When an electric discharge passes through a tube filled with hydrogen and the emitted light is separated by a prism, 4 prominent lines appear: red, blue-green, blue, and violet.
Adjust conditions:
The sliders adjust the values
n=2→1で出る光子のエネルギーは、n=3→2の何倍か？
n=3から1へ直接移るとき、出る光の波長はいくらか？
The equations explain the phenomenon
1We write the equation of motion for the electron's circular motion.m v²r=k e²r²……
2We express the electron's energy in terms of the radius.E=−k e²2 r……
3We find the rate at which the radius decreases as electromagnetic waves are emitted.drdt=−(4/3) re² cr²……
4We integrate to find the time it takes the electron to fall into the proton.tfall≒1.6×10⁻¹¹ s……
Classical theory predicts that atoms are unstable (Step 4).We now assume that the electron emits no electromagnetic waves while it remains in certain orbits (stationary states). The following quantum condition determines the allowed orbits.
5We impose the condition that the circumference is an integer multiple of the electron's wavelength.2π r=n hm v……
6We find the allowed radii from the 2 conditions.rn=n² a₀……
7We substitute the radius to find the energy levels.En≒−13.6 eVn²……
We now assume that when the electron changes energy levels, 1 photon carries away the energy difference.From here on, m and n label energy levels (m is the initial level, n is the final level, and m ＞ n). The electron's mass is written as me.
8We relate the energy difference between levels to the photon energy.h ν=Em − En……
9We compare the energies of the 2 photons.h ν₂h ν₁=(1 − 14)/(14 − 19)=5.4……
10We convert photon energy to wavelength.1λ=R (1n² − 1m²)……
11We calculate the visible spectral lines.Balmer series n=2→656 nm，486 nm，434 nm，410 nm……
Further topics: invisible light and ionization
12We also calculate ultraviolet and infrared wavelengths.Lyman: 121.6 nm，Paschen: 1876 nm……
13We find the energy required to remove the electron from the atom.E∞ − E₁=13.6 eV……
What this chapter has shown
An electron bound to an atom has discrete energies given by −13.6 eV/n². The energy of an emitted photon is determined by the difference between 2 levels, so the wavelengths are also discrete, producing spectral lines of specific colors.
5.4 timesRatio of photon energies
A direct transition 3→1 emits 1 photon with an energy of 12.09 eV and a wavelength of approximately 103 nm.
In everyday lifeThe colors of neon signs are also determined by differences between atomic energy levels.
Detailed summary
The electron in a hydrogen atom was modeled as undergoing circular motion, with the electrostatic force providing the centripetal force. Adding just 1 condition—that the circumference is an integer multiple of the de Broglie wavelength—restricts the radius and energy to discrete values. Since a photon's energy equals the difference between 2 energy levels, the emitted light also has discrete wavelengths, and the calculated values match the observed spectral lines. The energy levels follow a 1/n² dependence, so lower levels are spaced farther apart.