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© The scientific sentence. 2010

Schrdinger wave equation



1. The time-independent Schrdinger wave equation

From the 
classical wave equation:2ψ(x,t)/∂x2 - (1/v2) ∂2ψ(x,t)/∂t2 = 0 


and the de Broglie relationship p = h/λ,  we will derive 
the single-particle time-independent Schrdinger equation:
  
The general solution of the classical harmonic wave equation is :

Ψ(x,t) = A exp i{kx - ωt}

Indeed:2Ψ(x,t)/∂x2 = - k2 &psi(x,t), and 
∂2Ψ(x,t)/∂t2 = - ω2 &psi(x,t). 

Substituting these experessions in the left-hand side 
of the equation, we get: 
- k2 &psi(x,t)  + (ω2/v2) &psi(x,t)

Let's recall that k = 2π/λ, v = ν λ and ω = 2πν. Then: 
v2  =  (2π ν/k)2  = (ω/k)2. Thus: 
ω2/v2 = k2.  
The equation becomes:
 
- k2 Ψ(x,t) + k2 Ψ(x,t) = 0. Which is true if 
of course &psi(x,t) is not null.

  
We can express Ψ(x,t) = A exp i{kx - ωt} as: 
Ψ(x,t) = A exp i{ kx} exp i{- ωt}  = ψ(x) ƒ(t), by 
separating the variables of space and time.
  
The equation becomes: 
ƒ(t) ∂2ψ(x)/∂x2 - (1/v2) ψ(x) ∂2ƒ(t)/∂t2 = 0 
 
∂2ƒ(t)/∂t2 = - ω2 ƒ(t); 
Thus: 
ƒ(t) ∂2ψ(x)/∂x2 +  ω2 ƒ(t) (1/v2) ψ(x) = 0 
That is: 
∂2ψ(x)/∂x2 +  (ω2/v2) ψ(x) = 0 .
  
This equation describes the spacial amplitude of a matter-wave. 
Its total (kinetic + potential) energy is E = p2/2m + V(x). 
 
Using the de Broglie relationship for a nonrelativistic case, 
we get an expression for the momentum p of the particle-wave: 
p = h/λ = {2m[E - V(x)]}1/2.
 
With ω2/v2 = k2 = (2π/λ)2 =
4π22 = 4π2 {2m[E - V(x)]}/h2 = {2m[E - V(x)]}/ 2. 
 We have then:
∂2ψ(x)/∂x2 +  {2m[E - V(x)]}/ 2 ψ(x) = 0. Or:
- ( 2/2m)∂2ψ(x)/∂x2 + V(x) ψ(x) = E ψ(x).
 
Extended to the case of three dimensions, we obtain: 
      
The Schrdinger time-independant equation:      
- ( 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r)

2. The time-dependent Schrdinger wave equation

We have: ∂ψ(r,t)/∂t = - iω ψ(r,t). 
If we use the Planck law E = hν =   ω for a particle-wave, we 
can write: 
iω ψ(r) = i(E/ ) ψ(r,t). Thus 
∂ψ(r,t)/∂t = - i(E/ ) ψ(r,t). 
We get then an expression for the total energy: E = i  (1/ψ(r,t) ∂ ψ(r,t)/∂t.
Using this expression and the relation of de Broglie, we get:
- ( 2/2m)∂2ψ(r,t)/∂r2 + V(r) ψ(r,t) =  i  ∂ ψ(r,t)/∂t

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

3. The solution of the Schrdinger equation

1. Let's remark that we didn't evaluate the first or the second derivative with respect to the space variable (x or r) of the wave function Ψ(r,t) to derive the equation (we did it just to prove the validity of the solution Ψ(x,t) = A exp i{kx - ωt}). If we separate the varibales as Ψ(r,t) = ψ(r) ƒ(t); the spatial function could be anyting; wheras ƒ(t) has to be exactly of the form ƒ(t) = A exp i{- ωt}. With Ψ(r,t) = ψ(r) ƒ(t), the Schrdinger time-dependant equation is written as: i ψ(r) ∂ƒ(t)/∂t = - ( 2/2m) ƒ(t) ∂2ψ(r)/∂r2 + V(r) ψ(r) ƒ(t) = = [- ( 2/2m) ∂2ψ(r)/∂r2 + V(r) ψ(r) ]ƒ(t). Using the Schrdinger time-independant equation: - ( 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r), we get: i ψ(r) ∂ƒ(t)/∂t = [E ψ(r)]ƒ(t). Or i ∂ƒ(t)/∂t = Eƒ(t). Rearranging, we obtain: ∂ƒ(t)/ ƒ(t)= - (i E/ )∂t. Integrating we find: ln ƒ(t) = - (i E/ )t + constant. That is: ƒ(t) = A exp {- (i E/ )t} = A exp {- iωt}. Which is exactly the expressed used in Ψ(x,t) = A exp i{kx - ωt}.       The solution of the Schrdinger time-dependant equation is:      ψ(r,t) = ψ(r) exp {- i (E/ )t}      ψ(r) is any function. E is the total energy of the particle. The temporal component of the solution is henceforth known. It is exp {- i (E/ )t}. Wheras, the spatial component ψ(r) is to be derived. The time-independent Schrdinger equation is the most used. Indeed, when the expression of the potential V(r) is given, the time-independent equation allows us to derive the spacial component ψ(r) of the wave function, and the total energy E which is used in the expression of the temporal componenet exp {- i (E/ )t}. For a particle of mass "m" in a potential V(r), the time-independent Schrdinger wave equation is used to determine its spacial wave function and its total energy.

4. Properties of the solution of the Schrdinger equation

1.The Schrodinger equation is linear. The wave-packet is also solution of this equation. We can show easily that the general solution of the Schrodinger equation is of the form Ψ(r,t) = ΣAi Ψi(r,t), where the Ψi(r,t) is the wave function of wave-matter that is a solution of this equation. 2.

©: The scientificsentence.net. 2007.
  
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