Question

# An electron confined to a one-dimensional box has energy levels given by the equation En=n2h2/8mL2 where...

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

En=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.

Calculate the energies of the n=1,n=2, and n=3 levels for an electron in a box with a length of 180 pm .

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

In what region of the electromagnetic spectrum do these wavelengths lie?

Length of box = 180 pm = 1.80x10-10 m

Mass of electron = 9.1x10-31 kg

En = n2h2 / 8mL2

Fon n = 1

E1 = 12*(6.626x10-34)2 / 8*9.1x10-31 * (1.80x10-10)2

E1 = 1.86x10-18 J

Fon n = 2

E2 = 22*(6.626x10-34)2 / 8*9.1x10-31 * (1.80x10-10)2

E2 = 7.44x10-18 J

Fon n = 3

E3 = 32*(6.626x10-34)2 / 8*9.1x10-31 * (1.80x10-10)2

E3 = 1.67x10-17 J

n =1 n = 2

We know that,

1 / = R [1/(n1)2 - 1/(n2)2]

where, = wavelength

R = Rydberg's constant (1.097 x 107 m−1)

1 / = (1.097 x 107 m−1) [1/(1)2 - 1/(2)2]

= (1.097 x 107 m−1) [1 - 1/4]

= (1.097 x 107 m−1) * 0.75

= 0.82275 x107 m-1 = 1.215 x 10-7 m = 121.5 nm (ultraviolet region)

For, n =2 n = 3

1 / = (1.097 x 107 m−1) [1/(2)2 - 1/(3)2]

= (1.097 x 107 m−1) [1/4 - 1/9]

= (1.097 x 107 m−1) * 0.138

= 0.15138 x107 m-1 = 6.605 x 10-7 m = 660.5 nm (visible region)

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