Qm Harmonic Oscillator — Examples

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the kinematics and massive-particle model postulate P5, read through the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (a), (b), (c), and assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let ${\psi }_0$ be the ground eigenvector of the model's energy operator $H_1$, let $\mathrm{\ell }=\sqrt{\mathrm{\hslash }\mathrm{/}(m\omega )}$, and let $Q$, $P$ be the model's position and momentum observables. Then:

(a) the class ${\psi }_0$ has the representative ${\psi }_0(x)={\mathrm{\ell }}^{-1\mathrm{/}2}{\pi }^{-1\mathrm{/}4}{e}^{-x^2\mathrm{/}(2{\mathrm{\ell }}^2)},$ it is a unit vector, it is the model's coherent state of label $0$, and $H_1{\psi }_0=\frac{\mathrm{\hslash }\omega }{2}{\psi }_0$;

(b) the position outcome law of the assigned observable $(Q,\mathrm{m})$ in the ground preparation has the density $f(x)={\mathrm{\ell }}^{-1}{\pi }^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}{\mathrm{\ell }}^2},$ with ${\int }_{\mathbb{R}}f=1$, mean $0$ and second moment ${\mathrm{\ell }}^2\mathrm{/}2$; hence ${\mathrm{\Delta }}_{{\psi }_0}Q=\mathrm{\ell }\mathrm{/}\sqrt{2}$;

(c) the momentum outcome law of the assigned observable $(P,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$ has the density $g(p)=\frac{\mathrm{\ell }}{\mathrm{\hslash }\sqrt{\pi }}{e}^{-{\mathrm{\ell }}^2{p}^2\mathrm{/}{\mathrm{\hslash }}^2},$ with ${\int }_{\mathbb{R}}g=1$, mean $0$ and second moment ${\mathrm{\hslash }}^2\mathrm{/}(2{\mathrm{\ell }}^2)$; hence ${\mathrm{\Delta }}_{{\psi }_0}P=\mathrm{\hslash }\mathrm{/}(\mathrm{\ell }\sqrt{2})$;

(d) consequently ${\mathrm{\Delta }}_{{\psi }_0}Q{\mathrm{\Delta }}_{{\psi }_0}P=\mathrm{\hslash }\mathrm{/}2$: the ground preparation is one of the coherent preparations of Fluctuations of the oscillator coherent states and the position-momentum relation of the model, and it saturates the model's position-momentum relation;

(e) the computation is one preparation of the adopted model; it identifies no measured distribution with $f$ or $g$ and no laboratory state with the ground state.

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the kinematics and massive-particle model postulate P5, read through the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (a), (b), (c), and 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 $({\psi }_n)$ be the model's complete orthonormal eigenbasis, $a_{\psi },a_{\psi }^{†},N_{\psi }$ the transported ladder and number operators of The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clause (b), and $E_n=\mathrm{\hslash }\omega (n+\frac{1}{2})$. Then:

(a) the first ladder images and their norms are $a_{\psi }{\psi }_0=0,a_{\psi }{\psi }_1={\psi }_0,a_{\psi }{\psi }_2=\sqrt{2}{\psi }_1,a_{\psi }^{†}{\psi }_0={\psi }_1,a_{\psi }^{†}{\psi }_1=\sqrt{2}{\psi }_2,$ $\mathrm{\parallel }{a}_{\psi }{\psi }_1\mathrm{\parallel }=1,\mathrm{\parallel }{a}_{\psi }{\psi }_2\mathrm{\parallel }=\sqrt{2},\mathrm{\parallel }{a}_{\psi }^{†}{\psi }_0\mathrm{\parallel }=1,\mathrm{\parallel }{a}_{\psi }^{†}{\psi }_1\mathrm{\parallel }=\sqrt{2};$

(b) the matrix elements of the ladder operators in the eigenbasis are $⟨{\psi }_m,a_{\psi }{\psi }_n⟩=\sqrt{n}{\delta }_{m,n-1}(n\ge 1),⟨{\psi }_m,a_{\psi }^{†}{\psi }_n⟩=\sqrt{n+1}{\delta }_{m,n+1},⟨a_{\psi }{\psi }_n,a_{\psi }{\psi }_m⟩=n{\delta }_{nm};$

