A particle in a one-dimensional box of length L has a kineticenergy much greater

Question

A particle in a one-dimensional box of length L has a kineticenergy much greater than its rest energy. What is the ratio of thefollowing energy levels En: E2/E1,E3/E1, E4/E1? How doyour answers compare with the nonrelativistic case?

The answers are:
E2/E1 =([1+h2c2/(L2E02)]/[1+h2c2/(4L2E02)])1/2 E3/E1 =([1+9h2c2/(4L2E02)]/[1+h2c2/(4L2E02)])1/2 E4/E1 =([1+4h2c2/(L2E02)]/[1+h2c2/(4L2E02)])1/2 The formula I think you use is: En =(n2h2)/(8ml2), but I can't figureout where the 1+... or the square root comes from. A particle in a one-dimensional box of length L has a kineticenergy much greater than its rest energy. What is the ratio of thefollowing energy levels En: E2/E1,E3/E1, E4/E1? How doyour answers compare with the nonrelativistic case?

The answers are:
E2/E1 =([1+h2c2/(L2E02)]/[1+h2c2/(4L2E02)])1/2 E3/E1 =([1+9h2c2/(4L2E02)]/[1+h2c2/(4L2E02)])1/2 E4/E1 =([1+4h2c2/(L2E02)]/[1+h2c2/(4L2E02)])1/2 The formula I think you use is: En =(n2h2)/(8ml2), but I can't figureout where the 1+... or the square root comes from. The formula I think you use is: En =(n2h2)/(8ml2), but I can't figureout where the 1+... or the square root comes from.

Explanation / Answer

The link derives the following formula for the n-th energy level (though they use k, rather than n):
En = [(mc

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