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

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The relationship between wave mechanics and matrix mechanics is so natural in the language of Dirac's bra- and ket- language, and the theory of Hilbert space, that hardly anyone pays attention to the distinction, justifiably, and most people switch dialects in the middle of their discussion. Basically, anything requiring solving differential equations is termed "wave mechanics", and anything relying on matrix techniques and linear algebra is "matrix mechanics". So, for instance, for the harmonic oscillator, solving Hermite's differential equation to then produce Hermite function eigenfunctions of Schroedinger's hamiltonian is WM, but using the algebraic recursions of creation and annihilation operators to understand the quantized spectrum of the number operator is MM.

Today, most people blend the two dialects and speak in a hybrid, the result of a century of supplemental education that filled in the gaps of the educational system of the 1920s, strong on differential equations, and weak in linear algebra. The distinction should not concern you, but you must, must, hit the QM books and learn Dirac's bracket notation.

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.

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