Qm One Dimensional Potentials

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $m,\mathrm{\hslash }>0$, let $\mathrm{\Omega }\subseteq \mathbb{R}$ and $I\subseteq \mathbb{R}$ be nonempty open intervals (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $V:\mathrm{\Omega }\to \mathbb{R}$ be a function. Let $u,v:\mathrm{\Omega }\times I\to \mathbb{R}$ be functions whose partial derivatives ${\mathrm{\partial }}_{t}u,{\mathrm{\partial }}_{t}v,{\mathrm{\partial }}_{x}u,{\mathrm{\partial }}_{x}v,{\mathrm{\partial }}_x^{2}u,{\mathrm{\partial }}_x^{2}v$ exist and are continuous on $\mathrm{\Omega }\times I$ (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set) and satisfy the pair of equations $\mathrm{\hslash }{\mathrm{\partial }}_{t}u=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^{2}v+Vv,-\mathrm{\hslash }{\mathrm{\partial }}_{t}v=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^{2}u+Vu$ pointwise on $\mathrm{\Omega }\times I$. Write $\psi :=u+iv$ for the corresponding $\mathbb{C}$-valued function with component functions $u$ and $v$ (Real and imaginary parts, complex conjugation, and modulus), so that the two displayed equations are the componentwise reading of $i\mathrm{\hslash }{\mathrm{\partial }}_t\psi =-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^2\psi +V\psi$. Put $n(x,t):=\mathrm{\mid }\psi (x,t){\mathrm{\mid }}^2=u(x,t{)}^2+v(x,t{)}^2,j(x,t):=\frac{\mathrm{\hslash }}{m}\mathrm{Im}\text{\hspace{0.17em}⁣}(\overline{\psi (x,t)}{\mathrm{\partial }}_x\psi (x,t))=\frac{\mathrm{\hslash }}{m}(u{\mathrm{\partial }}_{x}v-({\mathrm{\partial }}_{x}u)v)(x,t)\mathrm{.}$ Then:

(a) Continuity identity. ${\mathrm{\partial }}_{t}n(x,t)+{\mathrm{\partial }}_{x}j(x,t)=0$ for every $(x,t)\in \mathrm{\Omega }\times I$.

(b) Constant flux of stationary solutions. Separately, let $u_{\mathrm{s}},v_{\mathrm{s}}:\mathrm{\Omega }\to \mathbb{R}$ have continuous second derivatives, and let $E\in \mathbb{R}$ and $V:\mathrm{\Omega }\to \mathbb{R}$ satisfy $-\frac{{\mathrm{\hslash }}^2}{2m}{u}_{\mathrm{s}}^{\mathrm{\prime }\mathrm{\prime }}+V{u}_{\mathrm{s}}=E{u}_{\mathrm{s}}$ and $-\frac{{\mathrm{\hslash }}^2}{2m}{v}_{\mathrm{s}}^{\mathrm{\prime }\mathrm{\prime }}+V{v}_{\mathrm{s}}=E{v}_{\mathrm{s}}$ pointwise on $\mathrm{\Omega }$. Then $j_{\mathrm{s}}(x):=\frac{\mathrm{\hslash }}{m}(u_{\mathrm{s}}{v}_{\mathrm{s}}^{\mathrm{\prime }}-u_{\mathrm{s}}^{\mathrm{\prime }}{v}_{\mathrm{s}})(x)$ is constant on $\mathrm{\Omega }$.

The assertion is an identity for the declared differentiable functions and the declared real pair $V$; it constructs no probability law, no state and no physical interpretation.

• Meaning of the hypotheses. The two real equations are exactly the real and imaginary parts of the complex equation displayed in the statement, and no differentiability of a $\mathbb{C}$-valued function beyond the stated continuous partial derivatives of $u$ and $v$ is used. The function $V$ need not be continuous for the identity (a); it is used only through the given pointwise equations, and in (b) only through the two stationary equations.
• What is not asserted. The lemma produces no normalized state, no probability, no boundary condition and no scattering or transmission coefficient. In particular the constancy in (b) is a statement about a differentiable size on the open interval $\mathrm{\Omega }$ and assumes nothing about the behaviour of $j$ at the endpoints of $\mathrm{\Omega }$ or at infinity.

