One of my teachers said that the Hamiltonian operator can be thought of as infinite dimensional matrix, but I could not understand why. Moreover, if that is true, then how can we write the Hamiltonian operator as a matrix?
P.S: Please don't use concepts much beyond high school level. I know the basics of matrices and am still in the process of learning linear algebra topics like "Vector spaces". So try to explain in simple words as much as possible. Thank you.
If you know how a derivative can be approximated in a finite space, you know the answer to your question.
Any derivative of a continuous function $f$, let it be for example $f''$ can be approximated using Taylor series for a discrete space: $$f''\approx \frac{f_{i+1}-2 f_{i}+ f_{i-1}}{h^2}$$ This equation permits a representation in matrix form: $$f'' \approx D^2f$$
Where $D^2$ is proportional to the following matrix: $$\left[\begin{matrix} 2 & -1 & ...& &\\ -1 & 2 &- 1 &...&\\ 0 & -1 & 2 &- 1 &...\\ ... &&&&\\\ ... &&-1&2&-1\\ ...\\ &&&-1&2 \end{matrix}\right]$$ Therefore if $h\to 0$ then the finite dimensional space turns into an infinite dimensional one, lesding to a infinite matrix $D^2$ representind the second derivative in continuum.