Assume that the de Broglie wave associated with an

Question

Assume that the de Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if that distance d between the atoms of the array is 2Å . A similar standing wave is again formed if d is increased to 2.5 Å but not for any intermediate value of d. Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can form.

Navjyot Kalra · Accepted Answer

Hello Student,

Please find the answer to your question

As nodes are formed at each of the atomic sites, hence

2Å = n (λ / 2) …(1)

[ ∵ Distance between successive nodes = λ / 2]

and 2.5 Å = (n + 1) λ /2

∴ 2.5 /2 = n + 1 / n , 5/ 4 = n + 1 / n or n = 4

Hence, from equation (1),

2Å = 4 λ / 2 i.e., λ = 1 Å

Now, de Broglie wavelength is given by

λ = h / √ 2mK or K = h² / λ² 2m

∴ K = (6.63 x 10^-34)²/ (1 x 10^-10 )² x 2 x 9.1 x 10^-31 x 1.6 x 10^-19 eV

= (6.63)² / 8 x 9. 1 x 1.6 x 10² eV = 151 eV

d will be minimum, when

n = 1, d_min = λ / 2 = 1Å / 2 = 0.5Å

Thanks
Navjot Kalra
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