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A particle of mass m is confined to a na...

A particle of mass m is confined to a narrow tube of length L.
(a)Find the wavelength of the de-Broglie wave which will resonate in the tube.
(b)Calculate the corresponding particle momenta
(c )Calculate the corresponding energies.

Text Solution

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(a) The de-Brogile waves will resonate with a node at each end of the tube.
Few of the possible resonance forms are as follows : `lambda_(n) = (2L)/(n), n = 1,2,3"…….."`

(b) Since d-Broigle wavelengths are `lambda_(n) = (h)/(p_(n))`
`p_(n) = (h)/(lambda_(n)) = (nh)/(2L) , n = 1,2,3"........."`
(c) The kinetic energy of the particles are `K_(n) = (p_(n)^(2))/(2m) - (n^(2)h^(2))/(8L^(2)m) , n = 1, 2, 3"....."`
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