3.9 Sspd_chapter1_part6_neil bohr's quantum theory of atoms.  (Page 2/2)

Total Energy = E = Kinetic Energy + Potential Energy;

Therefore E = p ^2 /(2m e ) – qV(r) 1.16

Where p= linear momentum of the electron;

m _e = mass of the electron;

q = the charge of an electron;

V(r) = potential at position r .

Wave number or wave propagation vector k = 2π/λ = (2π/h)p = p/ћ

Or p = k ћ 1.17

An electron in free space is an unbounded electron and can assume a continuum of energy values as p can assume a continuum of linear momentum values. p can assume a continuum of linear momentum values because in free space there are no boundary conditions to restrict the linear momentum values.

From Eq.(1.16) E vs p or E vs k is a parabolic curve as shown in Fig.(1.12).

Figure.1.12. E vs k curve for an electron in free space.

Where m _e ^* = effective mass of the electron.

When an electron is restricted to an infinite potential well [Figure 1.13.C] then the matter wave as a traveling wave is converted to a standing wave. In a potential well only those standing waves are permitted which satisfy the condition of nodes at the potential well boundaries just as a vibrating spring can vibrate only at those notes where the standing waves have nodes at the two ends of the string which are tied to the side walls. That is the wavelength of the standing wave must satisfy the condition:

W = (n . ½)λ 1.18

Where n= 0, 1, 2, 3 ; and W is the width of the potential well.

Fig(1.13.a and b and c) give the permitted standing wave pattern for a vibrating string tied on the two ends as well as for an electron in One-Dimensional Potential Well with infinite potential barrier at the two ends and the figure of an infinite potential well respectively..

Fig(1.13.A) The Standing Wave Pattern for a vibrating string tied on the two ends;

For a standing wave on a vibrating string tied on two ends, the following condition has to be satisfied:

(N).(1/2)(λ n ) = L 1.19

Fig.(1.13.B) The Standing Wave Pattern for an electron in 1-D infinte Potential Well in 1st,3rd,5th quantum states.

Figure 1.13.C. An infinite potential well of width W microns

For standing wave in an infinite potential well, the following condition has to be satisfied:

(n.1/2)(λ n ) = W 1.20

The positively charged nucleus of an Atom creates 3-D Potential Well of spherical symmetry. In this finite space, electrons can exist only as Standing Waves with the boundary condition:

2.π.r = n.(h/p n ) = n.λ n 1.21

where n = 1, 2, 3, 4,………..

Eq.(1.21) implies that electrons have physical existence only when the circumference of the permissible orbit is an integral multiple of the matter wave wavelength. This directly takes us to the Postulates of Neil Bohr.

Rewriting Eq.(1.15),

Iω = nћ = nh/(2π) 1.22

Where I = moment of inertia of the orbiting electron = mr ² ;

ω = orbital angular velocity = v/r ;

Substituting the values of I and ω in Eq.(1.22) we obtain the following Equation:

m.v.r = nh/(2π) 1.23

Reshuffling the terms we get:

2.π.r = n.h/(mv n ) = n.h/(p n ) = n.λ n

Thus the first Postulate of Neil Bohr is a restatement of the fact that electron is a real material particle as well as it is a matter wave and hence electron’s permitted orbits are radiationless and support the standing wave pattern only.

Neil Bohr inadvertently corroborated the wave nature of electrons..

When we are working at macroscopic level that is at cm and meter level, then Newtononian Mechanics and Maxwell’s Electromagnetic Theory suffice for neutral and charged particles. But when we are working at atomic scale i.e. at Angstrom scale then we have to invoke Quantum Mechanics.

When we increase the scale from Angstrom to meter then Quantum Mechanics results correspond to those of Classical Mechanics. This known as Correspondence Principle.