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $m,\mathrm{\hslash }>0$, put $c:=2{\pi }^2{\mathrm{\hslash }}^2\mathrm{/}m$, and let $H_0$ be the free one-dimensional operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) with $d=1$, this $c$ and $b=0$: $H_0:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H_0):=\{f\in L^2(\mathbb{R};\mathbb{C}):\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in L^2(\mathbb{R};\mathbb{C})\},$ where ${\mathcal{F}}_2$ is the unitary Plancherel transform of that theorem and $M_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}$ is multiplication by the function $\xi \mapsto c\mathrm{\mid }\xi {\mathrm{\mid }}^2$. Let $D^2$ denote the second weak derivative, defined componentwise on the locally integrable classes of $\mathbb{R}$ (Weak derivative of a locally integrable function), and let $W^{2,2}(\mathbb{R};\mathbb{C})$ be the Sobolev space (Integer-order Sobolev spaces and their norms). Then:

(a) $D(H_0)=W^{2,2}(\mathbb{R};\mathbb{C})$ as sets and $H_{0}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f\text{for every }f\in D(H_0);$ equivalently, ${\mathcal{F}}_2(H_{0}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ and ${\mathcal{F}}_2(-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ for almost every $\xi$, and the two $L^2$ classes agree.

(b) For every real bounded measurable $V:\mathbb{R}\to \mathbb{R}$, the operator $H_0+M_V$ of clause (c) of the cited theorem, with domain $D(H_0)$, coincides with the operator $f\mapsto -\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf$ on the same domain; in particular, for $E\in \mathbb{R}$ and $\psi \in W^{2,2}(\mathbb{R};\mathbb{C})$ the eigenvalue equation $(H_0+M_V)\psi =E\psi$ is equivalent to the weak equation $-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2\psi +V\psi =E\psi$ elementwise in $L^2(\mathbb{R};\mathbb{C})$.

The statement identifies operators on the same domain; it asserts neither smoothness of the classes in $W^{2,2}$ nor any physical interpretation.

• Why the factor $c$. With the Plancherel convention $\widehat{f}(\xi )=\int f(x)e^{-2\pi ix\xi }dx$ the weak Laplacian has multiplier $-4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$, so $-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2$ has multiplier $\frac{2{\pi }^2{\mathrm{\hslash }}^2}{m}\mathrm{\mid }\xi {\mathrm{\mid }}^2$: the constant $c$ of the cited theorem is fixed by this convention and not a convention-free physical statement.
• What is not asserted. The lemma compares two mathematical operators. It asserts no smoothness of $W^{2,2}$ classes, no boundary values, no self-adjointness statement beyond that of the cited theorem, and no identification with any measured Hamiltonian.

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space (Hilbert space), let $A:D(A)\subset H\to H$ be a densely defined self-adjoint operator (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator), and let $c>0$ be real. Write $cA$ for the operator with domain $D(cA):=D(A)$ and $(cA)x:=c(Ax)$. Then:

(a) $cA$ is self-adjoint.

(b) For every $z\in \mathbb{C}$ one has $z\in \rho (cA)$ if and only if $z\mathrm{/}c\in \rho (A)$, and then $(cA-z{)}^{-1}=c^{-1}(A-z\mathrm{/}c{)}^{-1}$; consequently $\rho (cA)=c\rho (A)$ and $\sigma (cA)=c\sigma (A)$.

(c) If some, hence every, resolvent of $A$ is compact (Resolvent and spectrum of an unbounded operator, Compact linear operator), then every resolvent of $cA$ is compact and $A$ and $cA$ have the same complete orthonormal eigenfamily $(e_j{)}_{j\in J}$. If $N:=\dim H<\mathrm{\infty }$, take $J=\{1,\dots ,N\}$; if $H$ is infinite dimensional, take $J=\mathbb{N}$, with $\mathrm{\mid }{\mu }_j\mathrm{\mid }\to \mathrm{\infty }$ and each eigenvalue repeated according to its finite multiplicity. In both cases $A{e}_j={\mu }_j{e}_j$, $(cA)e_j=c{\mu }_j{e}_j$, and $\sigma (A)=\{{\mu }_j:j\in J\},\sigma (cA)=\{c{\mu }_j:j\in J\}$ with no other spectral points. The finite-dimensional conclusion also holds without a separate compactness assumption, since bounded operators on such $H$ are compact.

(d) Eigenvectors correspond exactly: for $\lambda \in \mathbb{R}$ and $x\in H$, $x\mathrm{\ne }0$, one has $x\in D(A)$ and $Ax=\lambda x$ if and only if $x\in D(cA)$ and $(cA)x=(c\lambda )x$; and every eigenvalue of $cA$ has the same eigenspace as the corresponding eigenvalue of $A$.

(e) If $A\ge \gamma$ in the quadratic-form sense, meaning that $⟨Ax,x⟩\ge \gamma \mathrm{\parallel }x{\mathrm{\parallel }}^2$ for every $x\in D(A)$, then $cA\ge c\gamma$ in the same sense.

Only the scaling of the given operator is asserted; no domain, spectrum or eigenvector is claimed for any other rescaling, and no physical interpretation is attached.

