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4. Applications.
Two applications of Schrödinger equations are presented: the forecast of the tunnel effect and an
explanation of alpha radioactivity.
Tunnel effect.
Consider a particle, with a definite energy E, moving towards a potential barrier of height
U > E: According to classical mechanics, the particle cannot go through
that barrier. On the contrary, in quantum mechanics there is a not null probability that the particle
goes through the barrier. This is a consequence of first Schrödinger equation.
In the following, we will study the tunnel effect with respect to a particle moving in only one direction, to simplify the problem from a mathematical point of view.
A particle moving along the xaxis, with mass m and energy E, approaches
from the left a potential barrier with energy U > E. According to
classical mechanics, the particle cannot go inside the potential barrier: The grey area in the
draw is inaccessible. According to quantum mechanics, the probability that the particle
penetrates the potential barrier (blue line in the draw) is not null.
The first Schrödinger equation applied to that particle is:
Equation (4.1) is a linear differential equation with constant coefficients.
The auxiliary equation
has two real solutions
.
Thus two solutions of equation (4.1) are the real functions
φ_{1}(x) = e^{x k} and
φ_{2}(x) = e^{ x k}, where e is is the base
of natural logarithms. Since the first divergent solution φ_{1}(x) is not acceptable,
the general solution for equation (4.1) is the real function
φ(x) = A · e^{ x k},
where A is a real arbitrary constant and
.
The function φ(x) = A · e^{  x k}
describe the state of the particle inside the potential barrier. The square of its module gives
the probability P(x) that the particle is in a point x occupied by the potential barrier:
P(x) =  A · e^{ x k}  ^{2} = A^{2} · e^{  2 x k} .
The probability that the particle is in a point inside the potential barrier is not null.
Thus, according to Schrödinger quantum mechanics, a particle can penetrate in a potential barrier in
a manner prohibited by classical mechanics.
