Ciro Santilli @cirosantilli 34

 Incoming links: Generalized coordinate

Manifold  Updated 2024-12-15  +Created 1970-01-01

We map each point and a small enough neighbourhood of it to

R^{n}

, so we can talk about the manifold points in terms of coordinates.

Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.

Manifolds are cool. Especially differentiable manifolds which we can do calculus on.

A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.

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Noether's theorem  Updated 2024-12-15  +Created 1970-01-01

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For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.

Video 1.

The most beautiful idea in physics - Noether's Theorem by Looking Glass Universe (2015)

Source. One sentence stands out: the generated quantities are called the generators of the transforms.

Video 2.

The Biggest Ideas in the Universe | 15. Gauge Theory by Sean Carroll (2020)

Source. This attempts a one hour hand wave explanation of it. It is a noble attempt and gives some key ideas, but it falls a bit short of Ciro's desires (as would anything that fit into one hour?)

Video 3.

The Symmetries of the universe by ScienceClic English (2021)

Source. youtu.be/hF_uHfSoOGA?t=144 explains intuitively why symmetry implies consevation!

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Quantum electrodynamics Lagrangian  Updated 2024-12-15  +Created 1970-01-01

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L_{Q E D} = \overset{ˉ}{ψ} (i ℏ c D / - m c^{2}) ψ - \frac{1}{4 μ _{0}} F_{μ ν} F^{μ ν}

(1)

where:

$F$ is the electromagnetic tensor

Note that this is the sum of the:

Dirac Lagrangian, which only describes the "inertia of bodies" part of the equation
the electromagnetic interaction term $\frac{1}{4 μ _{0}} F_{μ ν} F^{μ ν}$ , which describes term describes forces

Note that the relationship between

ψ

and

F

is not explicit. However, if we knew what type of particle we were talking about, e.g. electron, then the knowledge of psi would also give the charge distribution and therefore

F

As mentioned at the beginning of Quantum Field Theory lecture notes by David Tong (2007):

by "Lagrangian" we mean Lagrangian density
the generalized coordinates of the Lagrangian are fields

Video 1.

Particle Physics is Founded on This Principle! by Physics with Elliot (2022)

Source.

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Schrödinger equation  Updated 2024-12-15  +Created 1970-01-01

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The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

fine structure
spontaneous emission coefficients

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

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