A.11 Checking units and scaling

In a general expression, multiply top and bottom by powers of $c$ until, as far as possible, $\hslash$ only occurs as $\hslash c$, $m$ as $m{c}^2$ and $\omega$ as $\hslash \omega$ or $\omega ∕c$. Then use the table

If the electric charge enters without external fields, write in terms of $\alpha$. Use the combinations $eE$ and ${\mu }_{B}B$ (both with dimensions of energy) when external fields enter. (See section A.12 for more on units in EM.)

In calculations use eV or MeV as much as possible (eg instead of using $m_e$ in kg, use $m_e{c}^2=0.511$ MeV). Remember $\hslash c=1973$ eV Å $=197.3$ MeV fm; also useful is $\hslash =6.582\times 10^{-22}$ eV s${}^{-1}$.

Often we need to cast the Schrödinger equation in dimensionless units to recognise the solution in terms of special functions. Suppose we have a potential of the form $V\left(x\right)=\beta x^n$ for some integer $n$. Then the dimensions of $\beta$ are [Energy][Length${}^{-n}$]. The other scales in the problem are ${\hslash }^2∕2m$ which has dimensions [Energy][Length${}^2$] and the particle energy. We proceed by forming a length scale $x_0={\left({\hslash }^2∕\left(2m\beta \right)\right)}^{1∕\left(n+2\right)}$ and an energy scale $ℰ={\left({\hslash }^2{\beta }^{2∕n}∕2m\right)}^{n∕\left(n+2\right)}$. Writing $x=x_{0}z$ and $E=ℰ\mu$, the Schrödinger equation for $y\left(z\right)\equiv \psi \left(x\right)$ reads

or its 3-d equivalent. For the harmonic oscillator, $n=2$ and $\beta =\frac{1}{2}m{\omega }^2$, so $x_0={\left(\hslash ∕m\omega \right)}^{1∕2}$ and $ℰ=\frac{1}{2}\hslash \omega$ as expected. For the hydrogen atom, $n=-1$ and $\beta =\hslash c\alpha$, so $r_0=\frac{1}{2}{a}_0$ and $ℰ=2m{c}^2{\alpha }^2=4E_{\text{Ry}}$, which illustrates the fact that we might have to play with numerical factors in the length scale to obtain the standard form.

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