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12.3.3: Kinetic Energy Corrections

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    Before claiming that this formula explains the fine structure of the hydrogen atom, however, one needs to be careful. The correction is of the order tex2html_wrap_inline1552 , which means it of the order tex2html_wrap_inline1554 , where v is the electron speed. The kinetic energy used in the Hamiltonian when solving the Schrödinger equation was just tex2html_wrap_inline1558 , which contributed to order tex2html_wrap_inline1560 . However, the next term in the expansion of the true relativistic kinetic energy is of order tex2html_wrap_inline1562 and hence will contribute to order tex2html_wrap_inline1552 . So if one wishes to quote the energy splittings of the hydrogen atom accurate to order tex2html_wrap_inline1552 , one had better include the contribution from this further correction.

    The relativistic kinetic energy of the electron can be expanded in terms of momentum as


    Therefore, the correction to the Hamiltonian is


    At first sight, this looks quite complicated, since it involves the operator tex2html_wrap_inline1568 . However, one can make use of the fact that


    to get


    With tex2html_wrap_inline1570 , applying first-order perturbation theory to this Hamiltonian reduces to the problem of finding the expectation values of tex2html_wrap_inline1572 and tex2html_wrap_inline1574 . This can be done with some effort, and the result is


    Combining equations 57 and 52 and using the fact that j=l-1/2,l+1/2, the complete energy correction to order tex2html_wrap_inline1578 may be written


    This energy correction depends only on j and is called the fine structure of the hydrogen atom, since it is of order tex2html_wrap_inline1582 times smaller than the principle energy splittings. This is why tex2html_wrap_inline1584 is known as the fine-structure constant. The fine structure of the hydrogen atom is illustrated in figure 4. Note that all levels are shifted down from the Bohr energies, and that for every n and l there are two states corresponding to j=l-1/2 and j=l+1/2, except for s states. Also note that states with the same n and j but different l have the same energies, though this will be shown later not to be true due an effect know as the Lamb shift. As an aside, these fine structure splittings were derived by Sommerfeld by modifying the Bohr theory to allow elliptical orbits and then calculating the energy differences between the different states due to differences in the average velocity of the electron. By using the wrong method he got exactly the right answer, a coincidence which caused much confusion at the time.

    Figure 4: The fine structure of the hydrogen atom. The diagram is not to scale.

    Strictly speaking, equation 58 has only been shown to be correct for tex2html_wrap_inline1602 states, although it turns out to be correct for all l. To do the calculation correctly for l=0, one needs to include the effect of an additional term in the Hamiltonian known as the Darwin term, which is purely an effect of relativistic quantum mechanics and can only be understood in the context of the Dirac theory. It is therefore appropriate to discuss the Dirac theory to achieve a more complete understanding of the fine structure of the hydrogen atom.

    Contributors and Attributions

    • Randal Telfer (JWST Astronomical Optics Scientist, Space Telescope Science Institute)

    This page titled 12.3.3: Kinetic Energy Corrections is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.