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題組內容

1. (22%) For classical particles following Boltzmann's distribution of energy, please answer the following questions:

(b) (4 %) Please show that the average energyis

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(a) (4%)Please show that the molecular energy distribution is with an energy (E), where kB is the Boltzmann constant and T is the temperature. Assume the total number of the gas particles is N in a volume of V.

(c) (14%) Express the Boltzmann distribution in terms of speed and show that the most probable speed (v) is the root-mean square speedHint: you may need

(a) (8%) Explain why a wavefunction and its first derivate need to be continuous and finite.

(b) (8%) Now we have the wavefunctions in this quantum well system as follows: Please show that

(c) (14%) By dividing Eq. (1) by Eq. (3) and dividing Eq. (2) by Eq. (4), please show that Then plot tanvs.x Also show that there always exists at least one quantum subband in this finite quantum well. tan()=

(a) (6%) concentration of ionized donors

(b) (4%) density of the implanted dopants ND

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