We generalize a space-time-symmetric (STS) extension of nonrelativistic quantum mechanics (QM) to describe a particle moving in three spatial dimensions. In addition to the conventional time-conditional (Schrödinger) wave function $\psi (x,y,z|t)$, we introduce space-conditional wave functions such as $\varphi (t,y,z|x)$, where $x$ plays the role of the evolution parameter. The function $\varphi (t,y,z|x)$ represents the probability amplitude for the particle to arrive on the plane $x=\text{constant}$ at time $t$ and transverse position $(y,z)$. More generally, within our framework, any coordinate $x^{\mu }\in \{t,x,y,z\}$ can be conveniently chosen as the evolution parameter, depending on the experimental context. This leads to a unified formalism governed by a generalized Schrödinger-type equation ${\widehat{P}}^{\mu }|{\varphi }^{\mu }\left(x^{\mu }\right)\rangle =i\hslash {\eta }^{\mu \nu }\frac{d}{d{x}^{\nu }}|{\varphi }^{\mu }\left(x^{\mu }\right)\rangle$. It reproduces standard QM when $x^{\mu }=t$, with $|{\varphi }^0\left(x^0\right)\rangle =|\psi \left(t\right)\rangle$, and recovers the STS extension when $x^{\mu }=x^i\in \{x,y,z\}$. For a free particle, we show that $\varphi (t,y,z|x)=\langle t,y,z|\varphi \left(x\right)\rangle$ naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states $|{\varphi }^{\mu }\left(x^{\mu }\right)\rangle$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^{\mu }\rangle$, leading to constraint equations of the form ${\widehat{\mathbb{P}}}^{\mu }|{\mathrm{\Phi }}^{\mu }\rangle =0$. Rather than privileging time as the sole evolution parameter, this formulation treats all coordinates on equal footing.

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• Quantum correlations, foundations & formalism
• Quantum formalism
• Quantum foundations
• Tunneling & traversal time