Units

The equations governing quantum mechanical systems, for example the time-dependent Schrödinger equation for a single particle, $ i \hbar \frac{\partial\psi(x)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} + V(x) \psi(x)$, include very small quantities when using the SI unit system, such as the reduced Planck constant $\hbar = 6.63 \cdot 10^{-34}\,\mathrm{m^2 kg / s}$, the Bohr radius $a_0 = 5.29 \cdot 10^{-11}\,\mathrm{m}$, and atomic masses like $m_\mathrm{^{87}Rb} = 1.44 \cdot 10^{-25}\,\mathrm{kg}$. This makes the equations unwieldy in numerical application, because computers represents numbers to a finite precision.

The solution to this is to non-dimensionalize the working equations, which is equivalent to changing the unit system. The non-dimensionalized equations are obtained by writing for each quantity $\alpha_{\mathrm{SI}} = \mu_{[\alpha]} \cdot \alpha_\mathrm{sim}$, where $\alpha_{\mathrm{SI}}$ is the value of the quantity in SI units, $\mu_{[\alpha]}$ is a scaling factor also carrying the dimension (the 'simulation unit'), and $\alpha_\mathrm{sim}$ is the dimensionless number representing the quantity in simulation units. For example,

  • $x_\mathrm{SI} = \mu_\mathrm{length} \cdot x_\mathrm{sim}$
  • $t_\mathrm{SI} = \mu_\mathrm{time} \cdot t_\mathrm{sim}$
  • $V_\mathrm{SI} = \mu_\mathrm{energy} \cdot V_\mathrm{sim}$
  • $m_\mathrm{SI} = \mu_\mathrm{mass} \cdot m_\mathrm{sim}$,
  • $\psi_\mathrm{SI} = \mu_{[\psi]} \cdot \psi_\mathrm{sim} = \mu^{-1/2}_\mathrm{length} \cdot \psi_\mathrm{sim}$

We can choose some of these units independently as 'base'-quantities, e.g. $\mu_\mathrm{mass} =m_\mathrm{^{87}Rb} = 1.44 \cdot 10^{-25}$ such that the mass of Rubidium 87 measures is $m_\mathrm{sim} = 1$ in simulation units. The number of independent units are limited, however, and we must define some in terms of the other, e.g. $\mu_{[\psi]} = \mu^{-1/2}_\mathrm{length}$. We also choose $\mu_\mathrm{energy} = \frac{\hbar^2}{\mu_\mathrm{mass} \mu^2_\mathrm{length}}$, the reason for which will be clear below.

The decompositions are substituted into the original expressions (where the initial SI units are now marked for clarity).

Left hand side: $ i \hbar \frac{\partial\psi_\mathrm{SI}}{\partial t_\mathrm{SI}} = \frac{\hbar}{\mu_\mathrm{time}} \mu^{-1/2}_\mathrm{length} \bigg(i \frac{\partial\psi_\mathrm{sim}}{\partial t_\mathrm{sim}}\bigg)$

Right hand side:

$$\bigg[-\frac{\hbar^2}{2m_\mathrm{SI}} \frac{\partial^2}{\partial x^2_\mathrm{SI}} + V_\mathrm{SI} \bigg] \psi_\mathrm{SI} = \bigg[- \bigg(\frac{\hbar^2}{\mu_\mathrm{mass} \mu^2_\mathrm{length}}\bigg) \frac{1}{2m_\mathrm{sim}} \frac{\partial^2}{\partial x^2_\mathrm{sim}} + \mu_\mathrm{energy} \cdot V_\mathrm{sim} \bigg] \psi_\mathrm{sim} \mu^{-1/2}_\mathrm{length}$$

$$=\bigg[- \mu_\mathrm{energy}\frac{1}{2m_\mathrm{sim}} \frac{\partial^2}{\partial x^2_\mathrm{sim}} + \mu_\mathrm{energy} \cdot V_\mathrm{sim} \bigg] \psi_\mathrm{sim} \mu^{-1/2}_\mathrm{length}=\bigg[-\frac{1}{2} \frac{\partial^2}{\partial x^2_\mathrm{sim}} + V_\mathrm{sim} \bigg] \psi_\mathrm{sim}\mu_\mathrm{energy}\mu^{-1/2}_\mathrm{length}$$

We then equate the two and divide by $\mu_\mathrm{energy}\mu^{-1/2}_\mathrm{length}$ on both sides

$$\frac{\hbar}{\mu_\mathrm{time} \mu_\mathrm{energy}} \bigg(i \frac{\partial\psi_\mathrm{sim}}{\partial t_\mathrm{sim}}\bigg) = \bigg[-\frac{1}{2} \frac{\partial^2}{\partial x^2_\mathrm{sim}} + V_\mathrm{sim} \bigg] \psi_\mathrm{sim}$$

Finally, we can use our freedom in choosing the units to require $\frac{\hbar}{\mu_\mathrm{time} \mu_\mathrm{energy}} = 1 \Rightarrow \mu_\mathrm{time}=\frac{\hbar}{\mu_\mathrm{energy}}$, and drop the 'sim' subcript to obtain the final, non-dimensionalized equation

$$i \frac{\partial\psi}{\partial t}= \bigg[-\frac{1}{2} \frac{\partial^2}{\partial x^2} + V \bigg] \psi$$

which is effectively identical to the initial equation with $\hbar=m=1$. The stationary equation is similarly non-dimensionalized.

As example numbers, we could choose as base quantities

  • $\mu_\mathrm{length} = 532 nm$ (the site spacing of a $1064 nm$ laser lattice),¬†
  • $\mu_\mathrm{mass} = m_\mathrm{^{87}Rb} = 1.44 \cdot 10^{-25}\,\mathrm{kg}$ (the mass of Rubidium 87).

The energy and time time scalings are then measured implicitly in

  • $\mu_\mathrm{energy}=\frac{\hbar^2}{\mu_\mathrm{mass} \mu^2_\mathrm{length}}=2.76\cdot 10^{-31}J$
  • $\mu_\mathrm{time}=\frac{\hbar}{\mu_\mathrm{energy}}=0.38ms$

In the case of Bose-Einstein condensates, the non-dimensionalization is a bit more tricky because of the interaction term $\beta|\psi|^2$, see Appendix in the QEngine article. In that terminology, Composer always has $\kappa=1/2$ and only one of the two conditions in Eq.(A.1) can be chosen freely.