• Where positivity is used. $c>0$ is used in step 2.1 (boundedness of the pairing is preserved under multiplication by $c\mathrm{\ne }0$, but the adjoint computation needs $c$ real, and the resolvent identity needs $c\mathrm{\ne }0$), and in step 5.1 for the direction of the form inequality. For $c<0$ the operator $cA$ is still self-adjoint when $A$ is, but the form inequality reverses; no such statement is used here.
• What is not asserted. The lemma transfers self-adjointness, resolvents, spectra, eigenfamilies and form lower bounds along positive scaling of a fixed operator. It does not construct a self-adjoint operator from a symmetric one, does not assert any min-max formula, and does not identify any operator with a physical energy.

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a mass parameter $m>0$ and a spin label $s=0$, so that the framework's one-particle model The massive spin-s particle model of the framework: state space, operators, units and frame readings reads its state space as $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with the pairing linear in the first argument, its componentwise position family $Q_1,Q_2,Q_3$, its componentwise momentum family $P_1,P_2,P_3$ and the action constant $\mathrm{\hslash }>0$; the spin-zero internal factor carries the trivial representation of P6. Units are those of Dimensions, SI unit conventions and quantity types of the framework: position is read in metres, momentum in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy and potential energy in joules, and $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared effective one-dimensional model. As an additional model assumption of the pair, adopt the line model. Its declared state space is $H_{\mathrm{l}\mathrm{i}\mathrm{n}}:=L^2(\mathbb{R};\mathbb{C})=\{f:\mathbb{R}\to \mathbb{C}\text{ measurable}:{\int }_{\mathbb{R}}\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx<\mathrm{\infty }\}/\text{\hspace{0.17em}⁣}\sim ,$ the complex Hilbert space of almost-everywhere classes with the pairing $⟨f,g⟩={\int }_{\mathbb{R}}f\bar{g}dx$ linear in the first argument (The space $L^p(\mu )$ as the quotient by null functions). Its declared position assignment is the multiplication assignment $Q$, defined on the classes with $xf\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, with the declared outcome law ${\mu }_{\psi }^Q(B):={\int }_B\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx(B\subseteq \mathbb{R}\text{ Borel})$ for a unit class $\psi$. Indeed, the positive sigma-finite Lebesgue measure space $\mathbb{R}$ and the finite measurable multiplier $x\mapsto x$ satisfy Spectral measure of a real L2 multiplication operator clauses (a)-(c) under the choice assumptions of (a): $Q$ is self-adjoint on the displayed domain, with spectral PVM $E_Q(B)=M_{{\mathbf{1}}_B}$ and $\mathrm{\parallel }{E}_Q(B)\psi {\mathrm{\parallel }}^2={\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^2$. The probability reading of this mathematical law is adopted under P2. Its declared momentum assignment is $P:=-i\mathrm{\hslash }D$ on the classes whose weak derivative $D\psi$ lies in $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$ (Weak derivative of a locally integrable function). The pair asserts no derivation of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, $Q$ or $P$ from the three-dimensional model $H_0$ of P5: the reduction to one spatial dimension, for instance by an idealized strong transverse confinement or by a declared one-dimensional sector, is an assumption of the model. An assignment of a concrete material system to the line model is an identification in the sense of clause (iii) of P7 and carries the corresponding uncertainty.

(c) The assigned energy operator. Let $V:\mathbb{R}\to \mathbb{R}$ be real, bounded and measurable; write $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{x\in \mathbb{R}}\mathrm{\mid }V(x)\mathrm{\mid }$. The assigned energy operator of the line model with potential $V$ is $H_V:=H_0^{\mathrm{l}\mathrm{i}\mathrm{n}}+M_V,H_0^{\mathrm{l}\mathrm{i}\mathrm{n}}:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,c:=\frac{2{\pi }^2{\mathrm{\hslash }}^2}{m},$ with $M_V$ the multiplication operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a), on the common domain $D(H_V):=D(H_0^{\mathrm{l}\mathrm{i}\mathrm{n}})=\{f\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}:\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}\}$, where ${\mathcal{F}}_2$ is the unitary Plancherel transform of that theorem (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (a)-(c)). The declared energy unit is the joule. By The one-dimensional free Schr"odinger operator as the weak second-derivative operator the domain $D(H_V)$ is the Sobolev space $W^{2,2}(\mathbb{R};\mathbb{C})$ and $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf(f\in D(H_V)),$ $D^2$ the second weak derivative, so the model's time-independent equation inside any open interval on which $V$ is (say) constant is the ordinary differential equation $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}+V\psi =E\psi$. The model adopts $(H_V,\mathrm{J})$ as its assigned energy observable in the sense of Assigned sharp observables and the outcome law of the framework clause (a); its self-adjointness is established by the line-model energy theorem of this page and is not presupposed here. A declared potential is compactly supported when $V$ vanishes almost everywhere outside some bounded interval.

