It is useful to consider the result for the energy eigenvalues for the one-dimen

Question

It is useful to consider the result for the energy eigenvalues for the one-dimensional box E_n = h^2 n^2/8 ma^2, n = 1, 2, 3, .... as a function of n, m, and a. By what factor do you need to change the box length to decrease the zero point energy by a factor of 400 for a fixed value of m? By what factor would you have to change n for fixed values of a and m to increase the energy by a factor of 400? By what factor would you have to increase a at constant n to have the zero point energies of an electron be equal to the zero point energy of a proton in the box?

Explanation / Answer

a.)

E = h2n2/8ma2

n = 1 is the quantum number corresponding to the zero point energy.

h is planck 's constant, m is constant (given), 8 is a number.

E is inversly proprtional to a2.

The box has to be decreased by a factor of 20, so that a2 becomes 400.

b.)

E = h2n2/8ma2

For a fixed values of a and m, n has to be increased by a factor of 20, to increase the energy by a factor of 400.

c) The mass of proton is 1837 times heavier than an electron. The factor by which one has to increase a at constant n to have a equal to square root of 1837, which is equal to 42.86 or 43 times.

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