Model of Hydrogen Atom
|* To prove L=angular momentum=mvr=nh/2π
mv2/r=kze2/r2 or (mvr)v=kze2 or Lv=kze2
Ek =mv2 /2 =kze2 /2r
Ep =-kze2 /r =-mv2
TE =-kze2 /2r =-mv2 /2 =-mk2z2e4 /2L2 .
Differentiating, d(TE) =(mk2z2e4 /L3 )dL=(mv2L2 /L3)dL=mv2dL/L =hν=h/T=hv/2πr or mvr*dL/L = h/2π or dL=h/2π
* To find out value of r, v, TE,λ
Lv=kze2 or (nh/2π)v= kze2 or v=2πkze2 /nh = z*( 2πkze2/h)*(1/n)=z*α /n
r= kze2 / mv2 or r =n2h2 / 4π2mkze2
TE=-mv2 /2 =-2π2mk2z2e4 /n2h2
ΔTE=hc/ λ =(2π2mk2z2e4/h2) [1/n12 - 1/n22]
or 1/λ =(2π2mk2z2e4/ch3) [1/n12 - 1/n22]=(2π2mk2e4/ch3) [1/n12 - 1/n22] when z=1
or 1/λ =R [1/n12 - 1/n22] where R=(2π2mk2z2e4/ch3)
* To find dimension of the atom
From de broglie hypothesis, λ = h/p = h/mv
We know that when a string of length l is tied to a rigid support at both ends and is stretched, it vibrates in such a manner so that the nodes are at both ends and the condition that is satisfied is sin (2πl/λ) =0 or 2πl/λ =nπ where n is any integer for standing wave to be formed.
λ = 2l/n or l=n (λ/2)
Hence, 2l/n=h/mv or p= nh/2l Taking l=2πr, we get 4πr =nh/mv or mvr=nh/4π
E=p2/2m = n2h2/8ml2 .....(a). This is the total energy of a vibrating particle trapped in a box of length l.
But E=2π2mk2z2e4 /n2h2 ....(b) ;
So equating (a) & (b), we get l=n2h2 /4πmkze2
whose dimension is of the order of 10-10 m.If in the Bohr equation, we substitute mass of electron by mass of neutron, the dimension comes to the order of 10-14m which is roughly the size of nucleus.
* To prove mvr=nh/2π from de broglie wavelength
We already know that the condition for standing wave in case of a stretched string of length l tied rigidly to both ends is
de Broglie considered the circumference of Bohr's circular orbit as a string of length l=2πr where standing waves are formed. He assumed that these are not ordinary waves but matter waves of energy E=pc where p is the momentum & E = hc/λ . This follows from Einstein's Special theory of relativity E=√(m02c4 +p2c2). For entities with zero rest mass E=pc
Equating E from both cases, λ=h/p
Here de broglie enters the scenrio , where he postulated the symmetry between mass less entities and entities having non-zero rest mass and extended the concept to λ=h/p where p=mv for these entities. Hence , this was a postulate and conclusion was not reached through derivation.
Now, l= nh/(2mv) or 2πr= nh/(2mv) or mvr=nh / 4π. or angular momentum = h/4π, 2h/4π,3h/4π, .......
de Broglie assumed l=n λ . Why ???
possible explanation is that when matter is converted to energy, there is equal contribution of matter and anti-matter.
Hence anti-matter also contributes anti-matter angular momentum= h/4π, 2h/4π,3h/4π, .......
Hence total angular momentum of matter wave= h/2π, 2h/2π,3h/2π, .......nh/2π or mvr=nh/2π (this is not in any text book but my assumption)
* de broglie wavelength is velocity dependant whereas compton wavelength is not velocity dependant.