The Particle in a Box

Particle in a one-dimensional box

Consider a particle when it is constrained to move in a rectangular box of dimensions a, b, and c in length. Within the box, i.e., between x = 0 and a, y = 0 also b,  z, c = 0 the potential energy is zero, but at the boundaries, it increases suddenly to infinity and remains everywhere outside the box at this value.

If we represent U now as the sum of its components along the three coordinate axes, namely,

 U = U_x +U_y+U_{z       .....(1)}

then, by proceeding in the same manner as above, it is feasible to isolate the Schrödinger condition into three differential equations of the structure for every one of the factors.

 ....(2)

Arrangement of Eq. (2) yields between x = 0 and x= a the eigenfunctions obtained are 

  .......(3)  

and the eigenvalues for the energies 

E_x =n_x² h²/8ma² ……... (4)

where n_x is a quantum number taking on the value n_x =1,2,3 etc. similarly the other differential equations give 

 .....(5)

   .......(6)

E_y = n_y² h²/8mb² ……... (7)

E_z = n_z² h²/8mc² ……... (8)

where n_x and n_z are again quantum numbers equal to 1,2,3 etc .  from (3) (5) (6) follows as

Ψ₌ Ψ_x Ψ_y Ψ_z 

 .......(9)

Fig.1 Plot of  Ψ_x vs x  a=2

Fig.2   Plot of  Ψ²_x vs x for a=2

and the total energy from the equation (4) (7) (8) 

                              E=E_x+E_y+E_z

E=h²/8m(n²_x/a²₊ n²_y/b² n²_z/c²)

It is of interest to observe that whereas the energies of a free particle can vary continuously, those for a particle in a box are quantized. in this energy, equation numerators reduce the value which in turn reflects a small gap between energy levels.

Figure 1-2, plots of Ψ_x and Ψ²_x vs x for several values of n_x and a =2 From Figure 1. it may be seen that the number of half-waves given by Ψ_x is identical with the value of n_x Similarly. Figure .2) shows that n_x also the number of maxima exhibited by Ψ²_x