Question

Particle in a three-dimensional box: a. Give the equation for a particle in a three-dimensional box b. How does the density of states (i.e., number of states per unit of energy) change with increasing energy? Explain the answer.

Answer 1

pave funetion of Particle in BD box copies sti.Vs .LX Ly XX.in volume of box Total energy equation The energy of the particle in a 3-D cube/box in the ground state given by mx=1 ny = 1 and m₂=1 El,l,i = nome The ground state hare only one wave function and no other state has this specific energy. The ground

State and the energy level are said to be non- degenerate. Howe evero in 3-D box: potential energy of a state dependro upon the Sum of the squares of the quantum numbers (eq-). The particle having a particular value of energy in the excited state may haro seberal different Stationary stalão or wave funetion. For the first exicited state, three combinations of the quantum numbers (x, ny, na) are (2, 1, 1); (12,1),(1,12). The sum of square of quantum number in each combination is nsame (=6). Each ware function har same energy. - E, E21, 1 = {1,2, 1 = $1,1,28 m2 :

corresponding to these combinations of quantum number, three different wave function and three different states are possible. Hence, first excited state called three-fold or triply (degenerate. - The number of independent wave functions for the stationary staters of an energy level is called and the degree of degeneracy. The value of energy level with the corresponding combination and sum of square of the quantum numbers. รา? . . + + 2 Example - 3rd excited state nă + my+ nă = a energy state = (2,2,1) (1,2,2) (2,12) - degeracy Total energy = an lae Degree of degene recy = 3