Another Hydrogen Question

 

Reference: John S. Rigden, Hydrogen: The Essential Element, Harvard University Press, 2002

 

One line in hydrogen’s spectrum is actually a set of distinct lines. To explain this fine structure, Arnold Sommerfeld, in 1916, modified Bohr’s theory by including elliptical orbits for electrons. (to which he also applied Einstein’s theory of relativity). The resulting energy states of the hydrogen atom were expressed in terms of the dimensionless constant  = 1/137.03599976, and it was named the fine structure constant.

It turns out that  surfaces in many contexts, in so many, that along with the speed of light, c; Newton’s universal constant, G; and Planck’s constant, h, considered by many physicists to be one of the fundamental constants of nature[1]. For example,

a is related to the charge of the electron*2 by

 

Equation(1)    ,

 where k = Coulomb’s constant = 8.9875 X 109 Nm2/C2         

                                    e = charge of the electron = 1.60217733 X10-19 C

 

                                    h = Planck’s constant = 6.6260755 X 10-34 Js

 

                                    c = speed of light = 2.99792458 X 108 m/s,

 

and the fine structure constant is also related to the mass of the neutron by

 

Equation(2)    , where            rn = radius of the neutron= 2.87799 X 10-14 m

                                                                        mn = mass of neutron = 1.6749543 X 10 –27 kg.

                                               

 

Question:        Given equation(3):     re ( radius of electron) = ke2/(mec2),

where me = electron’s mass = 9.1093897 X 10-31 kg,

use basic algebra to express the product of the electron’s mass and radius in terms of the neutron’s mass and radius.

 

Answer:          Isolate ke2 in equations (1) and (3). Then substitute for ke2 to obtain:

 

                        : equation(4).

 

                        Substitute equation(4) into eq(2) :

 

                       

 

                        h2 = 4p2remec2rnmn

 

                       

 

                       

 

 [1] In Scientific American June 2005, Barrow and Webb argue that during the universe’s matter dominated era (we are now in a dark-energy dominated era), the fine structure constant’s value actually increased. The idea came from comparing highly accurate measurements of present spectra to those of quasars, a window into conditions  found 7.5 billion years ago.

 [2] The fine structure constant is also expressed as a= e2/(2e0hc), where e0 is the vacuum permitivity constant, which is related to Coulomb’s constant according to   e0= 1/(4pk)

 

                       

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[1] In Scientific American June 2005, Barrow and Webb argue that during the universe’s matter dominated era (we are now in a dark-energy dominated era), the fine structure constant’s value actually increased. The idea came from comparing highly accurate measurements of present spectra to those of quasars, a window into conditions  found 7.5 billion years ago.