Justification for Squared $ {L}_{2} $ Data and Smoothness Term as an Error Bound
One of the motivations to use the $ {L}_{2} $ norm comes from the Maximum a Posteriori Estimation (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 Gaussian you'd get the exact model you posted above.

An example of the derivation of MAP model to the above can be seen in my answer to Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples.