  In this paper we compare three different orthogonal systems in
$\mathrm{L}_2(\mathbb{R})$ which can be used in the construction of a spectral
method for solving the semi-classically scaled time dependent Schr\"odinger
equation on the real line, specifically, stretched Fourier functions, Hermite
functions and Malmquist--Takenaka functions. All three have banded
skew-Hermitian differentiation matrices, which greatly simplifies their
implementation in a spectral method, while ensuring that the numerical solution
is unitary -- this is essential in order to respect the Born interpretation in
quantum mechanics and, as a byproduct, ensures numerical stability with respect
to the $\mathrm{L}_2(\mathbb{R})$ norm. We derive asymptotic approximations of
the coefficients for a wave packet in each of these bases, which are extremely
accurate in the high frequency regime. We show that the Malmquist--Takenaka
basis is superior, in a practical sense, to the more commonly used Hermite
functions and stretched Fourier expansions for approximating wave packets
