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# Have you ever realized that you have forgot the boundary conditions?

Once in a while, you will forget about the boundary conditions. Today, I am going to be talking about the Time dependent Schrodinger Equation for a particle in a box as an example to show that boundary conditions really matter when solving certain equations. For those who aren't familiar with the Time dependent Schrodinger Equation, it follows: $$i\hbar \frac { \partial \Psi }{ \partial t } =-\frac { \hbar ^{ 2 } }{ 2m } \nabla ^{ 2 }\Psi +V(x)\Psi$$. The boundary conditions of the box are $$0\le x\le L$$. We could solve the Schrodinger Equation using separation of variables so we get $$\Psi (x,t)=Asin(kx)$$ but if we forgot about the boundary conditions, we would get $$\Psi (x,t)=Ae^{ i(kr-\omega t) }$$. We would also get different energy eigenvalues. For $$\Psi (x,t)=Asin(kx)$$, we would get $${ E }_{ n }=\frac { \hbar ^{ 2 }n^{ 2 }\pi ^{ 2 } }{ 2mL^{ 2 } }$$ whereas, for $$\Psi (x,t)=Ae^{ i(kr-\omega t) }$$, you would get that there are no certain energy eigenvalues. In conclusion, always remember to include boundary conditions in your courses such as differential equations and other courses. I am 12 and many people in my school really don't care about the conditions that are given anyways so that is why I am posting this.

Note by Sunny Sahu
2 years, 8 months ago

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