Linked Questions

2
votes
0answers
287 views

Accidental degeneracy in Hydrogen Energies [duplicate]

The energy of Hydrogen electron ground state should depend on $n$ and $\ell$, but it only depends on $n$. What is the reason behind this accidental degeneracy; I know that the reason lies in symmetry; ...
2
votes
0answers
44 views

What symmetry operation mixes states with different $\ell$ in hydrogen atom? [duplicate]

We can mix states with different $m$ in hydrogen atom by rotating it around some axis (not coinciding with $z$). Thus rotation is the symmetry operation which mixes states with different $m$. As ...
16
votes
2answers
3k views

Is there a kind of Noether's theorem for the Hamiltonian formalism?

The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
15
votes
2answers
4k views

Invariance of Lagrangian in Noether's theorem

Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
3
votes
1answer
5k views

Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing ...
5
votes
1answer
439 views

Physical implication of conservation of Laplace-Runge-Lenz vector

I'm having trouble wrapping my mind around the Laplace-Runge-Lenz vector. Conservation of momentum can be visualized as an object moving in a straight line with constant speed. One can even visualize ...
2
votes
2answers
129 views

What operator lowers the total angular momentum?

Assume states $|j,m\rangle$, say $j\in\{3,2,1,0\}$, initially at $3$. Is there any "lowering" operator I could apply such that $L_-|j,m\rangle = |j-1,m\rangle $? How to express it in the $J_z$ ...
0
votes
0answers
582 views

Showing The Laplace–Runge–Lenz vector (per unit mass) is constant

Given an inverse square law $\ddot{\vec{r}}=-\frac{\mu}{r^2}\hat{r}$, I define the Angular momentum per unit mass as $\vec{H}=\vec{r}\times\dot{\vec{r}}$. Showing it's constant is strightfoward. Then ...
3
votes
1answer
261 views

Feynman's Lost Lecture: what is the significance of $\frac{d\mathfrak{v}}{d\theta}=-\frac{GMm}{\left|\mathfrak{L}\right|}\hat{\mathfrak{r}}?$

My question pertains to a fact used by Richard Feynman in his so-called Lost Lecture. http://books.wwnorton.com/books/Feynmans-Lost-Lecture/. I have only skimmed the book, so I have much more to learn ...
4
votes
1answer
95 views

Why are some symmetries invisible to the configuration space Lagrangian $L(q, \dot q,t)$?

Usually, when people talk about Lagrangians they are talking about a function of configuration space variables $q_i$ and their time derivatives $\dot q_i$. This is a function $L = L(q_i, \dot q_i,t)$. ...
1
vote
0answers
68 views

Legendre transformation and correspondance between Noether charges and quasi-symmetries

I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...