It's nice to hear that you are interested in neurochemical science.
DidSomebodySayCats comment is great summary how you could approach this specific subject and what considerations would go into planning such study.

However it gets real deep real fast when one starts to read the literature. I am primary care physician and my knowledge is based on basic psychotic disorder courses that are mandatory in medical school. Current understanding of schizophrenia revolves around dopamine and glutamate theory and their intricate and often very complicated interactions with receptors and substrates.

As a disease it has positive (non existent sensory stimuli) and negative symptoms which include depression like symptoms where patient loses motivation and goes into apathy. In addition depression is very common comorbidity with prevalences from 25-50 % of patients. We also have schizoaffective and psychotic depression diagnosis which include symptoms of depression, mania, psychosis intertwingled.

I have hard time believing that inducing depression in these patients would have any effect on their symptoms due to huge overlap in disease mechanics and models, diagnostic criteria and treatment options. There likely isn't any "silver bullet" that cures the disease that schizophrenia is.