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© The scientific sentence. 2010

Free-Particle Wave Function



1. Schrödinger time-independent equation:

  
The Schrödinger time-independent equation: 
- ( 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r)    (1.1)


If the particle is free, then V(r) = 0, therefore, the equation (1.1) becomes:

- ( 2/2m)∂2ψ(r)/∂r2  = E ψ(r)    (1.2)
Or:
∂2ψ(r)/∂r2 + [2mE /( 2)]ψ(r) = 0    (1.2)
This equation is the form:
∂2ψ(r)/∂r2  + [k2]ψ(r) = 0    , 
with k2 = 2mE /2

This equation has the solution of the traveling wave of the form:
 
ψ(r) = A cos(kr) + B sin (kr) 

A and B are constants. The constant B can be written in the following 
complex form : B = iA, then ψ(r) = A [cos(kr) + i sin (kr)]. Using 
Euler formula : exp{ix} = cos (x) + i sin (x) , we can write: 
 
ψ(r) = C exp{ikr}    (1.3)

2. Schrödinger time-dependent equation:

 
The Schrödinger time-dependant equation is :
i ∂ψ(r,t)/∂t = - ( 2/2m)∂2ψ(r,t)/∂r2 + V(r) ψ(r,t)    (2.1)

If the particle is free, then V(r) = 0, therefore, the equation (2.1) becomes:

i ∂ψ(r,t)/∂t = - ( 2/2m)∂2ψ(r,t)/∂r2     (2.2)

The technique is to separate the solution to a dependent and independent 
factors as follows: 

ψ(r,t) = ψ(r)  x ψ(t). But ψ(r) is already found and expressed 
by the equation (1.3). Therefore, the equation (2.2) becomes:

i  C exp{ikr} ∂ψ(t)/∂t = - C ( 2/2m) ψ(t) ∂2[exp{ikr}]/∂r2     (2.2)

Or:

i  C exp{ikr} ∂ψ(t)/∂t + C ( 2/2m) (ik)2 exp{ikr} ψ(t)   = 0    (2.2)

According to the De Broglie relationship : P =  k,
We obtain:

i ∂ψ(t)/∂t - P2/2m) ψ(t)   = 0    (2.2)

Since the particle is free KE = P/2m = E =  ω

Therefore (2.2) becomes:

∂ψ(t)/∂t = - i ω ψ(t)      (2.2)


∂ψ(t)/ψ(t)  = - i ω  ∂t      (2.2)
The solution of this equation is :  ψ(t) = D exp {- i ωt}

Finally, the solution of the Schrödinger time-dependent equation 
is: 
ψ(r,t) = ψ(r) x ψ(t) =  C exp{ikr} x D exp {i ωt} = A exp{ikr - i ωt}

ψ(r,t)  = A exp{ikr - i ωt}    (2.3)


  


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