Physlib.QuantumMechanics.OneDimension.HilbertSpace.Basic

The Hilbert space for a one-dimensional quantum system is defined as the space $L^2(\mathbb{R}, \mathbb{C})$ of equivalence classes of square-integrable functions from the real numbers $\mathbb{R}$ to the complex numbers $\mathbb{C}$. This space consists of measurable functions $f: \mathbb{R} \to \mathbb{C}$ such that the integral of the square of their magnitude is finite, i.e., $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$, where functions are identified if they are equal almost everywhere.

The mapping `toBra` is an anti-linear (conjugate-linear) isomorphism from the Hilbert space $\mathcal{H} = L^2(\mathbb{R}, \mathbb{C})$ to its strong dual space $\mathcal{H}^*$. For any vector $f \in \mathcal{H}$, it returns a continuous linear functional that maps any vector $g \in \mathcal{H}$ to the complex inner product $\langle f, g \rangle$. In the context of quantum mechanics, this corresponds to the mapping of a state vector (ket) to its corresponding functional (bra).

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ for a one-dimensional quantum system. For any two vectors $f, g \in \mathcal{H}$, the application of the functional $\text{toBra}(f)$ to the vector $g$ is equal to the complex inner product $\langle f, g \rangle$.

The anti-linear mapping $\text{toBra} : \mathcal{H} \to \mathcal{H}^*$ from the Hilbert space $\mathcal{H} = L^2(\mathbb{R}, \mathbb{C})$ to its strong dual space $\mathcal{H}^*$, which maps a state vector (ket) to its corresponding linear functional (bra), is surjective.

Let $\mathcal{H} = L^2(\mathbb{R}, \mathbb{C})$ be the Hilbert space for a one-dimensional quantum system. The anti-linear mapping $\text{toBra} : \mathcal{H} \to \mathcal{H}^*$, which maps a state vector (ket) to its corresponding linear functional (bra) in the dual space such that $\text{toBra}(f)(g) = \langle f, g \rangle$, is injective.

For a function $f: \mathbb{R} \to \mathbb{C}$, the proposition `MemHS f` is the condition that $f$ belongs to the $L^2$ space over the real line with respect to the Lebesgue measure. Specifically, it asserts that $f$ is almost everywhere strongly measurable and that the integral of the square of its norm is finite, i.e., $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$. This property characterizes functions that can be represented as elements of the Hilbert space for one-dimensional quantum mechanics.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function. If $f$ is an element of the Hilbert space for one-dimensional quantum mechanics (i.e., $f$ satisfies the property $f \in L^2(\mathbb{R}, \mathbb{C})$), then $f$ is almost everywhere strongly measurable.

A function $f: \mathbb{R} \to \mathbb{C}$ satisfies the property `MemHS f` if and only if it is almost everywhere strongly measurable and square-integrable, which means that the integral of the square of its norm $\int_{-\infty}^{\infty} |f(x)|^2 \, dx$ is finite.

The zero function $f(x) = 0$ belongs to the Hilbert space for one-dimensional quantum mechanics, i.e., $0 \in L^2(\mathbb{R}, \mathbb{C})$. This means that the zero function is almost everywhere strongly measurable and satisfies $\int_{-\infty}^{\infty} |0|^2 \, dx < \infty$.

The constant function $f: \mathbb{R} \to \mathbb{C}$ defined by $f(x) = 0$ for all $x \in \mathbb{R}$ belongs to the Hilbert space for one-dimensional quantum mechanics, i.e., $f \in L^2(\mathbb{R}, \mathbb{C})$. This means that the function is almost everywhere strongly measurable and satisfies the square-integrability condition $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$.

Let $f, g: \mathbb{R} \to \mathbb{C}$ be functions. If $f$ and $g$ are square-integrable functions belonging to the $L^2$ space (i.e., $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$ and $\int_{-\infty}^{\infty} |g(x)|^2 \, dx < \infty$), then their sum $f + g$ is also square-integrable and belongs to the $L^2$ space.

For any function $f: \mathbb{R} \to \mathbb{C}$ and any complex scalar $c \in \mathbb{C}$, if $f$ satisfies the property of being in the Hilbert space for one-dimensional quantum mechanics (i.e., $f \in L^2(\mathbb{R}, \mathbb{C})$, meaning $f$ is almost everywhere strongly measurable and $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$), then the scalar multiple $c \cdot f$ also satisfies this property.

