  The single well 1D harmonic oscillator is one of the most fundamental and
commonly solved problems in quantum mechanics. Traditionally, in most
introductory quantum mechanics textbooks, it is solved using either a power
series method, which ultimately leads to the Hermite polynomials, or by ladder
operators methods. We show here that, by employing one straightforward variable
transformation, this problem can be solved, and the resulting state functions
can be given in terms of parabolic cylinder functions. Additionally, the same
approach can be used to solve the Schr\"odinger equation for the 1D harmonic
oscillator in a uniform electric field. In this case, the process yields two
possible solutions. One is the well-known result where the 1D oscillator
eigenvalues are reduced by a frequency-dependent term, which can have any
positive value. The other is where the field term is restricted to be an
integer and the eigenvalues are in the same form as for the field-free case. We
show how the results can be used to create a harmonic approximation for the
bound states of a Lennard-Jones potential.
