Path integral Monte Carlo on a lattice. II. Bound states

The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and the atoms are restricted to sites on a one-dimensional lattice. A path integral formalism developed within the context of the canonical ensemble is utilized, where the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp in the context of a square well potential, forcing the qp to occupy bound states. We consider a one-dimensional square well potential where all atoms on the lattice are occupied with one atom with an on-site potential except for a contiguous set of sites of various lengths centered at the middle of the lattice. Comparison of the potential energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the numerical calculations.

Figure 1

Diagram depicting a finite square well of depth $V_0$ and ranging along on the one-dimensional lattice from $-j*$ to $j*$. The lattice itself ranges from $-N$ to $N$.

Figure 2

Energy level diagram as determined by finding the eigenvalues of the matrix $M$ for the Schrödinger difference equation for a well of width $w=31$. In this case, $j^{*}=15, a=10.0$, and $b=0.0$.

Figure 3

Energy level diagram as determined by finding the eigenvalues of the matrix $M$ for the Schrödinger difference equation for a well of width $w=71$. In this case, $j^{*}=35, a=10.0$, and $b=0.0$.

Figure 4

Energy level diagram as determined by finding the eigenvalues of the matrix $M$ for the Schrödinger difference equation for a well of width $w=201$. In this case, $j^{*}=100, a=10.0$, and $b=0.0$.

Figure 5

Average potential energy versus inverse temperature, $\beta$, for a quantum particle moving on a one-dimensional lattice with a square well configuration of width $w=31$, depth $\epsilon =10.0$, and centered at the midpoint of the lattice. The numerical curve is a plot of the theoretical results. The other curves represent the Monte Carlo simulations for $p=10, p=100$, and $p=200$. The agreement is strong. Error bars are too small to be seen on this scale.

Figure 6

Radius of gyration versus inverse temperature, $\beta$, for a quantum particle moving on a one-dimensional lattice with a square well configuration of width $w=31$, depth $\epsilon =10.0$, and centered at the midpoint of the lattice. The curves represent the Monte Carlo simulations for $p=10, p=100$, and $p=200$. Error bars are too small to be seen on this scale.

Figure 7

2D plot of a sample density matrix, which is for a well of width $w=31$ and for $\beta =0.10$. Note that the entire density matrix is practically zero except a line representing the diagonal elements, with particular concentration of relatively higher values near the center of the well.

Figure 8

2D plot of a sample density matrix for a well of width $w=31$ and for $\beta =5.0$, which is a significantly higher value of $\beta$ than that for Fig. 7. This density matrix for higher $\beta$ is showing practically the entire matrix to be zero except for a small region near the center of the well.

Figure 9

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.01$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 10

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.05$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 11

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.10$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 12

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.50$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 13

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =1.0$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 14

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =5.0$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 15

The numerical and computational self-correlation (or qp-qp correlation) function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =10.0$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 16

The numerical and computational qp-qp correlation function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=71$ and for $\beta =10.0$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 17

The numerical and computational qp-qp correlation function $G_1\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=201$ and for $\beta =10.0$. The computational $G_1\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 18

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.01$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 19

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.05$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 20

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.10$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 21

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =0.50$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 22

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =1.0$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 23

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =5.0$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 24

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=31$ and for $\beta =10.0$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 25

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=71$ and for $\beta =10.0$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.

Figure 26

The numerical and computational atom-qp correlation function $G_2\left(n\right)$ of the qp on the lattice with the subject square well configuration for a well of width $w=201$ and for $\beta =10.0$. The computational $G_2\left(n\right)$ is for $p=10, p=100$, and $p=200$.