1.

Classically, anelectron can be in any orbit around the nucleus of an atom. Then whatdetermines the typical atomic size? Why is an atom not, say, thousandtimes bigger than its typical size? The question had greatly puzzledBohr before he arrived at his famous model of the atom that you havelearnt in the text. To simulate what he might well have done beforehis discovery, let us play as follows with the basic constants ofnature and see if we can get a quantity with the dimensions of lengththat is roughly equal to the known size of an atom (~ 10−10m).(a) Construct a quantity with the dimensions of length fromthe fundamental constants e, me, andc. Determine its numerical value.(b) You will find that the length obtained in (a) is manyorders of magnitude smaller than the atomic dimensions. Further, itinvolves c. But energies of atoms are mostly innon-relativistic domain where c is not expected to play anyrole. This is what may have suggested Bohr to discard c andlook for ‘something else’ to get the right atomic size.Now, the Planck’s constant h had already made itsappearance elsewhere. Bohr’s great insight lay in recognisingthat h, me, and e will yieldthe right atomic size. Construct a quantity with the dimension oflength from h, me, and e andconfirm that its numerical value has indeed the correct order ofmagnitude.

Answer»

Classically, an
electron can be in any orbit around the nucleus of an atom. Then what
determines the typical atomic size? Why is an atom not, say, thousand
times bigger than its typical size? The question had greatly puzzled
Bohr before he arrived at his famous model of the atom that you have
learnt in the text. To simulate what he might well have done before
his discovery, let us play as follows with the basic constants of
nature and see if we can get a quantity with the dimensions of length
that is roughly equal to the known size of an atom (~ 10−10
m).



(a) Construct a quantity with the dimensions of length from
the fundamental constants e, me, and
c. Determine its numerical value.



(b) You will find that the length obtained in (a) is many
orders of magnitude smaller than the atomic dimensions. Further, it
involves c. But energies of atoms are mostly in
non-relativistic domain where c is not expected to play any
role. This is what may have suggested Bohr to discard c and
look for ‘something else’ to get the right atomic size.
Now, the Planck’s constant h had already made its
appearance elsewhere. Bohr’s great insight lay in recognising
that h, me, and e will yield
the right atomic size. Construct a quantity with the dimension of
length from h, me, and e and
confirm that its numerical value has indeed the correct order of
magnitude.


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