What is the relationship like between Thomas and Ann Putnam in The Crucible?
I would say that the relationship between Ann and Thomas Putnam is not a very good one. I think that they feel that their relationship is fine, but in their association, Miller is making a fundamental point about how relationships are not able to progress if the people in them are not willing to progress from an emotional standpoint. Both Thomas and Ann are fundamentally unhappy people because they are filled with so much in terms of resentment of others.
Thomas Putnam is driven by the desire to be wealthy and to own more land. It is something that has taken root in him and represents an end that is never satisfied. We see this in Act I when Proctor and Corey enter and Putnam engages in a heated discussion about land rights and lumber. In Act III, Corey accuses Putnam of taking advantage of those who are accused in the attempts to consolidate his own wealth and increase his land ownership. At a historical moment when the land charter of Salem had been revoked, Putnam's desire for more land and greater economic wealth knows no real limits. It is for this reason that he is shown to be a very unhappy or at least unlikeable person.
Ann Putnam is not much better. Her resentment stems from the fact that so many of her children have died at such young ages. Ann cannot fully accept this reality or the idea that somehow, she has experienced death more than others. This resentment takes its greatest form in her hatred and envy of Sarah Good, who has experienced bounty in childbirth and rearing. Ann is someone who is shown to be searching for someone, anyone, to blame for her condition of her children's death. To a great extent, her support of Sarah Good's prosecution as a witch is fed by this need to blame.
Both Thomas and Ann never experience moments alone in the play where their relationship is examined. However, it becomes fairly clear that both of them together do not help ease the pain and bitterness of the other. If anything, Miller seems to be suggesting that their own conditions of anger, misapplied hatred, and envy of others corrupt or taint their ability to interact properly with society. It would only make sense that this would taint or corrupt their interactions with one another, as well.