Schrödinger wave equation

1. The time-independent Schrödinger 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 Schrödinger 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 =
4π2/λ2 = 4π2 {2m[E - V(x)]}/h2 = {2m[E - V(x)]}/ℏ 2. 
 We have then:
∂2ψ(x)/∂x2 +  {2m[E - V(x)]}/ℏ 2 ψ(x) = 0. Or:
- (ℏ 2/2m)∂2ψ(x)/∂x2 + V(x) ψ(x) = E ψ(x).
 
Extended to the case of three dimensions, we obtain: 
      
The Schrödinger time-independant equation:      
- (ℏ 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r)

2. The time-dependent Schrödinger wave equation

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

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

3. The solution of the Schrödinger 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 Schrödinger time-dependant equation is written as:

iℏ  ψ(r) ∂ƒ(t)/∂t = - (ℏ 2/2m) ƒ(t) ∂2ψ(r)/∂r2 + V(r) ψ(r) ƒ(t) = 
= [- (ℏ 2/2m) ∂2ψ(r)/∂r2 + V(r) ψ(r) ]ƒ(t).

Using the  Schrödinger time-independant equation: 
- (ℏ 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r), we get:
iℏ  ψ(r) ∂ƒ(t)/∂t = [E ψ(r)]ƒ(t). Or 
iℏ  ∂ƒ(t)/∂t = Eƒ(t).   
Rearranging, we obtain:
∂ƒ(t)/ ƒ(t)= - (i E/ℏ )∂t. Integrating 
we find: ln ƒ(t) = - (i E/ℏ )t + constant. That is:

ƒ(t) = A exp {- (i E/ℏ )t} = A exp {- iωt}.  
Which is exactly the 
expressed used in Ψ(x,t) = A exp i{kx - ωt}.

      
The solution of the Schrödinger time-dependant equation is:     
ψ(r,t) = ψ(r)  exp {- i (E/ℏ )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/ℏ )t}. Wheras, the spatial component 
ψ(r) is to be derived. The time-independent Schrödinger 
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/ℏ )t}.
 
For a particle of mass "m" in a potential V(r), the 
time-independent Schrödinger wave equation is used to 
determine its spacial wave function and its total energy.

4. Properties of the solution of the Schrödinger
equation Ψ(r,t)

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 Ψi(r,t) is the wave function of 
wave-matter that is a solution of this equation.

2. The wavefunction Ψ(r,t), in general, is a complex function 
of r and t, contains information , as an amplitude of probability, 
about where and when the quantum system is somewhere.

3. The wavefunction Ψ(r,t) is continuous and its derivative 
dΨ(r,t)/dt is continuous. This function must vanish at infinity.
its square-integrable is equal to 1.






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