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Of course, the ground state is an eigenstate of ##\hat{b}## with eigenvalue 0. Just calculate the derivative to see it.
Now I am somewhat puzzled. Are you sure about that? Maybe I just missed something, but isn't ##\hat{b}## the equivalent of the creation operator? I always thought that the annihilation operator had the plus sign.Of course, the ground state is an eigenstate of ##\hat{b}## with eigenvalue 0. Just calculate the derivative to see it.