We obtained that this is Four times two times the mass and this elevated to one divided by two. So by doing this integral, this integral is quite easy to obtain from this. In these times two times the mass divided by The energy and all these elevated to one divided by two and integrated over the energy. So in here we know that since the value of the function in the number of electrons, it is different from zero between the interval of zero and the for me energy then we will have in here just simply two times L divided by clams constant. That is the integral from zero to infinity of the number of electrons, that depends on the energy times the number and function times the differential in the energy. Now, with that said, um for part a the number of particles is given by the following integral. When the energy is greater than the fermi energy. Now the party of two is when the energy, it is less or equal to the fermi energy in 80 0. So we know that then also the value of E contains the value of two Electrons or zero. So to solve this problem, we are given that this expression here is the number of um yes, it's the number of electrons and that depends on the energy. So for part a of this problem, we need to determine the fermi energy at a temperature of zero silk kelvin degrees, zero kelvin degrees. So there are and electrons in the sample and each state can be occupied by two electrons. Is the length of the sample and M is the mass of the electron. And these times the square root of two times the mass divided by the energy.
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So for this problem we are told in a one dimensional system, the number of energy states per unit length is equal to L divided by clams constant.