Physlib.QuantumMechanics.Hydrogen.Rydberg

The Rydberg energy $Ry$ of a hydrogen-like atom is defined as $$Ry = \frac{m k^2}{2 \hbar^2}$$ where $m$ is the mass, $k$ is the potential constant, and $\hbar$ is the reduced Planck constant.

The Rydberg energy $Ry$ of a hydrogen-like atom is given by the formula $$Ry = \frac{m k^2}{2 \hbar^2}$$ where $m$ is the mass, $k$ is the potential constant, and $\hbar$ is the reduced Planck constant.

For a natural number $n$, the level index $L_n$ (also known as the effective principal quantum number) of the $n$-th Bohr level in a $d$-dimensional hydrogen atom is defined as: $$L_n = n + \frac{d - 1}{2}$$ In the specific case of $d = 3$, this corresponds to the standard principal quantum number $n + 1$.

For a $d$-dimensional hydrogen atom, the level index $L_n$ (the effective principal quantum number) for a given natural number $n$ is defined by the formula: $$L_n = n + \frac{d - 1}{2}$$ where $d$ is the dimension of the atom.

For a natural number $n$, the $n$-th Bohr energy level $E_n$ of a $d$-dimensional hydrogen atom is defined as: $$E_n = -\frac{Ry}{L_n^2}$$ where $Ry$ is the Rydberg energy and $L_n = n + \frac{d - 1}{2}$ is the effective principal quantum number (or level index). These values represent the expected bound-state energies of the hydrogen atom Hamiltonian. In the case where $d = 1$ and $n = 0$, the level index $L_n$ is zero and the energy level $E_n$ is defined to be $0$.

For a $d$-dimensional hydrogen atom and a natural number $n$, the energy of the $n$-th Bohr level $E_n$ is given by: $$E_n = -\frac{Ry}{L_n^2}$$ where $Ry$ is the Rydberg energy and $L_n$ is the level index (the effective principal quantum number), defined as $L_n = n + \frac{d - 1}{2}$.

The Rydberg energy $Ry$ is non-negative, meaning $Ry \ge 0$.

Let $k$ be the potential constant and $Ry$ be the Rydberg energy of a hydrogen-like atom. If $k \neq 0$, then the Rydberg energy is strictly positive ($Ry > 0$).

If the potential constant $k = 0$, then the Rydberg energy $Ry = 0$.

For a $d$-dimensional hydrogen atom where the dimension $d \neq 0$, the level index $L_n$ (also known as the effective principal quantum number) is non-negative for any $n \in \mathbb{N}$. That is, $0 \leq L_n$, where the level index is defined as $L_n = n + \frac{d - 1}{2}$.

For a $d$-dimensional hydrogen atom, if the dimension satisfies $d \geq 2$, then for any natural number $n \in \mathbb{N}$, the level index $L_n = n + \frac{d - 1}{2}$ is strictly positive ($L_n > 0$).

For a $d$-dimensional hydrogen atom, the level index (effective principal quantum number) defined as $L_n = n + \frac{d - 1}{2}$ is strictly increasing with respect to the natural number $n$. That is, for any $n_1, n_2 \in \mathbb{N}$, if $n_1 < n_2$, then $L_{n_1} < L_{n_2}$.

The level index $L_n$ (effective principal quantum number) of a $d$-dimensional hydrogen atom, defined as $L_n = n + \frac{d - 1}{2}$ for $n \in \mathbb{N}$, is monotone. That is, for any natural numbers $n_1$ and $n_2$, if $n_1 \le n_2$, then $L_{n_1} \le L_{n_2}$.

For any natural number $n$, the $n$-th Bohr energy level $E_n$ of the $d$-dimensional hydrogen atom is non-positive, i.e., $E_n \leq 0$.

For a $d$-dimensional hydrogen atom with potential constant $k$, if the dimension satisfies $d \geq 2$ and the potential constant $k \neq 0$, then for any natural number $n$, the $n$-th Bohr energy level $E_n$ is strictly negative, i.e., $E_n < 0$.

Suppose the potential constant $k$ of a $d$-dimensional hydrogen atom is $0$. Then for any natural number $n$, the $n$-th Bohr energy level $E_n$ is equal to $0$.

In a $d$-dimensional hydrogen atom with potential parameter $k \neq 0$ and dimension $d \geq 2$, the sequence of Bohr energy levels $E_n$ is strictly increasing with respect to $n$. That is, for any natural numbers $n_1 < n_2$, the corresponding energy levels satisfy $E_{n_1} < E_{n_2}$.

For a $d$-dimensional hydrogen atom, if the dimension $d \ge 2$, then the sequence of Bohr energy levels $E_n$ is monotonically increasing. That is, for any natural numbers $n$ and $m$, if $n \le m$, then $E_n \le E_m$.

For two natural numbers $n_1$ and $n_2$, the transition energy is the difference between the $n_2$-th and $n_1$-th Bohr energy levels, defined as $E_{n_2} - E_{n_1}$. This represents the energy associated with a transition between these two levels.

