$\frac{\beta x ^{2}}{ kT }$ is dimensionless

$[\beta]\left[ L ^{2}\right]=\left[ ML ^{2} T ^{-2}\right]$

$[\beta]=\left[ MT ^{-2}\right]$

$\alpha^{2} \beta$ has dimensions of work

$\left[\alpha^{2}\right]\left[ MT ^{-2}\right]=\left[ ML ^{2} T ^{-2}\right]$

$[\alpha]=\left[ M ^{0} LT ^{0}\right]$