New answers tagged

6 votes

How to write completeness of wavefunctions without bra ket notation?

The completeness relation reads: $$\sum_{I=1}^{\infty} \psi^{*}_{I}(x') \psi_{I}(x)=\delta(x'-x ).$$ Proof: Suppose any wave function $\Psi$ can be expanded using the $\psi_i$: $$ \Psi(x)=\sum_{I=1}^{\...
user avatar
  • 267
10 votes

How to write completeness of wavefunctions without bra ket notation?

One way you can show the completeness relation without bra-ket notation is just $$\sum_{i} \langle \psi_i , v \rangle \psi_i = v \qquad \forall v\in\mathcal{H},$$ where $\mathcal{H}$ is the Hilbert ...
user avatar
1 vote
Accepted

What $\mu$ represents in Gryzinski's free fall atomic model equation?

It almost certainly represents the magnetic dipole moment of the electron — this is standard notation and (as noted in the wiki article) the last term is meant to account for interactions between the ...
user avatar
1 vote

Why are Hamilton's equations sometimes written with a gradient?

The simplest answer is, that it's a far more compact/applicable form of notation generalization, especially when dealing with various types of field theories (both classical or quantum). In general, ...
user avatar
  • 184
3 votes
Accepted

Why are Hamilton's equations sometimes written with a gradient?

Your best guess is correct. Let's take the example of an $N$ particle system in $\text{3D}$. If I wanted to define my generalized coordinates as just the position of each particle, then one way I ...
user avatar
1 vote
Accepted

What does $\sin(a,b)$ mean in the absorber theory of radiation?

It stands for sine of the angle between the two vectors in the braces, in this case, between the retarded radius vector $r_k$ and the charged particle acceleration $\mathscr{U}$.
user avatar
3 votes
Accepted

Commutator between covariant derivative and a field

It may help to evaluate the commutator on a function, i.e. $$[\partial_\mu,\Phi]f=\partial_\mu(\Phi f)-\Phi\partial_\mu f=(\partial_\mu \Phi)f+\Phi \partial_\mu f-\Phi \partial_\mu f.$$ The last step ...
user avatar
  • 561
1 vote

In functional derivative the starting point confusion

That's not really the definition of the functional derivative. One way to define the functional derivative is $$\frac{\delta F}{\delta f} (x_0) \equiv \lim_{\epsilon\rightarrow0}\frac{1}{\epsilon} (F[...
user avatar
1 vote

In an $n$ particle system, why is the Hamiltonian summed over $n$?

The index $i=1,\ldots n$ is a particle index; not a coordinate index. The $i$th particle carries a 3-momentum $p_i\in\mathbb{R}^3$. In the Hamiltonian $p_i^2=p_i\cdot p_i$ is a dot/scalar product. ...
user avatar
  • 170k
1 vote
Accepted

In an $n$ particle system, why is the Hamiltonian summed over $n$?

The hamiltonian is a scalar quantity as it represents total energy of a system. Each particle momenta in your equation $(1)$ is the magnitude of said particle's momenta $p_i = \sqrt{\sum\limits_i^d \...
user avatar
  • 165
2 votes

Question about integration limits in the special relativistic action

Yes, it's a mistake and they should be changed, they become proper time limits in the next step.
user avatar
  • 4,565
4 votes
Accepted

Notation for contracting vectors using metric tensors

If we take a vector $A$, which has three components, my understanding is that we can write this using Einstein notation as $A_{u}$ where this is actually $A_1+A_2+A_3$. No. $A_\mu$ is the $\mu^{th}$ ...
user avatar
  • 52k
6 votes
Accepted

What is the difference between the metric (tensor), $g_{\mu\nu}$, and the invariant interval, ${ds}^2$?

If you want to be super systematic about language and not overloading the terminology, you can say the following. Fix a smooth $n$-dimensional manifold $M$. A pseudo-Riemannian metric tensor field on ...
user avatar
  • 2,930
4 votes

What is the difference between the metric (tensor), $g_{\mu\nu}$, and the invariant interval, ${ds}^2$?

The interval is just a convenient way to give a metric, since from $$ds^2 = g_{\mu\nu} dx^\mu dx^\nu$$ you can just read off the metric components. It's especially useful when the metric has a lot of ...
user avatar
  • 25.7k
1 vote

Confusion about raising and lowering indices

While it is true that $$ \eta_{\mu\nu}~U^\mu~U^\nu=c^2 $$ in the ${+}{-}{-}{-}$ metric, there is no way to use that to reduce the expression you are starting with. So for example you can align your $w=...
user avatar
  • 35k
2 votes
Accepted

Confusion about raising and lowering indices

No. Both $\mu$ and $\nu$ are already dummy indices. Remember that in the Einstein summation convention, each index may appear at most twice - once upstairs and once downstairs.
user avatar
  • 52k
1 vote
Accepted

Divergence theorem in index notation

Maybe it helps you visualize if we expand things out. Let $A_{il} = \epsilon_{ijk}r_{j}\sigma_{kl}$, such that the first integral is: $$\tau_{i} = \displaystyle{\int A_{il}n_{l} dA}$$ Notice that we ...
user avatar
2 votes
Accepted

What were the $r$ and $n$ of $\theta$s (Polchinski String theory section 8.6 page 265)?

The numbers $r_i\in\mathbb{N}$ are the degeneracies/multiplicies of the eigenvalues $\theta_i$ of the diagonal background gauge field $A_{25}$ in the compactified 25th direction $$ A_{25}~=~-\frac{1}{...
user avatar
  • 170k

Top 50 recent answers are included