We have to calculate (a) the fractional energy difference between two adjacent energy levels of a particle confined in a one-dimensional well of infinite depth; (b) and discuss the result in terms of the correspondence principle.

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(a)

Let a particle of mass m be confined in a square well of width L and infinite depth. The energy levels of the particle are given by the equation

The difference energy of levels with quantum numbers and will be

Therefore, the fractional energy difference between two adjacent energy levels of a particle confined in a one-dimensional well of infinite depth will be

(b)

The correspondence principle states that in the limit of large quantum numbers the quantum system behaves like a classical system. We calculate the fractional shift in energy levels having successive quantum numbers in the limit of large quantum number. That is we calculate

The classical energy of a particle of mass m and momentum p is

.

Therefore,

We recall that momentum of a particle in an infinite square well in state with quantum number n is

Therefore,

We thus note that

which is what we would expect for a free particle in classical mechanics. We have thus verified the correspondence principle.