Question

This is a pde problem. We will cover pde's in class after the break if you want to wait. The pde to solve is the 10 Schroedinger's equation:

iℏ∂/∂tΨ(x, t) = [-ℏ²/2m.∂²/∂x² + V (x, t)]Ψ(x, t).
Use units so that h and m are = 1, and do this for a free particle (V = 0) in a box of length L = 1. Your boundary condition is then that ψ is zero at x = 0 and x = L. Your solution for ψ will be complex. Plot the probability= |ψ²|vsx and t.


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Answer 1

To solve the provided time-dependent Schrödinger's equation for a free particle in a box of length L=1 and V=0, we can use separation of variables method to find the wave function Ψ(x, t). The solutions for the x and t parts yield X(x) and T(t) respectively, which can be combined to obtain the complete wave function Ψ(x, t). The probability density |Ψ(x, t)|² can then be calculated and plotted as a function of x and t.

Step-by-step explanation:

The given equation is the time-dependent Schrödinger's equation, which describes the behavior of quantum systems. It represents the motion of a free particle in a box of length L=1, where the potential energy V(x, t) is zero. To solve this equation and find the wave function Ψ(x, t), we can use separation of variables method.

1. Assume the wave function can be written as Ψ(x, t) = X(x)T(t)
2. Substitute Ψ(x, t) and the potential energy V(x, t) = 0 into Schrödinger's equation
3. Divide the equation by Ψ(x, t) to separate the x and t parts
4. The x part becomes -ℏ²/2m * d²X(x)/dx² = E, where E is the energy of the particle
5. The t part becomes iℏ * dT(t)/dt = -E
6. Solve the x part equation, which is a second-order differential equation, with the boundary conditions X(0) = X(L) = 0
7. Solve the t part equation, which is a first-order differential equation
8. Combine the solutions for X(x) and T(t) to obtain the complete wave function Ψ(x, t)
9. Calculate the probability density |Ψ(x, t)|² = |X(x)|² * |T(t)|²
10. Plot the probability density |Ψ(x, t)|² as a function of x and t