Free particle in 1 D box:
Here Free particle refers to electron and 1D box can be an atom
or a piece of metal wire. As the electron is completely free to
move within the box, its potential energy is zero. And the electron
moves only along x-axis, the following partical differential
equation (from Shroedinger’s time independent equation) will be
written as compete differential equation.
𝜕 2 Ψ 2m
+ 2 EΨ = 0
𝜕𝑥 2
ђ
𝑑2 Ψ
+ 𝑘2 Ψ = 0
2
𝑑𝑥
is the wave equation for electron′ s travel along one dimension.
X=0 and x=L are the boundaries of the one dimensional potential
box. In the following figure the potential energy inside the box is
zero; and the electron may face up to infinite potential by the
walls of the box in such a way that the electron never gets
outside the box.
𝑘2 =
2mE
ђ2
The solution of the second order differential equation is as
follows,
Ψ(x) = A sin kx + B cos kx
The wave function is zero at the boundaries so,
Ψ = 0 at x = 0
After applying the first boundary condition,
0= A×0+B×1
=0+B
⟹B=0
Thus, Ψ(x) = A sin kx
Ψ = 0 at x = L (the second boundary condition)
After applying the second boundary condition
0 = A sin kL
kL = nπ, n = 1, 2, 3, … …
k=
nπ
L
2mE
nπ
k = √ 2 , and also k =
ђ
L
k 2 L2 =
2mEn L2
= n2 π2
2
ђ
n2 π2 ђ2
En =
, n = 1,2,3 …
2mL2
The following figure explains the wave functions, probability
densities and energy values of the particle at different energy
levels.
Eigen functions:
Eigen means special or particular. After including particular
energy value function, the wave function is called as Eigen
function.
2mEn
Ψn = A sin kx = A sin √ 2 x , where, n = 1,2,3 ….
ђ
Ψn = A sin
nπx
L
+∞
∫−∞ |Ψn |2 dx = 1 (within the box, the hundred percentage of
the particle is existing, since boundaries prevent them strongly
from escaping outside)
L
L
∫|Ψn |2 dx = ∫ A2 sin2 (
0
0
L
nπx
) dx = 1
L
nπx
∫0 sin2 ( L ) dx = 1
L
nπx
∫0 (A2 /2)(1 − 𝑐𝑜𝑠2 ( L ))dx = 1
Solving
L
∫|Ψn |2 dx = A2
0
L
=1
2
2
A =
L
2
2
A = √ is the normalised constant
L
Thus by applying the normal condition (100% existence of the
particle within the box) the constant A has been found out. So
the constant A is called as normalised constant.
Normalised Eigen functions:
Eigen function with the normalised constant is called as
normalized Eigen function.
2
nπ
Ψn (x) = A sin kx = √ sin ( x) , where, n = 1,2,3 ….
L
L