For any two natural numbers $n_1$ and $n_2$, the transition energy between the corresponding Bohr energy levels is the difference between the $n_2$-th energy level and the $n_1$-th energy level: $$\Delta E(n_1, n_2) = E_{n_2} - E_{n_1}$$ where $E_n$ denotes the $n$-th energy level `energyLevel n` and $\Delta E(n_1, n_2)$ denotes the transition energy `transitionEnergy n₁ n₂`.

For two natural numbers $n_1$ and $n_2$ representing energy levels of a hydrogen atom, the transition frequency is defined as the ratio of the transition energy $E_{n_2} - E_{n_1}$ to the Planck constant $h$: $$\nu = \frac{E_{n_2} - E_{n_1}}{h}$$ This value is positive when $n_1 < n_2$ (corresponding to the emission of a photon), negative when $n_2 < n_1$ (corresponding to the absorption of a photon with frequency $|\nu|$), and zero when $n_1 = n_2$.

For any two natural numbers $n_1$ and $n_2$ representing energy levels of a hydrogen atom, the transition frequency $\nu$ between these levels is equal to the transition energy $\Delta E(n_1, n_2)$ divided by the Planck constant $h$: $$\nu = \frac{\Delta E(n_1, n_2)}{h}$$ where $\Delta E(n_1, n_2) = E_{n_2} - E_{n_1}$ is the energy difference between the $n_2$-th and $n_1$-th Bohr levels.

For any two natural numbers $n_1$ and $n_2$ representing energy levels of a hydrogen atom, let $\nu(n_1, n_2)$ denote the transition frequency. Exchanging the two levels changes the sign of the transition frequency: $$\nu(n_2, n_1) = -\nu(n_1, n_2)$$

For any energy level index $n \in \mathbb{N}$, the transition frequency $\nu$ between the level $n$ and itself is $0$.

The Rydberg frequency $R_\nu$ is the real number defined as the ratio of the Rydberg energy $Ry$ to the Planck constant $h$: $$R_\nu = \frac{Ry}{h}$$ It represents the frequency of a photon whose energy is equal to the Rydberg energy.

For a given speed of light $c$ and two natural numbers $n_1$ and $n_2$ representing energy levels of a $d$-dimensional hydrogen atom, the wavelength $\lambda$ associated with the transition from level $n_2$ to level $n_1$ is defined as: $$\lambda = \frac{c}{\nu}$$ where $\nu$ is the transition frequency between these levels. When $n_2 > n_1$ (and assuming $d \ge 2, k \neq 0$), $\lambda$ represents the wavelength of the emitted photon. If $n_2 < n_1$, the transition corresponds to absorption, and the resulting wavelength is negative, with its absolute value $|\lambda|$ representing the physical wavelength of the absorbed photon. For $n_1 = n_2$, where the frequency is zero, the wavelength is defined as $0$.

For any speed of light $c$ and natural numbers $n_1$ and $n_2$ representing indices of energy levels, the transition wavelength $\lambda$ between these levels is given by the ratio of the speed of light to the transition frequency $\nu$: $$\lambda = \frac{c}{\nu}$$

For any two natural numbers $n_1$ and $n_2$, the transition energy $E_{n_2} - E_{n_1}$ between the $n_2$-th and $n_1$-th Bohr energy levels of a $d$-dimensional hydrogen atom is given by the Rydberg formula: $$E_{n_2} - E_{n_1} = Ry \left( \frac{1}{L_{n_1}^2} - \frac{1}{L_{n_2}^2} \right)$$ where $Ry$ is the Rydberg energy and $L_n = n + \frac{d - 1}{2}$ is the level index (effective principal quantum number) for the $n$-th level.

The Rydberg frequency $R_\nu$ is given by the expression: $$R_\nu = \frac{m k^2}{4 \pi \hbar^3}$$ where $m$ is the mass of the particle, $k$ is the coupling constant of the potential, and $\hbar$ is the reduced Planck constant.

The Rydberg frequency $R_\nu$ is non-negative, that is, $R_\nu \ge 0$.

If the potential parameter $k$ is non-zero ($k \neq 0$), then the Rydberg frequency $R_\nu$ is strictly positive ($R_\nu > 0$).

If the potential parameter $k$ is equal to $0$, then the Rydberg frequency $R_\nu$ is equal to $0$.

For any natural numbers $n_1$ and $n_2$, the transition frequency $\nu$ between the Bohr levels $n_2$ and $n_1$ is given by the Rydberg formula: $$\nu = R_\nu \left( \frac{1}{L_{n_1}^2} - \frac{1}{L_{n_2}^2} \right)$$ where $R_\nu$ is the Rydberg frequency and $L_n = n + \frac{d - 1}{2}$ is the level index (effective principal quantum number) for the $n$-th energy level in $d$ dimensions.

