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Wave Properties of particles. Schrodinger Equation.

Problem 6.83

A particle of mass mm is located in a three-dimensional cubic potential well with absolutely impenetrable walls. The side of the cube is equal to a.a . Find: (a) the proper values of energy of the particle; (b) the energy difference between the third and fourth levels; (c) the energy of the sixth level and the number of states (the degree of degeneracy) corresponding to that level.

Reveal Answer
6.83. (a) E˙=(n12+n22+n32)π2ℏ2/2ma2,\dot{E}=\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right) \pi^{2} \hbar^{2} / 2 m a^{2}, where n1,n2,n3n_{1}, n_{2}, n_{3} are integers not equal to zero: (b) ΔE=π2ℏ2/ma2;\Delta E=\pi^{2} \hbar^{2} / m a^{2} ; (c) for the 6-th level n12+n22+n32=14n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=14 and E=7π2ℏ2/ma2;E=7 \pi^{2} \hbar^{2} / m a^{2} ; the number of states is equal o six (it is equal to the number of permutations of a triad 1,2,3.)1,2,3 .)
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