Derive an expression for calculating energy values corresponding n = 3 for a particle in one dimensional box.
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The particle in the box model system is the simplest non-trivial application of the Schrödinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics. For a particle moving in one dimension (again along the x- axis), the Schrödinger equation can be written
−â„22mψ′′(x)+V(x)ψ(x)=Eψ(x)
(3.5.1)−â„22mψ″(x)+V(x)ψ(x)=Eψ(x)
Assume that the particle can move freely between two endpoints x=0x=0 and x=Lx=L , but cannot penetrate past either end. This is equivalent to a potential energy dependent on x with
V(x)={0∞0≤x≤Lx<0andx>L
(3.5.2)V(x)={00≤x≤L∞x<0andx>L
This potential is represented in Figure 3.5.13.5.1 . The infinite potential energy constitutes an impenetrable barrier since the particle would have an infinite potential energy if found there, which is clearly impossible.
The particle is thus bound to a "potential well" since the particle cannot penetrate beyond x=0x=0 or x=Lx=L
ψ(x)=0forx<0andx>L
(3.5.3)ψ(x)=0forx<0andx>L
By the requirement that the wavefunction be continuous, it must be true as well that
ψ(0)=0andψ(L)=0
(3.5.4)ψ(0)=0andψ(L)=0
which constitutes a pair of boundary conditions on the wavefunction within the box. Inside the box, V(x)=0V(x)=0 , so the Schrödinger equation reduces to the free-particle form:
−â„22mψ′′(x)=Eψ(x(3.5.5)−â„22mψ″(x)=Eψ(x)
with 0≤x≤L0≤x≤L .
We again have the differential
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