Wave Equation (de Broglie)

Write the one dimensional wave equation and its solution.

Answer

The one dimensional associated de Broglie wave f(x) equation for a quantum particle of mass m is

-\hbar/(2m) *d^2 f(x)/dx + U(x)*f(x) = En*f(x)     (Schrodinger Equation)

where $hbar$ is the Planck constant, U(x) the potential energy and En the eigenvalues of energy of that body (with which is associated the wave)

for a classical mechanical oscillator the equation rewrites as
d² f(x)/dx²  +(k/m) *f(x) = En*f(x)

where m is the mass of the oscillator and k the elastic constant of the spring (medium)

The general solution of the above equation is the wave function

f(x) = A*exp(i*k*x)= A*(cos(kx) +i*sin(kx))

where A is a constant value, k is the oscillator wave vector and x the position of the oscillator.