(d) Bound states and stationary vocabulary. A stationary solution of the model with potential $V$ and energy $E\in \mathbb{R}$ is a nonzero class $\psi \in D(H_V)$ with $H_V\psi =E\psi$; the pair $(\psi ,E)$ is an eigenpair of the model. A bound state of the model is a stationary solution with energy $E<0$ whose representative is normalized, $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; its eigenvalue $E$ is its signed binding energy, read in joules, and its positive binding-energy magnitude is $-E$. The value $0$ is the reference energy in this negative-eigenvalue definition of a bound state. For a general bounded measurable $V$, this convention makes no assertion about the location of any spectral threshold. The model reads the energy outcome law of the pure preparation of a unit eigenstate $\psi$ with eigenvalue $E$ through clause (c) of Assigned sharp observables and the outcome law of the framework and Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a)-(b): for self-adjoint $H_V$ on the nonzero Hilbert space $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, its spectral projection at $E$ fixes the unit eigenvector, so the outcome in $\{E\}$ has probability one and the law is the point mass ${\delta }_E$. The position density of the preparation is $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ by the one-dimensional multiplication-PVM construction in clause (b) above. These readings are declarations of the model under P2 and construct no sample, detector or measured value.

(e) Declared generalized stationary solutions. Let $-\mathrm{\infty }=x_0<x_1<\cdots <x_n=\mathrm{\infty }$ be a finite partition of $\mathbb{R}$, let $E\in \mathbb{R}$, and let $\psi :\mathbb{R}\to \mathbb{C}$ be bounded and classically $C^1$ on $\mathbb{R}$ with a continuous derivative, and twice continuously differentiable on each open interval $(x_j,x_{j+1})$ restricted to which $V$ is continuous, satisfying $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}(x)+V(x)\psi (x)=E\psi (x)$ there. In addition, require the stationary equation on the whole line in the weak sense: $-\frac{{\mathrm{\hslash }}^2}{2m}{\int }_{\mathbb{R}}\psi (x){\phi }^{\mathrm{\prime }\mathrm{\prime }}(x)dx+{\int }_{\mathbb{R}}V(x)\psi (x)\phi (x)dx=E{\int }_{\mathbb{R}}\psi (x)\phi (x)dx$ for every smooth compactly supported real test function $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{R})$; the equality is complex, equivalently componentwise. All integrals are finite because $V$ and $\psi$ are bounded and the tests have compact support. This is exactly the assertion $D^2\psi =(2m\mathrm{/}{\mathrm{\hslash }}^2)(V-E)\psi$ with the weak derivative convention of Weak derivative of a locally integrable function; it is required even when $V$ is continuous on no open interval. For a piecewise constant potential and a $C^1$ piecewise $C^2$ solution of the displayed classical equations, integration by parts twice on the pieces gives this identity: the internal boundary terms cancel by continuity of $\psi$ and ${\psi }^{\mathrm{\prime }}$, and the external terms vanish by compact support of the test. Such a $\psi$ is a declared generalized stationary solution of the model at energy $E$. It need not be an element of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$ (for example, a function $x\mapsto A{e}^{ikx}$ with $A\in \mathbb{C}\setminus \{0\}$ and $k\in \mathbb{R}$ has constant density $\mathrm{\mid }A{\mathrm{\mid }}^2>0$, which is not integrable on $\mathbb{R}$). When it is not in $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, it does not represent a preparation of the model and clause (d) does not apply to it; its scattering reading uses only the declared flux quantities computed from it, and every such use states the idealization. Square-integrable solutions are not excluded: if the nonzero class of $\psi$ belongs to $D(H_V)$, the weak equation above is the eigenvalue equation of clause (d), and its unit normalization has the preparation and outcome-law readings of that clause. Merely declaring a generalized stationary solution imposes no such normalization. For dimensional bookkeeping, assign $\psi$ and its amplitudes the one-dimensional position-wavefunction dimension ${\mathrm{\ell }}^{-1\mathrm{/}2}$ of Dimensions, SI unit conventions and quantity types of the framework clause (d)1 (the SI reference unit is ${\mathrm{m}}^{-1\mathrm{/}2}$); this convention does not normalize $\psi$ or make it an element of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$. The pair's scattering reading assigns to $\psi$ the probability current $j(x)=\frac{\mathrm{\hslash }}{m}\mathrm{Im}(\overline{\psi (x)}{\psi }^{\mathrm{\prime }}(x))$, read in ${\mathrm{s}}^{-1}$ by clause (d)5 of Dimensions, SI unit conventions and quantity types of the framework, and reads a ratio of currents at infinity as a transmitted or reflected fraction of the incident current, conditional on the declared solution and on the asymptotics displayed in each statement; no such fraction is ascribed to a single particle and no count is predicted.