Given N
bosons moving in one-dimension on the x -axis defined from [0,L] with
periodic boundary conditions, a state of the
N-body system must be described by a many-body
wave function \psi(x_1, x_2, \dots, x_j, \dots,x_N). The
Hamiltonian, of this model is introduced as : H = -\sum_{i=1}^N \frac{\partial^2}{\partial x_i^2} + 2c \sum_{i=1}^N\sum_{j>i}^N \delta(x_i-x_j)\ , where \delta is the
Dirac delta function. The constant c denotes the strength of the interaction, c>0 represents a repulsive interaction and c an attractive interaction. The hard core limit c\to\infty is known as the
Tonks–Girardeau gas. For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., \psi(\dots, x_i,\dots, x_j, \dots) = \psi(\dots, x_j,\dots, x_i, \dots) for all i \neq j and \psi satisfies \psi( \dots, x_j=0, \dots ) =\psi(\dots, x_j=L,\dots ) for all j. The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say x_1 and x_2 are equal. The condition is that as x_2 approaches x_1 from above (x_2 \searrow x_1), the derivative satisfies :\left.\left(\frac{\partial}{\partial x_2} - \frac{\partial}{\partial x_1} \right) \psi (x_1, x_2)\right|_{x_2=x_1+}= c \psi (x_1=x_2). == Solution ==