14

Changing $i$ to $-i$ is not the same as changing the arrow of time. Changing $i$ to $-i$ is entirely irrelevant, it has no effect on the physics. It would be like inventing a new symbol for the number 3, and using that symbol instead of "3". The symbol itself is just a convention. Nothing really depends on it. The formulas look slightly different ...


10

Let's use the matrix representation of complex numbers. In this representation real numbers are proportional to the unit matrix, while the imaginary unit has the form $$i=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}.\tag1$$ Now, consider the Schrödinger's equation $$i\partial_t |\psi\rangle-H|\psi\rangle=0.\tag2$$ Rewrite it in the following form, ...


7

This is a good intuition! Yes, you can always take the complex conjugate of everything and thereby get a new theory whose predictions are the same as the old theory. In this case the waves rotate the opposite way around the complex plane, so a state with energy $E_n$ becomes a phasor $\exp({+i}E_nt/\hbar)$ rather than ${-i}.$ Arguably this is more intuitive, ...


3

First, to be clear, physics, with its five sigma threshold, has a higher standard of "proof" than almost any other discipline. In the social sciences, for example, two sigma is considered credible and three sigma is considered something close to divine truth. This is simply the wisdom of collective experience, although it has a solid basis that is ...


2

The second-to-last paragraph in the question starts with $$ p^\mu\to i\partial^\mu. $$ This implies $$ p_\mu\to i\partial_\mu, $$ because the index on both sides is lowered using the same metric (regardless of the sign convention for the metric). This equation can be separated into \begin{align*} p_0 &\to i\partial_0 \\ p_j &\to i\partial_j. \...


1

The terms are not exactly equivalent, although both likely hold in this context. To work in the time domain is (broadly) to consider a signal's strength as it changes as a function of varying time (as opposed to varying frequency, for example). Parametrizing a signal in terms of e^jwt means working in the time domain because one can plug in the time t (in ...


1

I don't think there is a problem using coordinates $\quad\left ( t, \displaystyle \left(\frac{x}{c}\right) \right)\quad$ vs $\quad\left ( ct, \vphantom{\displaystyle\left(\frac{x}{c}\right) } x \right)$. In fact, I prefer $\left ( t, \displaystyle \left(\frac{x}{c}\right) \right)$ because the time axis is more important to me... especially if I use a clock ...


1

I remember that I have also struggled with the sign of the minimal coupling term several times in the past. Therefore, I feel compassionate with your attempts to derive it. Would you accept the non-relativistic Schrödinger equation for the bare electric field of the nucleus (i.e. without magnetic potential) as "some principled way to understand what's ...


1

You first must differentiate between the component and the magnitude of a vector. A magnitude is always positive, think of it as a length and it does not matter which way (left to right or right to left) you measure the length. If the component is positive then that is also equal to the magnitude. Let's look at you equation $m\,g-c\, v=m\, a$ but in its ...


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