Stochastic Theories and Deterministic Differential Equations

Figure 6

Quantum solution without friction as a function of space x and time t. 𝜌^𝑄_𝑋(𝑡₀,𝑥)=𝜌^𝐿_𝑋(𝑡₀,𝑥)=(1/2)[(2𝜋𝑎²)^−1/2𝑒^{−(𝑥−𝑥01)2/(2𝑎2)}+(2𝜋𝑎²)^−1/2𝑒^{−(𝑥−𝑥02)2/(2𝑎2)}],𝑎²=1/2,𝑥₀₁=1.5,𝑥₀₂=−1.5, and 𝜀=0.