Let $f, g: \mathbb{R} \to \mathbb{C}$ be functions. If $f$ belongs to the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ (meaning $f$ is almost everywhere strongly measurable and $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$) and $f$ is equal to $g$ almost everywhere with respect to the Lebesgue measure, then $g$ also belongs to $L^2(\mathbb{R}, \mathbb{C})$.

Let $f: \mathbb{R} \to \mathbb{C}$ be an almost everywhere strongly measurable function, and let $[f]$ denote its equivalence class under almost everywhere equality. Then $[f]$ is an element of the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ if and only if $f$ satisfies the property `MemHS f`, which means that $f$ is square-integrable, i.e., $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$.

Given a function $f: \mathbb{R} \to \mathbb{C}$ and a proof $hf$ that $f$ satisfies the square-integrability condition (i.e., $f$ is almost everywhere strongly measurable and $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$), this definition constructs the corresponding element in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$. The resulting element represents the equivalence class of functions that are equal to $f$ almost everywhere.

For any element $f$ in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ for one-dimensional quantum mechanics, its representation as a function $f: \mathbb{R} \to \mathbb{C}$ satisfies the property $\text{MemHS} f$, meaning that it is almost everywhere strongly measurable and square-integrable, such that $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$.

For every element $f$ in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ for one-dimensional quantum mechanics, there exists a function $g: \mathbb{R} \to \mathbb{C}$ and a proof $h_g$ that $g$ is square-integrable (satisfying the property $\text{MemHS} \ g$) such that the construction $\text{mk} \ h_g$ of the Hilbert space element from $g$ is equal to $f$.

For any function $f: \mathbb{R} \to \mathbb{C}$ that is square-integrable (i.e., $f$ satisfies the property $\text{MemHS } f$), the function representation of the element in the Hilbert space constructed from $f$ (denoted as $\text{mk } f$) is equal to $f$ almost everywhere with respect to the Lebesgue measure.

For any two functions $f, g: \mathbb{R} \to \mathbb{C}$ that are square-integrable (satisfying the property `MemHS`), let $[f]$ and $[g]$ denote the corresponding elements in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ constructed via the mapping `mk`. The inner product of these two elements in the Hilbert space is given by the integral over the real line of the product of the complex conjugate of $f(x)$ and $g(x)$: $$\langle [f], [g] \rangle = \int_{-\infty}^{\infty} \overline{f(x)} g(x) \, dx$$ where $\overline{f(x)}$ denotes the complex conjugate of $f(x)$.

For any square-integrable function $f: \mathbb{R} \to \mathbb{C}$ (i.e., $f$ satisfies the property `MemHS f`), let $[f]$ be the corresponding element in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ constructed via the mapping `mk`. The $L^2$ norm of the Hilbert space element $[f]$ with respect to the Lebesgue measure is equal to the $L^2$ norm of the function $f$, i.e., $\|[f]\|_2 = \|f\|_2$.

Let $f: \mathbb{R} \to \mathbb{C}$ be an almost everywhere strongly measurable function. The equivalence class of $f$ belongs to the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ if and only if the function $x \mapsto |f(x)|^2$ is integrable, i.e., $\int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty$.

Let $f, g: \mathbb{R} \to \mathbb{C}$ be square-integrable functions belonging to the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$. Let $[f]$ denote the element (equivalence class) in the Hilbert space corresponding to the function $f$. Then the element representing the sum of the functions $f + g$ is equal to the sum of the elements representing $f$ and $g$ in the Hilbert space, i.e., $[f + g] = [f] + [g]$.

Let $f: \mathbb{R} \to \mathbb{C}$ be a square-integrable function belonging to the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$, and let $c \in \mathbb{C}$ be a complex scalar. Let $[f]$ denote the corresponding element (equivalence class) in the Hilbert space. Then the element representing the scalar multiple function $c \cdot f$ is equal to the scalar multiple of the element representing $f$, i.e., $[c \cdot f] = c \cdot [f]$.

Let $f, g: \mathbb{R} \to \mathbb{C}$ be square-integrable functions belonging to the Hilbert space for one-dimensional quantum mechanics, and let $[f]$ and $[g]$ denote their corresponding elements (equivalence classes) in $L^2(\mathbb{R}, \mathbb{C})$. Then $[f] = [g]$ in the Hilbert space if and only if $f = g$ almost everywhere with respect to the Lebesgue measure.

Let $f$ and $g$ be elements of the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ for one-dimensional quantum mechanics. Then $f = g$ if and only if their representing functions are equal almost everywhere with respect to the Lebesgue measure, denoted as $f = g$ a.e.