The Heisenberg operators can be written in terms of Schrödinger operators as ψ ψ ( x , t ) = e i K t ψ ψ ( x ) e − − i K t , and the creation operator is ψ ψ ¯ ¯ ( x , t ) = [ ψ ψ ( x , t ) ] † † , where K = H − − μ μ N is the grand-canonical Hamiltonian.