One of the motivations to use the $ {L}_{2} $ norm comes from the [Maximum a Posteriori Estimation][1] (MAP) framework.

If you model $ \psi \left( u \right) \sim \mathcal{N} \left( 0, \alpha \right) $ then if you derive the MAP Estimator in case the added noise is Guassian you'd get the exact model you posted above.


  [1]: https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation