[math]\left(\frac{1}{k}\frac{\partial}{\partial t} - \nabla^2\right)u(x,t) = f(x,t)[/math]
[math]u(x,t) \rightarrow 0 \ \text{as} \ |x| \rightarrow \infty, \ \ \ t > 0[/math]
[math]u(x,0) = f(x), \ \ \ -\infty < x < \infty[/math]
[math]\text{where} \ k = \frac{i\hbar}{2m}[/math]
[math]u(x,t) = \frac{1 - i}{2\sqrt{2\pi bt}}\int_{-\infty}^{\infty} f(y) e^{i(x - y)^2/(4bt)} \ dy[/math]
First obstacle.
When I am asked to find the steady state, I get rid of the time variable [imath]t[/imath] and I solve the equation normally. But in this problem, the boundary, initial conditions, and the final solution suggests that time is part of the solution. Why?
Second obstacle.
If I assume that I solved the problem with taking time into account, the solution would be in terms of summation, not integrals. This is another problem.
Third obstacle.
Do I have to assume [imath]f(x,t) = 0[/imath], in the steady state equation? Or do I have to expand it? And why is the domain of [imath]x[/imath] not fixed between two points?
Fourth obstacle.
Why does the final solution contain the variable [imath]y[/imath] and the constant [imath]b[/imath] (may be variable) while they are not part of the main problem?
What is happening in this Schrödinger equation? Everything looks awkward.
Find the steady state of Schrödinger equation subjected to the boundary and initial conditions.
[math]u(x,t) \rightarrow 0 \ \text{as} \ |x| \rightarrow \infty, \ \ \ t > 0[/math]
[math]u(x,0) = f(x), \ \ \ -\infty < x < \infty[/math]
[math]\text{where} \ k = \frac{i\hbar}{2m}[/math]
The final solution is.
[math]u(x,t) = \frac{1 - i}{2\sqrt{2\pi bt}}\int_{-\infty}^{\infty} f(y) e^{i(x - y)^2/(4bt)} \ dy[/math]
First obstacle.
When I am asked to find the steady state, I get rid of the time variable [imath]t[/imath] and I solve the equation normally. But in this problem, the boundary, initial conditions, and the final solution suggests that time is part of the solution. Why?
Second obstacle.
If I assume that I solved the problem with taking time into account, the solution would be in terms of summation, not integrals. This is another problem.
Third obstacle.
Do I have to assume [imath]f(x,t) = 0[/imath], in the steady state equation? Or do I have to expand it? And why is the domain of [imath]x[/imath] not fixed between two points?
Fourth obstacle.
Why does the final solution contain the variable [imath]y[/imath] and the constant [imath]b[/imath] (may be variable) while they are not part of the main problem?
What is happening in this Schrödinger equation? Everything looks awkward.