Dimension bookkeeping for the quantities and laws of the framework

The coordinates have $[t]=\tau$ and $[x_i]=\mathrm{\ell }$, so a three-dimensional position wave function has $[\psi ]={\mathrm{\ell }}^{-3\mathrm{/}2}$, its squared modulus has $[\mathrm{\mid }\psi {\mathrm{\mid }}^2]={\mathrm{\ell }}^{-3}$, and the product $\mathrm{\mid }\psi {\mathrm{\mid }}^2{\mathrm{d}}^{3}x$ is dimensionless, as required of a probability density.

For the operators of [L1], $[Q_j\psi ]=[x_j][\psi ]=\mathrm{\ell }\cdot {\mathrm{\ell }}^{-3\mathrm{/}2}={\mathrm{\ell }}^{-1\mathrm{/}2}=[\psi ]\cdot \mathrm{\ell }$ gives $[Q_j]=\mathrm{\ell }$, and $[P_k\psi ]=[\mathrm{\hslash }]\cdot [{\mathrm{\partial }}_k\psi ]=\mu {\mathrm{\ell }}^2{\tau }^{-1}\cdot {\mathrm{\ell }}^{-5\mathrm{/}2}=\mu {\mathrm{\ell }}^{-1\mathrm{/}2}{\tau }^{-1}=[\psi ]\cdot \mu \mathrm{\ell }{\tau }^{-1}$ gives $[P_k]=\mu \mathrm{\ell }{\tau }^{-1}$; the commutator identity is consistent, since $[Q_j{P}_k]=\mathrm{\ell }\cdot \mu \mathrm{\ell }{\tau }^{-1}=\mu {\mathrm{\ell }}^2{\tau }^{-1}=[\mathrm{\hslash }]=[i\mathrm{\hslash }{\delta }_{jk}]$.

For the Hamiltonian of [L2], the kinetic term acts as $[(-{\mathrm{\hslash }}^2\mathrm{/}2m)\mathrm{\Delta }\psi ]={\mu }^2{\mathrm{\ell }}^4{\tau }^{-2}{\mu }^{-1}{\mathrm{\ell }}^{-2}{\mathrm{\ell }}^{-3\mathrm{/}2}=\mu {\mathrm{\ell }}^2{\tau }^{-2}{\mathrm{\ell }}^{-3\mathrm{/}2}=[E][\psi ]$, and the potential term satisfies $[V\psi ]=[V][\psi ]$ with $[V]=\mu {\mathrm{\ell }}^2{\tau }^{-2}=[E]$; both terms therefore carry the dimension of energy times wave function, as required for their sum to define an operator.

The Schroedinger equation is dimensionally consistent: $[i\mathrm{\hslash }{\mathrm{\partial }}_t\psi ]=\mu {\mathrm{\ell }}^2{\tau }^{-1}\cdot {\tau }^{-1}\cdot {\mathrm{\ell }}^{-3\mathrm{/}2}=\mu {\mathrm{\ell }}^2{\tau }^{-2}{\mathrm{\ell }}^{-3\mathrm{/}2}=[H_{\mathrm{o}\mathrm{p}}\psi ]$, and the phase $H_{\mathrm{o}\mathrm{p}}t\mathrm{/}\mathrm{\hslash }$ of the evolution operator is dimensionless because $[E][t][\mathrm{\hslash }{]}^{-1}=1$.

The current of [L3] has $[j]=[\mathrm{\hslash }][m{]}^{-1}[\mathrm{\mid }\psi {\mathrm{\mid }}^2][\mathrm{\nabla }\psi ][\psi {]}^{-1}={\mathrm{\ell }}^2{\tau }^{-1}\cdot {\mathrm{\ell }}^{-3}\cdot {\mathrm{\ell }}^{-1}={\mathrm{\ell }}^{-2}{\tau }^{-1}$, which is ${\mathrm{\ell }}^{1-d}{\tau }^{-1}$ for $d=3$: a probability flux per unit area and time, not a charge current.

The expression $mv\cdot x\mathrm{/}\mathrm{\hslash }$ in [L4] is dimensionless, $[mv\cdot x]\mathrm{/}[\mathrm{\hslash }]=\mu \mathrm{\ell }{\tau }^{-1}\cdot \mathrm{\ell }\cdot (\mu {\mathrm{\ell }}^2{\tau }^{-1}{)}^{-1}=1$. Thus it can serve as a dimensionless argument if a separately specified model assigns it as a phase; this dimensional calculation alone proves no frame-transformation or covariance law. Combining with steps 1.1-1.5, every quantity and displayed law checked here is dimensionally consistent under the stated conventions.