jats:titleAbstract</jats:title>jats:pIn this paper we compare three different orthogonal systems in jats:inline-formulajats:alternativesjats:tex-math$$\textrm{L}_2({\mathbb {R}})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mtextL</mml:mtext> mml:mn2</mml:mn> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:miR</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger 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 jats:inline-formulajats:alternativesjats:tex-math$$\textrm{L}_2({\mathbb {R}})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mtextL</mml:mtext> mml:mn2</mml:mn> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:miR</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>