Quantum Mechanics
Schrodinger equation
Quantum Mechanics
Propagators : Pg
Quantum Simple Harmonic Oscillator QSHO
Quantum Mechanics
Simulation With GNU Octave
© The scientific sentence. 2010
|
| Free-Particle Wave Function
1. Schrödinger time-independent equation:
The Schrödinger time-independent equation:
- (ℏ 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:
- (ℏ 2/2m)∂2ψ(r)/∂r2 = E ψ(r) (1.2)
Or:
∂2ψ(r)/∂r2 + [2mE /(ℏ 2)]ψ(r) = 0 (1.2)
This equation is the form:
∂2ψ(r)/∂r2 + [k2]ψ(r) = 0 ,
with k2 = 2mE /ℏ2
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 :
iℏ ∂ψ(r,t)/∂t = - (ℏ 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:
iℏ ∂ψ(r,t)/∂t = - (ℏ 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:
iℏ C exp{ikr} ∂ψ(t)/∂t = - C (ℏ 2/2m) ψ(t) ∂2[exp{ikr}]/∂r2 (2.2)
Or:
iℏ C exp{ikr} ∂ψ(t)/∂t + C (ℏ 2/2m) (ik)2 exp{ikr} ψ(t) = 0 (2.2)
According to the De Broglie relationship : P = ℏ k,
We obtain:
iℏ ∂ψ(t)/∂t - P2/2m) ψ(t) = 0 (2.2)
Since the particle is free KE = P/2m = E = ℏ 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)
|
|