Considering that the particle can only hop between points in the lattice, we write x = n a , y = m a , where n , m are integers. Hofstadter makes the following ansatz : ψ ψ ( x , y ) = g n e i ν ν m , where ν ν depends on the energy, in order to obtain Harper's equation (also known as almost Mathieu operator for λ λ = 1 ):