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© The scientific sentence. 2010

Free Particle in a finite-walled box of length L
Finit well


The Schrödinger time-dependant equation is :

i ∂ψ(r,t)/∂t = - ( 2/2m)∂2ψ(r,t)/∂r2 + V(r) ψ(r,t)    

The energy values E of the particle are less than the value of potential energy 
for this finite potential well: E < Vo.

We have two regions:
1. Where the particle is free:  - L/2 < x < + L/2 , V(x) = 0 
2. Where the particle is linked by a constant potential V(x) Vo : 
- ∞ < x< - L/2 and + L/2 < x < ∞

Within the first region, the solution of the Schrödinger time-dependant equation 
is the one of the free-particle wavefunction:

ψ(r,t)  = A exp{ikr - i ωt}

Within the second region, the Schrödinger time-independant equation is the 
following:

- ( 2/2m)∂2ψ(r)/∂r2 + Vo ψ(r) = E ψ(r), Or:
2ψ(r)/∂r2 +  [2m(E - Vo)/  ( 2)]ψ(r) = 0 

Equation of the form: 
∂2ψ(r)/∂r2 - α2 ψ(r) = 0 
where α2 = 2m(Vo - E)/  ( 2)

This equation has the solution of the form:
 
ψ(r) = A exp{αr} + B exp{- αr} 

For a finite potential well, the solution to the Schrodinger equation 
is a wavefunction that gives an exponentially decaying 
penetration into the classicallly forbidden region. This is because in Classical 
Physics, we can not have a value of potential Vo greater that the 
total energie E (value of the Hamiltonian). 

In Quantum Mechanics, V(x) is the potential wherein the 
particule moves, not the one of the particle itself.




  


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