More Precisely 4.1 - The Energy Levels of the Hydrogen Atom

Posted by Andri Fadillah Martin on Saturday, March 17, 2012


By observing the emission spectrum of hydrogen and using the connection between photon energy and color first suggested by Einstein (Sec. 4.2), Niels Bohr determined early in the twentieth century what the energy differences between the various energy levels must be. Using that information, he was then able to infer the actual energies of the excited states of hydrogen.

A unit of energy often used in atomic physics is the electron volt (eV). Its name derives from the amount of energy gained by an electron when it moves through an electric potential of one volt. For our purposes, however, it is just a convenient quantity of energy, numerically equal to 1.60 x 10-19J (joule)—roughly half the energy carried by a single photon of red light. The minimum amount of energy needed to ionize hydrogen from its ground state is 13.6 eV. Bohr numbered the energy levels of hydrogen starting at the ground state, with level 1 the ground state, level 2 the first excited state, and so on. He found that by assigning zero energy to the ground state, the energy of any state (the nth, say) could be written as follows:
 

Thus, the ground state has energy E1 = 0 (by our definition), the first excited state has energy    the second excited state has energy    and so on. Notice that there are infinitely many excited states between the ground state and the energy at which the atom is ionized, crowding closer and closer together as n becomes large and En approaches 13.6 eV.


EXAMPLE: Using Bohr’s formula for the energy of each electron orbital, we can reverse his reasoning and calculate the energy associated with a transition between any two given states. To boost an electron from the second state to the third, an atom must be supplied with E3 - E2 = 1.89 eV of energy, or 3.03 x 10-19 J. Using the formula E = hf presented in the text, we find that this corresponds to a photon with a frequency of 4.57 x 1014 Hz, having a wavelength of 656 nm and lying in the red portion of the spectrum. (A more precise calculation gives the value 656.3 nm reported in the text.) Similarly, the jump from level 3 to level 4 requires   eV of energy, corresponding to an infrared photon with a wavelength of 1880 nm, and so on. A handy conversion between photon energies E in electron volts and wavelengths λ in nanometers is
 
The accompanying diagram summarizes the structure of the hydrogen atom. The various energy levels are depicted as a series of circles of increasing radius, representing increasing energy. The electronic transitions between these levels (indicated by arrows) are conventionally grouped into families, named after their discoverers, that define the terminology used to identify specific spectral lines. (Note that the spacings of the energy levels are not drawn to scale here, to provide room for all labels on the diagram. In reality, the circles should become more and more closely spaced as we move outward.)


Transitions starting from or ending at the ground state (level 1) form the Lyman series. The first is Lyman alpha (Lyα), corresponding to the transition between the first excited state (level 2) and the ground state. As we have seen, the energy difference is 10.21 eV, and the Lyα photon has a wavelength of 121.6 nm (1216 Å). The Lyβ (beta) transition, between level 3 (the second excited state) and the ground state, corresponds to an energy change of 12.10 eV and a photon of wavelength 102.6 nm (1026 Å). Lyγ (gamma) corresponds to a jump from level 4 to level 1, and so on. The accompanying table shows how we can calculate the energies, frequencies, and wavelengths of the photons in the Lyman series using the formulae given previously. All Lyman-series energies lie in the ultraviolet region of the spectrum.

The next series of lines, the Balmer series, involves transitions down to (or up from) level 2, the first excited state. All the Balmer series lines lie in or close to the visible portion of the electromagnetic spectrum. Because they form the most easily observable part of the hydrogen spectrum and were the first to be discovered, these lines are often referred to simply as the "Hydrogen" series and denoted by the letter H. As with the Lyman series, the individual transitions are labeled with Greek letters. An Hα photon (level 3 to level 2) has a wavelength of 656.3 nm and is red, Hß (level 4 to level 2) has a wavelength of 486.1 nm (green), Hγ (level 5 to level 2) has a wavelength of 434.1 nm (blue), and so on. The most energetic Balmer series photons have energies that place them just beyond the blue end of the visible spectrum, in the near ultraviolet.

The classification continues with the Paschen series (transitions down to or up from the second excited state), the Brackett series (third excited state), and the Pfund series (fourth excited state). Beyond that point, infinitely many other families exist, moving farther and farther into the infrared and radio regions of the spectrum, but they are not referred to by any special names. A few of the transitions making up the Lyman and Balmer (Hydrogen) series are marked on the figure. Astronomically, these are the most important sequences.






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