Question

An electron confined to a one-dimensional box has energy levels given by the equation

E_n=n²h²/8mL²

where n is a quantum number with possible values of 1,2,3,…,m is the mass of the particle, and L is the length of the box.
Part a
Calculate the energies of the n=1,n=2, and n=3 levels for an electron in a box with a length of 355 pm .

Enter your answers separated by a comma.

Part b
Calculate the wavelength of light required to make a transition from n=1→n=2 and from n=2→n=3.

Enter your answers separated by a comma.

Answer 1

part A )

E = n^2 h^2 / 8 m L^2

a)   n = 1

h = 6.626 x 10^-34 J s

L = 355 pm = 355 x 10^-12 m

mass of the electron = 9.10 x 10^-31 kg

E = n^2 h^2 / 8 m L^2

      = 1^2 x (6.626 x 10^-34)^2 / 8 x (9.10 x 10^-31) x (355 x 10^-12)^2

E    = 4.78 x 10^-19 J

n = 2

E    = 2^2 x (6.626 x 10^-34)^2 / 8 x (9.10 x 10^-31) x (355 x 10^-12)^2

E = 1.91 x 10^-18 J

n= 3

E    = 3^2 x (6.626 x 10^-34)^2 / 8 x (9.10 x 10^-31) x (355 x 10^-12)^2

E = 4.31 x 10^-18 J

part b )

n = 1 ----------> n= 2

E = (n²₂-n₁²) h² / 8 m L²

E = (2^2 - 1^1) x (6.626 x 10^-34)^2 / 8 x (9.10 x 10^-31) x (355 x 10^-12)^2

E = 1.434 x 10^-18 J

E = hc /

= h c / E

     = (6.626 x 10^-34) x ( 3 x 10^8) / (1.434 x 10^-18)

     = 1 .386 x 10^-7 m

      = 138.6 x 10^-9 m

      = 138.6 nm

wave length = 138.6 nm

n = 2 ----------> n= 3

E = (n²₂-n₁²) h² / 8 m L²

E = (3^2 - 2^2) x (6.626 x 10^-34)^2 / 8 x (9.10 x 10^-31) x (355 x 10^-12)^2

E = 2.393 x 10^-18 J

E = hc /

= h c / E

     = (6.626 x 10^-34) x ( 3 x 10^8) / (2.393 x 10^-18)

     = 8.307 x 10^-8 m

      = 83.07 x 10^-9 m

      = 83.07 nm

wave length = 83.07 nm