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Coming back to this very late! I’m not sure I understand what you mean, the question was asking for help with proving that in a TEM mode the electric and magnetic fields are perpendicular to each other, not vice-versa. The electric field you provide is indeed not transverse, but it’s still perpendicular to the magnetic field. It’s not a counter-argument to what I was trying to prove, but rather to its converse statement.
You are correct, if you specify that it’s along the same path it’s just a property of integrals. For a conservative force it is true that $W_{AB} = -W_{BA}$ regardless of what path is taken.
Of a particle? Experimentally? Usually by measuring the deflection radius in a magnetic field, obtaining the momentum through $p = qrB$ and dividing by the mass.
The QM understanding is that observables are operators whose eigenvalues provide the possible outcomes of the measurement process. It is true that not all Hermitian operators are observables, but only because not all infinite-dimensional Hermitian operators are diagonalisable. Whether or not the operator corresponds to a physical quantity that you are familiar with or can devise an experiment for is not relevant. If your question is about how to experimentally measure the velocity of a particle, you should rephrase it.
The operators are the observables. The result of a measurement will be simply taking the expectation value of the operator on a specific state. Nothing changes in the equation: $\frac{d}{dt} \langle \hat{x} \rangle = \frac{\langle \hat{p} \rangle}{m}$.
I feel that a Bayesian approach doesn’t really comply with the request to “assume the knowledge of a grade 12 or year 13/upper sixth student in your response if you can”.
Yes, it’s what I’m covering in the second part of the comment. Statistically speaking, it will (eventually, as $N \to \infty$) get closer. In practice, however, it cannot be guaranteed, especially if the increase in the number of measurements is small, such as 5 vs 8).
Isn’t that exactly what the confidence measures? The tighter your confidence interval is, the more confident you can be that your measurement is close to its “real” value. Obviously there’s always a chance of your value not being accurate due to a long string of “unlucky” measurements. You could, for example, have a very lucky first 5 measurements, and the following 95 could be very unlucky, drawing you further from the accurate value. Of course this becomes vanishingly unlikely the more measurements you do.
