappa a=\ln 10$ and $e^{\kappa a}=10$. Consider the scattering solution of Rectangular barrier: scattering solution, flux balance and transmission for this barrier and set $q:=e^{-2\kappa a}$. Then, as exact identities in the model:

(i) $C:=U^2\mathrm{/}(4E(U-E))=1$, $q=1\mathrm{/}100$, and $T=\frac{400}{10201},R=1-T=\frac{9801}{10201};$

(ii) with rational brackets, $\frac{3921}{100000}<T<\frac{3922}{100000},\frac{96078}{100000}<R<\frac{96079}{100000};$

(iii) the thick-barrier approximation value is $\frac{16E(U-E)}{U^2}q=\frac{1}{25}$, the exact difference is $T-\frac{1}{25}=-\frac{201}{255025},$ which lies between $-0.000789$ and $-0.000788$, and the error bound of the theorem reads $\mid T-\frac{1}{25}\mid \le \frac{201}{125000}=0.001608;$

(iv) since $q=1\mathrm{/}100>0$, the transmission $T$ is positive although the declared barrier height satisfies $U>E$; and by clause (b) of the barrier theorem the number $T$ is the transmitted fraction of the reference current $j_{\mathrm{i}\mathrm{n}}=(\mathrm{\hslash }k\mathrm{/}m)\mathrm{\mid }{A}_{\mathrm{i}\mathrm{n}}{\mathrm{\mid }}^2$, with $A_{\mathrm{i}\mathrm{n}}=1{\mathrm{m}}^{-1\mathrm{/}2}$, while $R$ is the reflected fraction.

The example is an exact computation in the declared barrier model with declared data $U,E,a,m,\mathrm{\hslash }$ related as above; it records no measurement and no apparatus value.

• Why this scaling is exactly solvable. At $U=2E$ the factor $U^2\mathrm{/}(4E(U-E))$ equals one and $\kappa =k$, so the general barrier formula collapses to ${\mathrm{sech}}^2(\kappa a)$, and the width $a=\ln (10)\mathrm{/}\kappa$ makes the hyperbolic cosine rational.
• What the check shows and what it does not. The computation verifies the exact formulas, the rational brackets and the published error bound of the barrier theorem at one declared parameter point. It does not establish the accuracy of the physical model, does not compare with data and produces no experimental conclusion.

Work in the declared line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with mass parameter $m>0$ and action constant $\mathrm{\hslash }>0$, energies, lengths and the depth $U$ read in the units of Dimensions, SI unit conventions and quantity types of the framework, and consider the finite square well $V(x)=-U$ for $\mathrm{\mid }x\mathrm{\mid }\le a$, $V(x)=0$ for $\mathrm{\mid }x\mathrm{\mid }>a$ of Finite square well: bound-state conditions, simplicity and count with half-width $a>0$ and depth $U>0$.

(i) Exact boundary case of the first branch. Declare $U:=\frac{{\mathrm{\hslash }}^2{\pi }^2}{16m{a}^2}\mathrm{.}$ Then $K=\sqrt{2mU}\mathrm{/}\mathrm{\hslash }=\pi \sqrt{2}\mathrm{/}(4a)$, the count parameter is $2aK\mathrm{/}\pi =\sqrt{2}\mathrm{/}2$, there is exactly one bound state, it is an even (cosine-type) state, and its binding energy is exactly $E=-\frac{U}{2}=-\frac{{\mathrm{\hslash }}^2{\pi }^2}{32m{a}^2}\mathrm{.}$ For this declared depth the root of the even condition is exactly $k=\kappa =\pi \mathrm{/}(4a)$.

(ii) Exact counts at declared depths. With $U:=\frac{25{\mathrm{\hslash }}^2{\pi }^2}{32m{a}^2}$ one has $K=5\pi \mathrm{/}(4a)$ and $2aK\mathrm{/}\pi =5\mathrm{/}2$, so there are exactly three bound states, with types even, odd and even in order of increasing energy. With $U:={\mathrm{\hslash }}^2{\pi }^2\mathrm{/}(8m{a}^2)$ one has $K=\pi \mathrm{/}(2a)$ and exactly one bound state; with $U:={\mathrm{\hslash }}^2{\pi }^2\mathrm{/}(2m{a}^2)$ one has $K=\pi \mathrm{/}a$ and exactly two bound states. For every declared depth $U>0$ there is at least one bound state.

(iii) Rational bracket of the first case. $\frac{707}{1000}<\frac{2aK}{\pi }=\frac{\sqrt{2}}{2}<\frac{708}{1000},$ so the count $\lceil 2aK\mathrm{/}\pi \rceil$ equals one in case (i).

The example is a conditional computation in the declared well model and asserts no measured binding energy; the identities in (i) and (ii) are exact in the model at the displayed declared data.

• Why the first case is exactly solvable. At the declared depth $K=\pi \sqrt{2}\mathrm{/}(4a)$ the radius circle of the even root condition passes through the point $k=\kappa =\pi \mathrm{/}(4a)$ of the first branch, where $\tan (ka)=1$, so the transcendental root condition is satisfied by an exact rational multiple of $1\mathrm{/}a$ and the signed binding energy is exactly $E=-U\mathrm{/}2$ (its positive magnitude is $U\mathrm{/}2$).
• What the counts show and what they do not. The comparisons verify the branch criterion and the ceiling formula at declared depths. They are not a numerical solution of the transcendental equations at generic depths, do not compare with data and assert nothing about any physical well.

