quantum mechanics - What is the conserved quantity of a scale-invariant universe?

Consider that we have a system described by a wavefunction $\psi(x)$. We then make an exact copy of the system, and anything associated with it, (including the inner cogs and gears of the elementary paticles, if any, aswell as the fabric of spacetime), but where all distances are multiplied by a number $k$, so $\psi(x) \to \psi(kx)$, we consider the case $k>1$ (if $k=-1$ this is just the parity operation, so for $k<0$ from the little I read about this we could express it as a product of P and "k" transformations).

Consider then that all observables associated with the new system are identical to the original, i.e. we find that that the laws of the universe are invariant to a scale transformation $x\to kx$.

According to Noether's theorem then, there will be a conserved quantity associated with this symmetry.

My question is: what would this conserved quantity be?

Edit: An incomplete discussion regarding the existence of this symmetry is mentioned here: What if the size of the Universe doubled?

Edit2: I like the answers, but I am missing the answer for NRQM!

Answer

The symmetry you are asking about is usually called a scale transformation or dilation and it, along with Poincare transformations and conformal transformations is part of the group of conformal isometries of Minkowski space. In a large class of theories one can construct an "improved" energy-momentum tensor $\theta^{\mu \nu}$ such that the Noether current corresponding to scale transformations is given by $s^\mu=x_\nu \theta^{\mu \nu}$. The spatial integral of the time component of $s^\mu$ is the conserved charge. Clearly $\partial_\mu s^\mu = \theta^\mu_\mu$ so the conservation of $s^\mu$ is equivalent to the vanishing of the trace of the energy-momentum tensor. It should be noted that most quantum field theories are not invariant under scale and conformal transformations. Those that are are called conformal field theories and they have been studied in great detail in connection with phase transitions (where the theory becomes scale invariant at the transition point), string theory (the two-dimensional theory on the string world-sheet is a CFT) and some parts of mathematics (the study of Vertex Operator Algebras is the study of a particular kind of CFT).