Modern Physics: Schro?dinger Equation Use the uncertainty principle involving ?x and ?p to explain why none...

Modern Physics: Schro?dinger Equation

Use the uncertainty principle involving ?x and ?p to explain why none of the potential wells investigated have a ground state energy of E = 0.

Solutions

Expert Solution

The Eigenfunctions and eigenvalues of a particle of mass m moving in rigid box of side L is given by

and where n is integer except zero.Now the exclusion of n=0 means that particle cannot have zero energy.This quantum mechanical result has no counterpart in classical mechanics where all energies including zero are possible.The lowest value of n is 1, and thus the lowest allowed value of energy is . This is consistent with uncertainity principle .Since particle is confined to the box, the uncertanity in its position is

. The uncertainity in its momentum must therefore be

and uncertainity in its kinetic energy would be .

The minimum energy must be of the order as uncertainity in energy.This is consistent with the value obtained by putting n=1 in the energy expression.Thus the uncertainity principle provides confirmation that E=0 is not admissible.

genius_generous answered 1 year ago

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