Question

It is useful to consider the result for the allowed P14.7 values of the total energy for the one-dimensional box E hn'18ma2,n 1, 2, 3,...as a function of n, m, and a. 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? values of a and m to increase the energy by a factor of 400? n to have the zero point energies of an electron be equal b. By what factor would you have to change n for fixed c. By what factor would you have to increase a at constant to the zero point energy of a proton in the box?

Answer 1

Energy is inversely proportional to the square of the box length E_n Proportional 1/a^2, for fixed values of n and m. So, by 20 factor you need to increase the box length to decrease the zero point energy by a factor of 400. E_n Proportional n^2, for a and m fixed By 20 factor you need to increase n to increase energy by a factor of 400. Mass of proton = 1837 times mass of electron Energy of proton = Energy of electron E_P = E_R or, n^2 h^2/8m_p a^2 = n^2 h^2/8me^2 a prime^2 or, (a prime/a)^2 = mp/me = 1837 me/me or, a prime = 42.86 a By 42.86 factor you have to increase a.