(c) every eigenvector is an eigenvector of the number operator, $N_{\psi }{\psi }_n=n{\psi }_n$, and the number and energy outcome laws of the eigenstate preparations are the point masses ${\mu }_{{\psi }_n}^{N_{\psi }}(\{n\})=1,{\mu }_{{\psi }_n}^{H_1}(\{E_n\})=1:$ the $n$-th eigenstate preparation carries exactly $n$ excitations and the energy $E_n=\mathrm{\hslash }\omega (n+\frac{1}{2})$;

(d) the annihilation and creation operators are not assigned sharp observables of the model: they are not self-adjoint and have no assigned sharp outcome laws. Their images and norms are operator-theoretic quantities; a norm may nevertheless equal an outcome coordinate of another assigned observable, as $\mathrm{\parallel }{a}_{\psi }{\psi }_1\mathrm{\parallel }=1$ equals the certain number outcome in ${\psi }_1$. The assigned sharp observables of the model are $(Q,\mathrm{m})$, $(P,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$, $(H_1,\mathrm{J})$ and $(N_{\psi },1)$, the dimensionless excitation-number observable of The quadratic-potential oscillator model of the framework clause (c), and only laws of those are read in clause (c);

(e) the computation is a finite low-level check of the ladder relations in the adopted model, not a measurement record and not a claim that a laboratory oscillator was prepared in an eigenstate.

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5, read through the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (a), (b), (c), and the coherent-state family of The coherent states of the oscillator model; assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix a label $z=z_1+i{z}_2$ with $\mathrm{\mid }z\mathrm{\mid }>0$, let $x_0:=\sqrt{2}\mathrm{\ell }{z}_1$, $p_0:=\frac{\sqrt{2}\mathrm{\hslash }}{\mathrm{\ell }}{z}_2$, let ${\mathrm{\Psi }}_z(t):=U(t){\mathrm{\Psi }}_z$ be the evolved coherent state, and write $x(t):=⟨Q{\mathrm{\Psi }}_z(t),{\mathrm{\Psi }}_z(t)⟩$, $p(t):=⟨P{\mathrm{\Psi }}_z(t),{\mathrm{\Psi }}_z(t)⟩$. Then:

(a) the preparation is periodic with period $\frac{2\pi }{\omega }$ in the sense that the prepared state is the same pure preparation at times $t$ and $t+\frac{2\pi }{\omega }$: $[U(2\pi \mathrm{/}\omega ){\mathrm{\Psi }}_z]=[{\mathrm{\Psi }}_z],[{\mathrm{\Psi }}_z(t+2\pi \mathrm{/}\omega )]=[{\mathrm{\Psi }}_z(t)](t\in \mathbb{R});$

(b) the mean pair is $x(t)=x_0\cos \omega t+\frac{p_0}{m\omega }\sin \omega t,p(t)=p_0\cos \omega t-m\omega x_0\sin \omega t,$ so that $x(0)=x_0$, $p(0)=p_0$, and at the quarter period $x\left(\frac{\pi }{2\omega }\right)=\frac{p_0}{m\omega }$, $p\left(\frac{\pi }{2\omega }\right)=-m\omega x_0$; the phase-space ellipse $\frac{x(t{)}^2}{2{\mathrm{\ell }}^2}+\frac{{\mathrm{\ell }}^{2}p(t{)}^2}{2{\mathrm{\hslash }}^2}=\mathrm{\mid }z{\mathrm{\mid }}^2$ is invariant along the motion;

(c) the position and momentum spreads and their product are constant along the motion: ${\mathrm{\Delta }}_{{\mathrm{\Psi }}_z(t)}Q=\frac{\mathrm{\ell }}{\sqrt{2}},{\mathrm{\Delta }}_{{\mathrm{\Psi }}_z(t)}P=\frac{\mathrm{\hslash }}{\mathrm{\ell }\sqrt{2}},{\mathrm{\Delta }}_{{\mathrm{\Psi }}_z(t)}Q{\mathrm{\Delta }}_{{\mathrm{\Psi }}_z(t)}P=\frac{\mathrm{\hslash }}{2}(t\in \mathbb{R});$

(d) the displayed $x(t)$ and $p(t)$ are preparation means of the model's position and momentum observables in the evolving coherent preparation; they are not the coordinates of a measured particle trajectory, and no measurement record, apparatus or empirical value is asserted.

