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1) Inline mathematics
For short expressions (like this $E = mc^2$) within a sentence, wrap the math in single dollar signs:
$E = mc^2$2) Displayed equations
For equations on their own line, wrap the math in double dollar signs:
$$
\frac{d}{dt}\langle A \rangle = \frac{i}{\hbar}\langle [H, A] \rangle
$$3) Greek letters and common symbols
Standard LaTeX commands are typically supported, for example:
\alpha,\beta,\gamma\partial,\nabla,\infty\langle \rangle,\sum,\int
Example:
$$
\psi(x,t) = A e^{i(kx - \omega t)}
$$4) Code or pseudocode
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for i in range(N):
evolve(state)5) References and context
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Feel free to ask questions ranging from introductory concepts to more advanced topics. If you can state what you already know and what exactly is confusing you, the discussion will be much more productive.
In non relativistic quantum mechanics why must observables be represented by Hermitian operators? In particular, how does Hermiticity guarantee real measurement outcomes, like what goes wrong physically if this condition is relaxed?
The short answer is: Hermitian operators are what make the measurement story actually work.
First, their eigenvalues are real, so the outcomes you measure are real numbers. If an operator isn’t Hermitian, you can get complex eigenvalues, which don’t really make sense as measurement results.
Second, they have nice eigenvectors (orthogonal and complete), so you can write any state as a sum of them. That’s what lets you say the probability of getting outcome $a$ is
$$|\langle a | \psi \rangle|^2$$
and have all the probabilities add up to 1.
If you drop Hermiticity, things start breaking:
eigenvalues can be complex
eigenvectors might not be orthogonal so probabilities stop behaving nicely
if the Hamiltonian isn’t Hermitian, time evolution isn’t unitary: $U(t) = e^{-iHt}$ isn’t unitary therefore total probability not conserved.
So yeah, without Hermiticity, the connection between the math and actual measurements kind of falls apart.
There are some edge cases (like PT-symmetric Hamiltonians) where you can still get real spectra, but then you have to change the inner product or reinterpret things.