Fourier Convention Translator

Fourier conventions change signs, factors of $2\pi$, and factors of $\hbar$. This page translates the most common cases.

Default Convention

In one dimension,

$$\psi(x) =\frac{1}{\sqrt{2\pi\hbar}} \int e^{ipx/\hbar}\phi(p)\,dp,$$

and

$$\phi(p) =\frac{1}{\sqrt{2\pi\hbar}} \int e^{-ipx/\hbar}\psi(x)\,dx.$$

Translation Table

Alternative	Translation
wave number $k$ instead of momentum $p$	use $p=\hbar k$ and $dp=\hbar\,dk$
phase $e^{ikx}$	this is the $k$ convention, not the $p$ convention
inverse transform has no prefactor	forward transform usually carries $1/(2\pi)$ or $1/(2\pi\hbar)$
both transforms have $1/\sqrt{2\pi}$	usually a $k$ transform, or $\hbar=1$
phase signs reversed	momentum operator or time-evolution sign may also be reversed

Where Signs Change

With the default convention, the position-space momentum operator is

$$p=-i\hbar\frac{d}{dx}.$$

If the phase convention is reversed without changing the operator convention, signs will conflict.

Formula Pages Affected

• Fourier Transforms
• Momentum Representation
• Plane Waves and Delta Normalization
• Probability Current

Common Mistakes

• Treating $\phi(p)$ and $\tilde\psi(k)$ as the same normalized function.
• Forgetting the Jacobian when changing from $p$ to $k$.
• Combining a symmetric normalization with an asymmetric inverse formula.
• Using FFT array conventions as if they were continuum Fourier conventions.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.