Question

A particle of mass m is placed in a three-dimensional rect- angular box with edge lengths 2L, L, and L. Inside the box the potential energy is zero, and outside it is in nite; therefore, the wave function goes smoothly to zero at the sides of the box. Calculate the energies and give the quan- tum numbers of the ground state and the rst ve excited states (or sets of states of equal energy) for the particle in the box.

Answer 1

the energy of groud state =
`(9h^(2) )/(32ml^(2) )`

Step-by-step explanation:

the energy of a 3 dimensional rectangular box is given by
`(h^(2) )/(8m)  ((n_(x) ^(2) )/(l_(x) ^(2)  )  +(n_(y)^(2)  )/(l_(y) ^(2) )+ (n_(z)^(2)  )/(l_(z) ^(2) ))`

where h is planks constant m is the mass of the particle
`n_(x)`,
`n_(y)` and
`n_(z)` are principal quantum number in x y and z direction. and
`l_(x)`,
`l_(y)` and
`l_(z)` are length of box in x y and z direction.

therefore the energy of ground state will be when
`n_(x)`,
`n_(y)` and
`n_(z)` = 1

therefore energy of ground state =
`(h^(2) )/(8m)  ((1 ^(2) )/(2l ^(2)  )  +(1^(2)  )/(l^(2) )+ (1^(2)  )/(l ^(2) ))`

=
`(9h^(2) )/(32ml^(2) )`