Consider a $d$-dimensional hydrogen atom with potential parameter $k$. If $d \geq 2$ and $k \neq 0$, then for any two energy level indices $n_1, n_2 \in \mathbb{N}$ such that $n_1 < n_2$, the transition frequency $\nu(n_1, n_2)$ is strictly positive.

For a $d$-dimensional hydrogen atom with dimension $d \ge 2$ and potential constant $k \neq 0$, the transition frequency $\nu(n_1, n_2)$ between the levels with indices $n_1$ and $n_2$ is strictly negative if $n_2 < n_1$. This corresponds to an upward transition (absorption of a photon) from level $n_2$ to level $n_1$.

For a given speed of light $c$ and two energy level indices $n_1, n_2 \in \mathbb{N}$ of a $d$-dimensional hydrogen atom, the inverse of the transition wavelength $\lambda$ (the wavenumber) is given by the Rydberg formula: $$\frac{1}{\lambda} = \frac{R_\nu}{c} \left( \frac{1}{L_{n_1}^2} - \frac{1}{L_{n_2}^2} \right)$$ where $R_\nu$ is the Rydberg frequency and $L_n$ is the level index (effective principal quantum number) defined as $L_n = n + \frac{d-1}{2}$.

Consider a hydrogen atom in dimension $d \ge 2$ with potential parameter $k \neq 0$. For a fixed energy level index $n_1 \in \mathbb{N}$, the transition frequency $\nu(n_1, n_2) = \frac{E_{n_2} - E_{n_1}}{h}$ is strictly increasing with respect to the level index $n_2$. That is, for any $n_2, n_3 \in \mathbb{N}$, if $n_2 < n_3$, then $\nu(n_1, n_2) < \nu(n_1, n_3)$.

Let $d$ be the dimension of a hydrogen atom such that $d \ge 2$. For a fixed lower energy level index $n_1 \in \mathbb{N}$, the transition frequency $\nu(n_1, n_2)$ is monotonically non-decreasing with respect to the upper energy level index $n_2 \in \mathbb{N}$. That is, for any $n_2, n_3 \in \mathbb{N}$, if $n_2 \le n_3$, then $\nu(n_1, n_2) \le \nu(n_1, n_3)$.

For a fixed lower energy level $n_1 \in \mathbb{N}$, as the upper energy level $n_2$ tends to infinity, the transition frequency $\nu(n_1, n_2)$ converges to the series limit: $$\lim_{n_2 \to \infty} \nu(n_1, n_2) = \frac{R_\nu}{L_{n_1}^2}$$ where $R_\nu$ is the Rydberg frequency and $L_{n_1} = n_1 + \frac{d-1}{2}$ is the effective principal quantum number (level index) for the level $n_1$ in $d$ dimensions.

For a natural number $n$, the Lyman frequency $\nu_{\text{Lyman}}(n)$ is defined as the frequency of a photon associated with the transition between the $n$-th energy level and the ground state (level $0$). It is calculated as the transition frequency from level $0$ to level $n$: $$\nu_{\text{Lyman}}(n) = \nu(0, n) = \frac{E_n - E_0}{h}$$ where $E_n$ is the energy of the $n$-th level and $h$ is the Planck constant.

For a natural number $n$, the Balmer frequency is defined as the frequency of the photon associated with the transition between the energy level of index $n$ and the energy level of index $1$. It represents the $n$-th line of the Balmer series for a $d$-dimensional hydrogen atom, calculated as: $$\nu = \frac{E_n - E_1}{h}$$ where $E_n$ and $E_1$ are the Bohr energy levels and $h$ is Planck's constant.

For a natural number $n$ representing an energy level, the Paschen frequency is the frequency of the photon involved in the transition between level $n$ and the level with index 2 in a $d$-dimensional hydrogen atom. It is defined as: $$\nu = \frac{E_n - E_2}{h}$$ where $E_n$ and $E_2$ are the Bohr energy levels and $h$ is the Planck constant.

In a $d$-dimensional hydrogen atom model with $2 \leq d \leq 5$ and a non-zero coupling constant $k$, the frequency of any line in the Balmer series is strictly less than the frequency of any line in the Lyman series. Formally, for any $n_2 \in \mathbb{N}$ and $n_3 \in \mathbb{N}^+$ (where $n_3 > 0$), the Balmer frequency $\nu_{\text{Balmer}}(n_2)$ and the Lyman frequency $\nu_{\text{Lyman}}(n_3)$ satisfy the inequality: $$\nu_{\text{Balmer}}(n_2) < \nu_{\text{Lyman}}(n_3)$$ where the frequencies are defined by the energy transitions $\nu_{\text{Balmer}}(n) = \frac{E_n - E_1}{h}$ and $\nu_{\text{Lyman}}(n) = \frac{E_n - E_0}{h}$ between the Bohr levels $E_n = -Ry / (n + \frac{d-1}{2})^2$. This separation of spectral series holds because for these dimensions, the Balmer series limit is less than or equal to the first Lyman line frequency.