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Ladder norms and excitation statistics of the oscillator eigenstates

Physical scope

The ladder operators of the one-dimensional oscillator model applied to the first excited eigenstates: the explicit ladder norms up to the second excited level, the outcome laws of the assigned dimensionless excitation-number observable in unit 1 and of the energy observable in the eigenstate preparations, and the warning that the annihilation and creation operators are not assigned sharp observables of the model. The example is a conditional computation inside the adopted model and asserts no measured excitation number, apparatus realisation or empirical value.

Example

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 ( ${A C}_{ω}$ ), The Axiom of Choice). Let $(ψ_{n})$ be the model's complete orthonormal eigenbasis, $a_{ψ}, a_{ψ}^{†}, N_{ψ}$ the transported ladder and number operators of The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clause (b), and $E_{n} = ℏ ω (n + \frac{1}{2})$ . Then:

(a) the first ladder images and their norms are $a_{ψ} ψ_{0} = 0, a_{ψ} ψ_{1} = ψ_{0}, a_{ψ} ψ_{2} = \sqrt{2} ψ_{1}, a_{ψ}^{†} ψ_{0} = ψ_{1}, a_{ψ}^{†} ψ_{1} = \sqrt{2} ψ_{2},$ $∥ a_{ψ} ψ_{1} ∥ = 1, ∥ a_{ψ} ψ_{2} ∥ = \sqrt{2}, ∥ a_{ψ}^{†} ψ_{0} ∥ = 1, ∥ a_{ψ}^{†} ψ_{1} ∥ = \sqrt{2};$

(b) the matrix elements of the ladder operators in the eigenbasis are $⟨ ψ_{m}, a_{ψ} ψ_{n} ⟩ = \sqrt{n} δ_{m, n - 1} (n \geq 1), ⟨ ψ_{m}, a_{ψ}^{†} ψ_{n} ⟩ = \sqrt{n + 1} δ_{m, n + 1}, ⟨ a_{ψ} ψ_{n}, a_{ψ} ψ_{m} ⟩ = n δ_{n m};$

(c) every eigenvector is an eigenvector of the number operator, $N_{ψ} ψ_{n} = n ψ_{n}$ , and the number and energy outcome laws of the eigenstate preparations are the point masses $μ_{ψ_{n}}^{N_{ψ}} ({n}) = 1, μ_{ψ_{n}}^{H_{1}} ({E_{n}}) = 1 :$ the $n$ -th eigenstate preparation carries exactly $n$ excitations and the energy $E_{n} = ℏ ω (n + \frac{1}{2})$ ;

(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 $∥ a_{ψ} ψ_{1} ∥ = 1$ equals the certain number outcome in $ψ_{1}$ . The assigned sharp observables of the model are $(Q, m)$ , $(P, k g m s^{- 1})$ , $(H_{1}, J)$ and $(N_{ψ}, 1)$ , 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.

Facts & Assumptions

Given: the Axiom of Countable Choice and the Axiom of Choice; the one-dimensional oscillator model with parameters $m, ω, ℏ > 0$ ; its complete orthonormal eigenbasis $(ψ_{n})$ with $H_{1} ψ_{n} = E_{n} ψ_{n}$ and $E_{n} = ℏ ω (n + \frac{1}{2})$ ; the closed ladder and number operators $a_{ψ}, a_{ψ}^{†}, N_{ψ}$ of the model's spectrum theorem clause (b).

[F1]

Eigenbasis and ladder operators. $(ψ_{n})$ is a complete orthonormal family of $H_{1}$ , $H_{1} ψ_{n} = E_{n} ψ_{n}$ , $N_{ψ} ψ_{n} = n ψ_{n}$ , and on the eigenbasis $a_{ψ} ψ_{n} = \sqrt{n} ψ_{n - 1}$ for $n \geq 1$ with $a_{ψ} ψ_{0} = 0$ , $a_{ψ}^{†} ψ_{n} = \sqrt{n + 1} ψ_{n + 1}$ and $a_{ψ}^{†} a_{ψ} = N_{ψ}$ , $a_{ψ} a_{ψ}^{†} = N_{ψ} + I$ on the common domain; the operators $a_{ψ}$ , $a_{ψ}^{†}$ are closed with $a_{ψ}^{*} = a_{ψ}^{†}$ and $(a_{ψ}^{†})^{*} = a_{ψ}$ , and $N_{ψ}$ is self-adjoint with eigenbasis $(ψ_{n})$ ; the number law of $N_{ψ}$ in the pure preparation of $ψ_{n}$ is $δ_{n}$ , and the energy law of $H_{1}$ is the atomic law $μ_{ψ_{n}}^{H_{1}} ({E_{n}}) = ∣ ⟨ ψ_{n}, ψ_{n} ⟩ ∣^{2} = 1$ (The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clauses (a), (b), (c); Closed ladder domains clauses (b), (c), (d), (e); Oscillator scaling and exact realization clauses (a), (b)).

