Ciro Santilli @cirosantilli 34

 Incoming links: Partial derivative

Einstein notation for partial derivatives  Updated 2024-12-15  +Created 1970-01-01

The Einstein summation convention works will with partial derivatives and it is widely used in particle physics.

In particular, the divergence and the Laplacian can be succintly expressed in this notation:

In order to expresse partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variales with indices such as

x_{0}

x_{1}

x_{2}

\partial_{0}

\partial_{1}

and

\partial_{2}

instead of the usual letters

x

y

and

z

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

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Let's start with the one dimensional case. Let the

q : R \to R

and a Functional

I

defined by a function of three variables

F : R^{3} \to R

I (q) = \int_{t_{0}}^{t} F (t, q (t), \overset{q}{˙} (t)) d t

(1)

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given

F

, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that

F

such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

\frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q ˙} = 0

(2)

\frac{\partial F}{\partial q}

and

\frac{\partial F}{\partial q ˙}

we simply mean "the partial derivative of

F

with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where

q

is the unknown, since:

the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q}$ is a function of $t, q (t), \overset{q}{˙} (t)$
the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q ˙}$ is a function of $t, q (t), \overset{q}{˙} (t)$ . And so it's derivative with respect to time will contain only up to $\overset{q}{¨}$

Now let's think about the multi-dimensional case. Instead of having

q : R \to R

, we now have

q : R \to R^{n}

. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case,

F

is going to be a function of 5 variables

F : R^{5} \to R

rather than 3 as in the one dimensional case, and the functional looks like:

I (q) = \int_{t_{0}}^{t} F (t, q_{1} (t), q_{2} (t), \overset{q_{1}}{˙} (t), \overset{q_{2}}{˙}) (t) d t

(3)

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions

q_{1}

and

q_{2}

of order up to 2 in both variables:

\frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ) )}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ) )}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{2} ˙} = 0

(4)

At this point, notation is getting a bit clunky, so people will often condense the

q_{i}

vector

\frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ) )}{\partial q _{2} ˙} = 0

(5)

or just omit the arguments of

F

entirely:

\frac{\partial F}{\partial q _{i}} - \frac{d}{d t} \frac{\partial F}{\partial q _{i} ˙} = 0

(6)

Video 1.

Calculus of Variations ft. Flammable Maths by vcubingx (2020)

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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