# Relation between Matrix mechanics and Wave mechanics [closed]

What is the relationship between Hamiltonian operator (matrix), position operator (matrix) and momentum operator (matrix) in Matrix mechanics and wave mechanics?

Anyway, to illustrate for the S H oscillator, in non-dimensionalized units of $$m,\omega,\hbar$$, ie, where these quantities are set equal to 1, always reinstatable uniquely by dimensional analysis, in the WM formulation, $$2\hat H= -\partial_x^2 + x^2, ~~~ \hat x = x ,~~~ \hat p = -i\partial_x, ~~\leadsto \\ \hat H \psi_n(x) =(n+1/2) \psi_n(x), ~~~ i\partial_t \psi(x,t) = \hat H \psi (x,t).$$
In the matrix formulation, Dirac defined annihilation and creation operators acting on a vacuum state vector, $$a={\hat x + i\hat p \over \sqrt{2} } , ~~~ a^\dagger ={\hat x - i\hat p \over \sqrt{2} }~~ \leadsto ~~ [a,a^\dagger ]=I,\\ a|0\rangle= 0 , ~~~ \leadsto \langle m|\hat x|n\rangle = x_{mn}=\sqrt{n/2}~ \delta_{n,m+1} + \sqrt{(n+1)/2}~\delta_{m,n+1}, \\ p_{mn}=-i\sqrt{n/2} ~\delta_{n,m+1} + i\sqrt{(n+1)/2} ~ \delta_{m,n+1}, \\ \langle m|\hat H|n\rangle = H_{mn} = (1/2 +n)\delta_{m,n}, \leadsto ~~~ \hat H |n\rangle = (1/2+n) | n\rangle .$$ all sparse, tasteful infinite-dimensional matrices. I bet when you learn about the derivation of this spectrum you'll curse your teachers for not teaching it to you first! Dirac left lots speechless in the mid 20s this way.
Anyway, the bridge between WM and MM is $$\psi_n(x) = \langle x| n\rangle, ~~~\Longrightarrow ~~~ \int\!\! dx ~~|x\rangle \psi_n(x) =|n\rangle,$$ which, unless you were a chemist, you'd use several times a week! You may convert wave to matrix expressions back and forth in the same phrase, indeed, thought, and never wonder about specious philosophical divisions of the 1920s.