Question

For a particle in a 1D box with a box length of 1.0 nm, a) Calculate the energy separation between states n-1 and n -2 in eV. b) Calculate the energy separation between states n 8 and n 9 in eV. c) Describe how the energy separation between adjacent energy levels (n and n+1) 4. changes as n increases.

Answer 1

energy of nth level is given as

where h=planck's constant

m=mass of the particle

L=box length=1 nm

as mass is not provided,using values of all other symbols,

energy of nth level=1.372*10^(-30)*n^2/m eV

part a:

difference of energy between n=1 and n=2 is given by

(1.372*10^(-30)*2^2/m )-(1.372*10^(-30)*1^2/m )

=4.116*10^(-30)/m eV

part b:

energy sepration between n=8 and n=9 is given by

(1.372*10^(-30)*9^2/m )-(1.372*10^(-30)*8^2/m )

=2.3324*10^(-29)/m eV

part c:

as n increases, energy sepration=(1.372*10^(-30)*(n+1)^2/m )-(1.372*10^(-30)*n^2/m )

=(1.372*10^(-30)/m)*(n^2+2*n+1-n^2)

=(1.372*10^(-30)/m)*(2*n+1)

so as n increases, the energy sepration increases.

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note: use the value of m for the mass of the particle to get particular answers.