[F2]

Assigned observables and their laws. The model assigns the status of assigned sharp observable to $(Q, m)$ , $(P, k g m s^{- 1})$ , $(H_{1}, J)$ and $(N_{ψ}, 1)$ , the dimensionless excitation number; their outcome laws are read from the spectral measures, and an operator that is not self-adjoint is not an assigned sharp observable of the framework (The quadratic-potential oscillator model of the framework clauses (b), (c); Assigned sharp observables and the outcome law of the framework clauses (a), (b); Dimensions, SI unit conventions and quantity types of the framework).

[F3]

Orthonormality. $⟨ ψ_{n}, ψ_{m} ⟩ = δ_{n m}$ and $∥ ψ_{n} ∥ = 1$ for every $n$ (Oscillator scaling and exact realization clause (a)).

Verification

Given: the data of the Example and [F1]-[F3].

1.1givenF1

[Ladder images] By [F1], $a_{ψ} ψ_{0} = 0$ , $a_{ψ} ψ_{1} = \sqrt{1} ψ_{0} = ψ_{0}$ , $a_{ψ} ψ_{2} = \sqrt{2} ψ_{1}$ , $a_{ψ}^{†} ψ_{0} = \sqrt{1} ψ_{1} = ψ_{1}$ and $a_{ψ}^{†} ψ_{1} = \sqrt{2} ψ_{2}$ .

1.2givenF1F3

[Matrix elements] By [F1] and [F3], for $n \geq 1$ and every $m$ , $⟨ ψ_{m}, a_{ψ} ψ_{n} ⟩ = \sqrt{n} ⟨ ψ_{m}, ψ_{n - 1} ⟩ = \sqrt{n} δ_{m, n - 1}$ , $⟨ ψ_{m}, a_{ψ}^{†} ψ_{n} ⟩ = \sqrt{n + 1} ⟨ ψ_{m}, ψ_{n + 1} ⟩ = \sqrt{n + 1} δ_{m, n + 1}$ , and $⟨ a_{ψ} ψ_{n}, a_{ψ} ψ_{m} ⟩ = \sqrt{n m} ⟨ ψ_{n - 1}, ψ_{m - 1} ⟩ = n δ_{n m}$ for $n, m \geq 1$ . At $n = 0$ , $a_{ψ}^{†} ψ_{0} = ψ_{1}$ gives $⟨ ψ_{m}, a_{ψ}^{†} ψ_{0} ⟩ = δ_{m, 1}$ . If $n = 0$ or $m = 0$ , one annihilation image vanishes by $a_{ψ} ψ_{0} = 0$ , so the pairing is zero, equal to $n δ_{n m}$ .

1.3givenF1F3

[Number and energy laws of the eigenstates] By [F1], $N_{ψ} ψ_{n} = n ψ_{n}$ and the number law of the unit eigenvector $ψ_{n}$ is $δ_{n}$ ; likewise the energy law of the unit eigenvector is $μ_{ψ_{n}}^{H_{1}} ({E_{n}}) = ∣ ⟨ ψ_{n}, ψ_{n} ⟩ ∣^{2} = 1$ by [F1] and [F3], so the preparation carries exactly $n$ excitations and the energy value $E_{n} = ℏ ω (n + \frac{1}{2})$ .

2.1step 1.1F3

[Clause (a)] Taking norms in the images of step 1.1 and using $∥ ψ_{k} ∥ = 1$ of [F3] gives $∥ a_{ψ} ψ_{1} ∥ = ∥ ψ_{0} ∥ = 1$ , $∥ a_{ψ} ψ_{2} ∥ = \sqrt{2} ∥ ψ_{1} ∥ = \sqrt{2}$ , $∥ a_{ψ}^{†} ψ_{0} ∥ = ∥ ψ_{1} ∥ = 1$ and $∥ a_{ψ}^{†} ψ_{1} ∥ = \sqrt{2} ∥ ψ_{2} ∥ = \sqrt{2}$ .

2.2step 1.2

[Clause (b)] The displays of clause (b) are exactly the identities computed in step 1.2.

3.1step 1.1step 1.3step 2.1F1F2

[Clause (d)] By [F1], $a_{ψ} ψ_{0} = 0$ and $a_{ψ}^{†} ψ_{0} = ψ_{1} \neq 0$ , so $a_{ψ} \neq a_{ψ}^{†}$ . Since $a_{ψ}^{*} = a_{ψ}^{†}$ and $(a_{ψ}^{†})^{*} = a_{ψ}$ by [F1], neither operator is self-adjoint. By [F2] neither is an assigned sharp observable, so neither has an assigned sharp outcome law; the computed images and norms alone do not supply such a law. This does not exclude numerical equality with an outcome coordinate of another observable: step 2.1 gives $∥ a_{ψ} ψ_{1} ∥ = 1$ , while step 1.3 gives $μ_{ψ_{1}}^{N_{ψ}} ({1}) = 1$ . The assigned observable pairs are those listed in [F2].

4.1step 1.1step 3.1given∎

[Clause (e)] The computation used the model's eigenbasis and ladder operators and the orthonormality of the eigenbasis; it produced no measurement record and no claim about a laboratory preparation.

Ladder norms and excitation statistics of the oscillator eigenstates

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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