Work in the declared Dirichlet interval model The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model clause (b) with $0<L<\mathrm{\infty }$, mass $m>0$, action constant $\mathrm{\hslash }>0$, and energies in joules under Dimensions, SI unit conventions and quantity types of the framework. Then, as exact identities in the model:

(i) Free box. For $V=0$ the eigenpairs of the assigned energy operator $H_{D,0}$ are exactly $({\mathcal{E}}_n,e_n),{\mathcal{E}}_n=\frac{{\mathrm{\hslash }}^2{\pi }^2{n}^2}{2m{L}^2},e_n(x)=\sqrt{\frac{2}{L}}\sin \text{\hspace{0.17em}⁣}\left(\frac{n\pi x}{L}\right),n\ge 1,$ the family $(e_n{)}_{n\ge 1}$ is a complete orthonormal family of $L^2(0,L;\mathbb{C})$, each eigenvalue is simple, and $e_n$ has exactly $n-1$ zeros in $(0,L)$, namely the points $x_k=kL\mathrm{/}n$, $k=1,\dots ,n-1$.

(ii) Constant shift. If $V=c$ is the constant potential of value $c\in \mathbb{R}$ (joules), then $H_{D,c}$ has exactly the same eigenfunctions $e_n$ and the eigenvalues ${\mathcal{E}}_n+c$; consequently the min-max comparison bound of the interval theorem is attained with equality, $\mathrm{\mid }{\mathcal{E}}_n+c-{\mathcal{E}}_n\mathrm{\mid }=\mathrm{\mid }c\mathrm{\mid }=\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$ for every $n$.

(iii) Gaps. ${\mathcal{E}}_{n+1}-{\mathcal{E}}_n=\frac{{\mathrm{\hslash }}^2{\pi }^2(2n+1)}{2m{L}^2}>0$, so the spectrum is strictly increasing.

The example is an exact conditional computation in the declared box model and asserts no measured transition energy.

• Attainment is not generic. The equality in (ii) is a property of the constant potential, for which the perturbation commutes with the free box operator. For a nonconstant $V$ the min-max bound is an inequality and its attainment is not claimed.
• What is not asserted. The example treats the free and constantly shifted Dirichlet box only. It does not treat Robin conditions, singular perturbations or a material wall, and it reports no data.

The three computations. Tunnelling through a rectangular barrier: an exactly solvable scaling and its thick-barrier comparison treats the rectangular barrier of Rectangular barrier: scattering solution, flux balance and transmission at the exactly solvable scaling $U=2E$ with $\kappa a=\ln 10$: it obtains $T=400\mathrm{/}10201$ and $R=9801\mathrm{/}10201$ as exact rationals, brackets them by $3921\mathrm{/}100000<T<3922\mathrm{/}100000$, compares the exact value with the thick-barrier value $1\mathrm{/}25$ (the difference is exactly $-201\mathrm{/}255025$) and verifies the theorem's error bound $201\mathrm{/}125000=0.001608$; the computation is a finite check of the model formulas, including the positivity of transmission through a classically forbidden barrier. Finite square well: an exactly solvable bound state and exact bound-state counts exhibits a declared well depth at which the unique bound state is exactly solvable (even state, $k=\kappa =\pi \mathrm{/}(4a)$, binding energy $E=-U\mathrm{/}2=-{\mathrm{\hslash }}^2{\pi }^2\mathrm{/}(32m{a}^2)$), verifies the even root condition with the special value $\tan (\pi \mathrm{/}4)=1$, and verifies the count and branch criterion at declared depths: exactly three bound states with types even, odd, even when $2aK\mathrm{/}\pi =5\mathrm{/}2$, exactly one when $2aK\mathrm{/}\pi =1$, exactly two when $2aK\mathrm{/}\pi =2$, and at least one for every $a,U>0$. The Dirichlet box: exact spectrum, nodes, constant shift and gaps records the exact free Dirichlet box eigenpairs ${\mathcal{E}}_n={\mathrm{\hslash }}^2{\pi }^2{n}^2\mathrm{/}(2m{L}^2)$ with sine eigenfunctions, their explicit $n-1$ nodal points $kL\mathrm{/}n$, the exact spectrum ${\mathcal{E}}_n+c$ of a constant potential with the min-max bound attained, and the positive gaps ${\mathrm{\hslash }}^2{\pi }^2(2n+1)\mathrm{/}(2m{L}^2)$.

What is and is not compared. The comparisons are internal: the exact barrier formula and its thick-barrier approximation, the exact well root condition and the exact branch count, and the exact box spectrum with the min-max bound. They verify the cited theorems at declared parameter points by exact rational and algebraic checks; they are not numerical approximations of generic transcendental roots, they contain no simulation presented as data, and they report no measured value, no uncertainty and no significance statement. The displayed decimals are rational bracket endpoints or exact terminating decimal expansions of rational values, including $201\mathrm{/}125000=0.001608$; each is justified in the verifications.

Evidence status. All three items are conditional computations in the declared line, well and box models of the pair's A page, under the same P1, P2, P4, P5 assumptions and the same cited mathematics; the empirical qualifications of the A page carry over unchanged. Nothing on this page is an observation, an apparatus model or a statistical inference, and agreement of the displayed identities with any measurement would be evidence about the combined model and apparatus assumptions rather than a proof of the model.