Tag Info

2

First, I want to say that different people use different notation and I welcome any comments. I also feel as if I am about to enter a minefield. Here the answer is made up with examples of use of $d$, $\partial$ and $\delta$. I would say for $d$ that $dV \over dx$ would be the total derivative in one dimension for $V(x)$ where the potential $V$ is a ...

19

Typically: $\rm d$ denotes the total derivative (sometimes called the exact differential):$$\frac{{\rm d}}{{\rm d}t}f(x,t)=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{{\rm d}x}{{\rm d}t}$$This is also sometimes denoted via $$\frac{Df}{Dt},\,D_tf$$ $\partial$ represents the partial derivative (derivative of $f(x,y)$ with respect to $x$ ...

2

$$\delta^{[\mu_1\mu_2\ldots \mu_n]}_{\nu_1\nu_2\ldots \nu_n}~=~ \frac{1}{n!}\sum_{\pi\in S_n}{\rm sgn(\pi)} \prod_{i=1}^n \delta^{\mu_{\pi(i)}}_{\nu_i}.$$ More generally, $$T^{[\mu_1\mu_2\ldots \mu_n]}~=~ \frac{1}{n!}\sum_{\pi\in S_n}{\rm sgn(\pi)} T^{\mu_{\pi(1)}\mu_{\pi(2)}\ldots \mu_{\pi(n)}}.$$ Here $S_n$ is the symmetric group of permutations, ...

1

Typically we use a superscript for the dimension in contravariant objects and a subscript for covariant objects. Traditionally greek super/subscripts are used to indicate all the dimensions and latin super/subscripts if we are considering only the spatial dimensions. So for example if $\bf x$ is a four vector we would write its components as $x^\alpha$, or ...

3

Warning: This post contains a really stupid but important error. I will fix it in a couple of hours, but for now please don't read it. There's a bit more to this than the other answers are covering. As you already noted, in expressions like this $$(c_1^\dagger c_2^\dagger \ldots) |0\rangle,$$ it is important to keep a consistent ordering. The reason is ...

2

I would interpret $1$ and $2$ in the function arguments as shorthands for the coordinates belonging to particles 1 and 2; but the subscripts $1, 2, i, k$ of the single-particle wave functions $\phi$ / $\psi$ and corresponding creation/annihilation operators $a^+$ / $a$ as specifying a single-particle state. The first part is quite a common convention: ...

2

As OP pointed out, one should stick to a single convention. Hence the order of the creation operators should be reversed in one of the two cited equations. Reversing the order of operators in which of the equations is purely a matter of choice. However, to obtain the simplest relation between the Slater determinant and the occupation number representations, ...

3

Since the states $|\psi\rangle$ and $-|\psi\rangle$ differ only by a phase factor, they really describe the same electron configuration. So, while it is a bit sloppy to have inconsistent sign conventions in different places in a text, it's not surprising to see it, and it's not really a big deal. (Unless you have inconsistent sign conventions within a single ...

6

The Lane-Emden is really a non-dimensional form of the Poisson's equation with spherical symmetry: $$\nabla^2f(r)\equiv\frac1{r^2}\frac{d}{dr}\left(r^2\frac{df}{dr}\right)$$ This should be clear to all that, since we have two factors of $d/dr$ on the right hand side of the above, it must be a 2nd order differential equation. This 2nd-order derivative ...

3

Given an inner product $(\dot{},\dot{})$, $\langle x \rvert$ is the linear functional defined by $(\lvert x\rangle, \dot{})$.

1

The density operator combines pure quantum states into a mixed quantum state. The basic idea is to take a system composed of many pure states and to represent them as a single object, which evolves in time, as a complete system. In this example, the mixed state is represented as a Block sphere, and the $\vec{\sigma}$ is a pauli matrix. The Bloch sphere is ...

Top 50 recent answers are included