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A particle of a mass m is located in a three-dimensional cubic potential well with absoulutely impenetrable walls. The side of the cube is equal to a Find: (a) the proper value of the energy of the particel, (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. |
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Answer» Solution :We PROCEED axactly as in (6.81). The wave function is chosen in the from `Psi(x,y,z)=A sin k_(1) sink_(2)y sink_(3)z`. (The ORIGIN is at one corner of the box and the axes of coordinates are ALONG the edges.)The boundary CONDITIONS are that `Psi=0` for `x=0,x=a,y=0,y=a,z=0,z=a` This gives `k_(1)=(n_(1)pi)/(a),k_(2)=(n_(2)pi)/(a)=k_(3)(n_(3)pi)/(a)` The energy eigenvalues are `E(n_(1),n_(2),n_(3))=(pi_(2) ħ^(2))/(2ma^(2))(n_(1)^(2)+n_(2)^(2)+n_(3))` The first level is `(1,1,1)`. The second has `(1,1,2),(1,2,1)`&`(2,1,1)`. The third level is `(1,2,2) or (2,1,2) or (2,2,1)`. Its energy is `(pi^(2) ħ^(2))/(2ma^(2))` The fourth energy level is `(1,1,3)` or `(1,3,1)` or `(3,1,1)` Its energy is `E=(11pi^(2) ħ^(2))/(2ma^(2))` (b) Thus `Delta=E_(4)-E_(3)=( ħ^(2)pi^(2))/(ma^(2))` (c )The FIFTH level is `(2,2,2)`. The sixth level is `(1,2,3),(1,3,2),(2,1,3),(2,1,3),(2,3,1),(3,1,2),(3,2,1)` `(7 ħ^(2)pi^(2))/(ma^(2))` and its degree of degeneracy is `6`(six). |
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