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I'm not sure why division of matrices would be useful, or unavailable, since these matrices are all easily invertible, but you have implicitly answered your own question but failed to recognize the an …

You might be trapping yourself in your notation. First, appreciate that $[\lambda_\alpha,{\cal H}]= [\lambda_\alpha, H ]$, so w.l.o.g. set $h_0=0$. Moreover, since $H|n\rangle=\epsilon_n|n\rangle$, …

You really don't need to overthink it. Just do it. Note that $$ J_3^2= \operatorname {diag} ( 1,1,0) \equiv K; ~~~~J_3^3=J_3. $$ You then expand the exponential of the matrix, as normally defined, $$ …

I am not sure I understand your possibly malformed question, but, ignoring the gaps, I rush to review basic facts, $$\hat{V}_{1,2}=\boldsymbol{\sigma}^{(1)}\cdot \boldsymbol{\sigma}^{(2)}= \tfrac{1}{ …

Just to make you aware of the "bridge" expressions, the real Hermite functions (not Hermite polynomials on which they are based), $\psi_n(x)=\langle x|n\rangle$. You then have $$\langle m| \hat x ^k|n …

This is a stark demonstration of the mess inflicted by failure to nondimensionalize. If your quantities are dimensional, write their units next to them, in your case: $$ M = \begin{pmatrix} 1 + xy & …

Your "according to my understanding" rotation around $\hat y$, (x goes towards z when you draw the diagram, as you should), is unlike the other two, so you paved your way to dyslexia purgatory. Normal …

Physical meaning? In simplified natural units, $$ H = {1 \over 2} (P^2 + X^2) ~, $$ is manifestly diagonal, I/2 + diag(0,1,2,3,...), for the standard matrix mechanics hermitian expressions $$ \sqrt{2 …

If you had not resolved the constraint, e.g., of a hyperspherical O(N) model, you'd use the standard Dirack bracket procedure, not needed here. Here, you only have Goldstone scalars, and no "σ", so yo …

I might expedite/explain the sound answer of @Thomas Fritsch, which the comment format makes all but impossible. With due apologies to the OP, an integral kernel (propagator) which is a delta function …

I'm not sure what hack you'd be asking for in your comment to the good answer. You are meant to apply the transformation laws of the impeccable answer already provided to your scaling, (3), a change o …

Indirectly. To start with, set the pesky and useless constants $\hbar/m\omega \mapsto 1 $, to work in natural units. You then recognize the matrix, as Eric emphasizes by identifying to his (5), as w …

Indeed, you are doomed. There is no such V. Suppose there were an equivalence (4,5,6). Then consider $\gamma_5 = i \gamma_0 \gamma_1 \gamma_2 \gamma_3$, as well as its transform, independently of ba …

As @Emilio points out, your premise is aggressively wrong. The two pictures are equivalent, and one does not dominate the other, any more than the left-hand side of an equation dominates the right-han …

Well, you may define your conventions, and physicists are perverse enough to actually do that (watch them...), any way you want. This is the dominant convention, and the logical "chain of custody", so …