6-4 ① This is a property of the sine function: d2sin(x)/dx2 = -sin(x)
and of the cosine function: d2cos(x)/dx2 = -cos(x)
So a linear combination of sine and cosine functions might provide a general solution to the differential equation:
ψ(x) = asin(kx) + bcos(kx)
② Starting from the equation:
We need to take our proposed solution, ψ(x) = asin(kx) + bcos(kx), and find the second derivative (use the chain rule, with the substitution u = kx):
So, the second derivative is equal to the original function multiplied by a constant. This means that it is, indeed, a solution to this free-particle Schrödinger equation.
③ Substituting this result into the equation at the top, we get:
Substituting k = 2π/λ into this energy expression, we get:
④ The de Broglie equation gives us: p = h/λ , so substituting into the energy expression gives:
which is the classical expression for the kinetic energy of the particle. Since there was no potential energy in this example, this is indeed equal to the total energy.
It works!
