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MCMC on a bounded parameter space?

I am trying to apply MCMC on a problem, but my priors(in my case they are $\alpha\in[0,1],\beta\in[0,1]$)) are restricted to an area? Can I use normal MCMC and ignore the samples that fall outside of the restricted zone(which in my case is [0,1]^2), i.e. reuse transition function when the new transition falls out of restricted(constrained) area?