Ground state. The Gaussian ground state: position and momentum laws and the saturated bound fixes the explicit Gaussian representative ${\psi }_0(x)={\mathrm{\ell }}^{-1\mathrm{/}2}{\pi }^{-1\mathrm{/}4}{e}^{-x^2\mathrm{/}(2{\mathrm{\ell }}^2)}$ of the ground vector, its unit norm, the eigenvalue equation $H_1{\psi }_0=\frac{\mathrm{\hslash }\omega }{2}{\psi }_0$, the position law with density ${\mathrm{\ell }}^{-1}{\pi }^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}{\mathrm{\ell }}^2}$ and variance ${\mathrm{\ell }}^2\mathrm{/}2$, the momentum law with density $\frac{\mathrm{\ell }}{\mathrm{\hslash }\sqrt{\pi }}{e}^{-{\mathrm{\ell }}^2{p}^2\mathrm{/}{\mathrm{\hslash }}^2}$ and variance ${\mathrm{\hslash }}^2\mathrm{/}(2{\mathrm{\ell }}^2)$, and the saturation ${\mathrm{\Delta }}_{{\psi }_0}Q{\mathrm{\Delta }}_{{\psi }_0}P=\mathrm{\hslash }\mathrm{/}2$ of the model's position-momentum relation.

Ladder and excitation structure. Ladder norms and excitation statistics of the oscillator eigenstates computes the first ladder images and their norms, the ladder matrix elements $⟨{\psi }_m,a_{\psi }{\psi }_n⟩=\sqrt{n}{\delta }_{m,n-1}$ and $⟨{\psi }_m,a_{\psi }^{†}{\psi }_n⟩=\sqrt{n+1}{\delta }_{m,n+1}$, and the excitation-number and energy laws ${\mu }_{{\psi }_n}^{N_{\psi }}(\{n\})=1$, ${\mu }_{{\psi }_n}^{H_1}(\{E_n\})=1$ of the eigenstate preparations; the number coordinate is read by the assigned observable $(N_{\psi },1\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t})$, and $a_{\psi }$ and $a_{\psi }^{†}$ are not assigned sharp observables.

Coherent motion. A coherent-state trajectory: period, phase-space ellipse and constant spreads shows that a nonzero-label coherent preparation is periodic with period $2\pi \mathrm{/}\omega$, that its mean pair traces the invariant phase-space ellipse $\frac{x^2}{2{\mathrm{\ell }}^2}+\frac{{\mathrm{\ell }}^2{p}^2}{2{\mathrm{\hslash }}^2}=\mathrm{\mid }z{\mathrm{\mid }}^2$ with the quarter-period values $x(\frac{\pi }{2\omega })=\frac{p_0}{m\omega }$, $p(\frac{\pi }{2\omega })=-m\omega x_0$, and that the position and momentum spreads and their product $\mathrm{\hslash }\mathrm{/}2$ are constant along the motion.

Every computation above is conditional on the adopted oscillator model of The quadratic-potential oscillator model of the framework and on the postulates P1, P2, P4 and P5 read through it; none of them reports a measurement. In particular: no numerical value of $m$, $\omega$ or $\mathrm{\hslash }$ is extracted; no laboratory state is identified with ${\psi }_0$, with an eigenstate ${\psi }_n$ or with a coherent state; the mean functions $x(t),p(t)$ are preparation means of the model's observables under the evolving preparation and are not the coordinates of a measured trajectory; and the periodicity of clause (a) of the coherent-trajectory example is a statement about the pure preparation (a ray), not about a measured signal. The harmonic approximation's range, anharmonic corrections, damping, driving, time-dependent parameters and coupling to electromagnetic fields or radiation are outside the examples, as is the displacement-operator presentation of the coherent family. The formal non-normalizable solution of the ground-state equation is authored on the mathematical companion examples page as its formal solution example, and is linked there rather than duplicated here.