In hydrogen’s characteristic spectra, each series - the Lyman, the Balmer, and so on – has a “series limit,” where the wavelengths at one end of the series tend to bunch up, approaching a single limiting value. Is it at the short-wavelength or the long-wavelength end of the series that the series limit occurs, and what is it about hydrogen’s allowed energies that leads to this phenomenon? Does the infinite well have series limits?

The allowed energy of the hydrogen atom is,

${\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{-}\frac{\mathbf{m}{\mathbf{e}}^{\mathbf{4}}}{\mathbf{2}{\left(4\pi {\epsilon }_0\right)}^{\mathbf{2}}{\sout{\mathbf{h}}}^{\mathbf{2}}\mathbf{n}}$

The allowed energy in a 3D potential well is,

${\mathit{E}}_{{\mathit{n}}_{\mathit{x}}\mathit{,}{\mathit{n}}_{\mathit{y}}\mathit{,}{\mathit{n}}_{\mathit{z}}}\mathit{=}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)\frac{{\mathit{\pi }}^{\mathit{2}}{\sout{\mathit{h}}}^{\mathit{2}}}{\mathit{2}\mathit{m}}$

Here n is the integer, m78C8 is the mass of the electron, e is the charge of the electron, n_x , n_y, n_z, are integers and ε₀is the permittivity of free space.

Taking a gander at the recipe for the permitted energies of the hydrogen particle from the condition,

$E_n=-\frac{m{e}^4}{2{\left(4\pi {\epsilon }_0\right)}^2{\sout{h}}^{2}n}$

Thus, the ground level has the best bad energy worth, and afterward, the worth declines by a component of . The distinction between the energy levels declines too, as gets increasingly elevated.

As approaches vastness here, the energy of the electron becomes zero, which implies that the electron is free and not bound to the particle any longer.

Taking a gander at the recipe for the permitted energies in a 3D-boundless likely well from the condition,

$E_{n_x\mathit{,}{n}_y\mathit{,}{n}_z}\mathit{=}\mathit{(}\frac{n_x^{\mathit{2}}}{L_x^{\mathit{2}}}\mathit{+}\frac{n_y^{\mathit{2}}}{L_y^{\mathit{2}}}\mathit{+}\frac{n_z^{\mathit{2}}}{L_z^{\mathit{2}}}\mathit{)}\frac{{\pi }^{\mathit{2}}{\sout{h}}^{\mathit{2}}}{\mathit{2}m}$

For a cubic well, and taking $n_x^2+n_y^2+n_z^2=n^2$this becomes,

$E_n=\frac{n^2{\pi }^2{\sout{h}}^2}{2mL}$

Thus, the ground level has the most reduced energy worth, and afterward, the worth increments by a variable of . The distinction between the energy levels also increases as n ever more elevated.

Here, the molecule cannot have zero energy. The most minimal energy is the ground state energy or the zero-point energy.

From the above figure, in the hydrogen molecule, the levels bundle up (draw nearer to one another) at the top. Thusly, as far as possible will relate to the most noteworthy energy progress in a specific series (ionization energy). Since energy is conversely relative to the frequency as indicated by $E=\frac{hc}{\lambda }$.

Consequently, as far as possible is at the short-wavelength end of the series. For the limitless well, there is no "bundling up" of the levels, as they get further separated from one another, not nearer as n increments. In like manner, the endless well doesn't have a series limit like that of the hydrogen iota.

Hence, the short-frequency end. No, the boundless well does not have series limits.