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5

These states represent intermediate coupling schemes that are halfway between the usual $LS$ coupling and the more extreme $jj$ coupling that happens in heavier atoms where relativistic effects mean that the spin-orbit coupling for each individual electron can match or exceed the orbit-orbit coupling between different electrons. The intermediate coupling ...


4

As Qmechanic pointed out in the comments, you're mixing Einstein and abstract index notation a bit. To make things absolutely clear, we will use early Latin indices for abstract indices $(abc)$ and Greek indices for component indices $(\mu\nu\rho)$ and will always indicate Einstein summation explicitly. First and foremost, an abstract index is nothing more ...


4

Just a coincidence. There are too many quantities and not enough letters. It probably does make a difference that the fields in which these two equations exist (material science and electromagnetism) are well enough separated that you typically won't see them both in the same papers or textbooks; if that weren't the case, people would start using different ...


3

No. The $p_x$ and $p_y$ orbitals are always real-valued, and the complex-valued $|m|=1$ orbitals are always denoted $p_{1}$ and $p_{-1}$. Depending on what scheme you're using, the third $l=1$ orbital can be denoted either $p_z$ or $p_0$, with both notations completely equivalent. Just because the $p_x$ and $p_y$ orbitals are linear combinations of $p_1$ ...


3

The idea behind the notation is that the operator $F$ is supposed to count the number of fermions in an expression, i.e. $$[F,A_n]= n A_n$$ if the operator $A_n$ contains $n$ fermions, what that means. Then $$[f(F),A_n]= f(n) A_n$$ for a sufficiently well-behaved function $f:\mathbb{C}\to \mathbb{C}$. In particular, for $f(x)=(-1)^x $, one has ...


3

This notation is mostly used in astronomy (or at least I haven't seen it elsewhere), and it is used to de-dimensionalize variables, or otherwise get rid of inconvenient units and exponents, whilst keeping track of what standard each variable is measured in. This is not really a unified notation; instead, it is a bunch of different variables the author have ...


3

We can talk about what the notation represents. Terms I introduce will be italicised. For 1, the ket $\left|\psi\right\rangle$ represents the state of a physical system. Quantum mechanics claims these are elements of a vector space. So far, it's all physical. However, everything afterwards will abstract from that. For 2 and 4, the bra ...


3

I think the easiest way to think about these objects is as follows: $|\psi\rangle$ is your physical state Your physical state comes with a machine (its dual) $\langle \psi |$, which when applied to any other physical state $|\phi \rangle$, spits out the overlap $\langle \psi | \phi \rangle$ between your state and $|\phi\rangle$ It also comes with a ...


2

You have stumbled upon a difference between how chemists and physicist denote orbitals. The $p_x, p_y, p_z$ notation is common in chemistry because the resulting orbitals are real. Physicist use $p_{-1}, p_0, p_1$ where the subscripts are the values of m and embrace the complexity of the resulting wave function. It is purely a matter of choice since all ...


2

Horizontal position of indices matters in principle because one might want to raise and lower indices on the Christoffel symbols. If the horizontal position of indices are not observed in a consistent manner, it becomes ambiguous which index was raised or lowered, and so forth, in particular if the connection is not torsionfree. Also note that different ...


2

Yeah, that's just a coincidence. The easy way to see this is that $\epsilon$ is a relatively static property of a dielectric but a totally dynamic property of a stretching material.


2

Engineers created that problem. ;) (probably not) Many physics books use $Y$ for Young's modulus (Symon, Knight, Young & Freedman). Taylor's Classical Mechanics uses YM. Halliday, Resnick & -fill-in-the-blank- state that engineers use $E$. I suspect that physicists started using $Y$ for exactly this reason: to highlight a difference in the meanings ...


2

The (Lie-)group $U(1)$ is the topological space $S^1$ (what we call a circle together with its standard open subsets) together with a rule how to multiply its points. In its representation as numbers in ${\mathbb C}$ with absolute value $1$, we have ${\mathrm e}^{{\mathrm i}\alpha}\bullet{\mathrm e}^{{\mathrm i}\beta}:={\mathrm e}^{{\mathrm ...


1

It seems to me that there should be a mistake. Let $A^{\mu}(x)=\partial^{\mu} \theta (x)$. Clearly you have $F^{\mu \nu}=0$. Thus, you just have to find a function $\theta(x)$ such that: $$(\partial_{\nu} \partial_{\mu} \theta)(\partial^{\nu} \partial^{\mu} \theta)\neq \frac{1}{2} (\partial_{\nu} \partial^{\nu} \theta)(\partial_{\mu} \partial^{\mu} \theta). ...


1

Here's a paper for you to ponder on: Teaching electromagnetic field theory using differential forms Excerpt from the abstract: computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by ...



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