Introduction to Differential Equations

llamaz 1 hour ago [-]

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of \displaystyle{\frac{dy}{dx} = 0}, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you’ve motivated ODEs and given the students a reason to solve ODEs with constant coefficients.

— Hacker News

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2019.10.02 Wednesday ACHK

Problem 13.6b

A First Course in String Theory | Topology, 2 | Manifold, 2

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13.6 Orientifold Op-planes

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In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for “orbit-manifold”) is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure.

The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group.

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In string theory, the word “orbifold” has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a quotient of \displaystyle{\mathbf{R}^n} by a finite group, i.e. \displaystyle{\mathbf{R}^n/\Gamma}. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space \displaystyle{M/G} where \displaystyle{M} is a manifold (or a theory), and \displaystyle{G} is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.

— Wikipedia on Orbifold

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an \displaystyle{n}-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension \displaystyle{n}.

— Wikipedia on Manifold

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In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints.

— Wikipedia on Topological space

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2019.09.26 Thursday ACHK

Pointer state, 2

Eigenstates 3.2

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Microscopically, a state can be definite or indefinite. Even if it is indefinite, the overlapping of superpositions of states of a lot of particles, or the superposition of a lot of system-microstates gives a definite macrostate.

If a state is definite, it is corresponding to one single system-macrostate directly.

I am referring to the physical definition, not the mathematical definition.

— Me@2012-12-31 09:28:08 AM

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If a microstate is definite, it is called an “eigenstate”. It is corresponding to one single system-macrostate directly.

However, the microstate is NOT the macrostate. The microstate is just corresponding to that macrostate.

— Me@2019-09-20 07:02:10 AM

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In quantum Darwinism and similar theories, pointer states are quantum states, sometimes of a measuring apparatus, if present, that are less perturbed by decoherence than other states, and are the quantum equivalents of the classical states of the system after decoherence has occurred through interaction with the environment. ‘Pointer’ refers to the reading of a recording or measuring device, which in old analog versions would often have a gauge or pointer display.

— Wikipedia on Pointer state

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In quantum mechanics, einselections, short for environment-induced superselection, is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions.

In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system.

After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain [in the sense of classical probability] state which can be decomposed into a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These ‘pointer states’ are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

Advocates of this approach argue that since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers). However, the basic program has been criticized as relying on a circular argument (e.g. R. E. Kastner). So the question of whether the ‘einselection’ account can really explain the phenomenon of wave function collapse remains unsettled.

— Wikipedia on Einselection

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Here I simply review the basic approach to ‘deriving’ einselection via decoherence, and point to a key step in the derivation that makes it a circular one.

— Ruth E. Kastner

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We should not derive einselection via decoherence. Instead, they should be regarded as different parts or different presentations of the same theory.

In other words, “einselection” and “decoherence” are synonyms.

— Me@2019-09-21 05:53:53 PM

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There has been significant work on correctly identifying the pointer states in the case of a massive particle decohered by collisions with a fluid environment, often known as collisional decoherence. In particular, Busse and Hornberger have identified certain solitonic wavepackets as being unusually stable in the presence of such decoherence.

— Wikipedia on Einselection

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2019.09.22 Sunday (c) All rights reserved by ACHK

事件實在論,更正

Event Realism | 事件實在論 6.1

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exist = can be found

無後果,就不再存在。

— Me@2013.09.25

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If the consequences of an event cannot be found anymore, that event no longer exists.

— Me@2019.09.05

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The surprising implications of the original delayed-choice experiment led Wheeler to the conclusion that “no phenomenon is a phenomenon until it is an observed phenomenon”, which is a very radical position. Wheeler famously said that the “past has no existence except as recorded in the present“, and that the Universe does not “exist, out there independent of all acts of observation”.

— Wikipedia on Wheeler’s delayed choice experiment

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「事件」並不完全「實在」。

實在 ~ 堅實地存在

仍然有後果的事件,才為之「仍然存在」。

永久地有後果的事件,才為之「實在」。

— Me@2019-09-05 09:08:41 PM

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2019.09.05 Thursday (c) All rights reserved by ACHK

Quantum observer 2

Consistent histories, 6.2 | Relational quantum mechanics, 2 | Eigenstates 2.3.2.2

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Would an observer see itself being in a superposition?

In a sense, tautologically, an observer is not a superposition of itself, because “an observer” can be defined as “a consistent history”.

an observer ~ a consistent history

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Because “state” is expressed in RQM as the correlation between two systems, there can be no meaning to “self-measurement”.

— Wikipedia on Relational quantum mechanics

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Would an observer see itself being in a superposition?

When we say that “before observation, observable B is in a superposition of some eigenstates”, you have to specify

  1. it is a superposition of what?

  2. it is a superposition with respect to what apparatuses or experimental setups?

— Me@2018-02-05 12:45 AM

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2019.09.03 Tuesday (c) All rights reserved by ACHK

Physical laws are low-energy approximations to reality, 1.6

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QM GR

too many particles

when you cool down the system, you see something that your equations cannot predict

only by experiment, you discover that you can go to there

the system state goes from one local minimum to another

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Since each theory is valid only when near a particular energy local minimum, we cannot use it to predict other local minima (other physical realities). That’s why we need to keep doing experiments that are designed for stress-testing existing theories. For example, one experiment can put a system in extreme high pressure. Another experiment can put a system in extreme high temperature.

Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. From there, we construct new theories to explain those phenomena.

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Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. In this sense, experimental physics is much richer.

Computational physics goes further in only one direction. Only experiments let you try randomly.

— Me@2019-08-18 07:51:05 PM

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2019.08.18 Sunday (c) All rights reserved by ACHK

From classical to quantum

From this viewpoint, the move from a classical to a quantum mechanical system is not a move from a comutative to a non-commutative algebra \displaystyle{\mathcal{A}} of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.

Of course, every non-commutative algebra determines an underlying partial commutative algebra and also its diagram of commutative subalgebras.

That fact that assuming the structure of a non-commutative algebra is the wrong assumption has already been observed in the literature (see, for example, [19]),

but it is often replaced by another wrong assumption, namely that of assuming the structure of a Jordan algebra.

These differing assumptions on the structure of \displaystyle{\mathcal A} affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on \displaystyle{\mathcal A}, the larger is its automorphism group).

— The Mathematical Foundations of Quantum Mechanics

— David A. Edwards

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2019.06.18 Tuesday ACHK

Unitarity (physics)

Unitarity means that if a future state, F, of a system is unique, the corresponding past point, P,  is also unique, provided that there is no information lost on the transition from P to F.

— Me@2019-05-22 11:06:48 PM

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In quantum physics, unitarity means that the future point is unique, and the past point is unique. If no information gets lost on the transition from one configuration to another[,] it is unique. If a law exists on how to go forward, one can find a reverse law to it.[1] It is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1.

Since unitarity of a theory is necessary for its consistency (it is a very natural assumption, although recently questioned[2]), the term is sometimes also used as a synonym for consistency, and is sometimes used for other necessary conditions for consistency, especially the condition that the Hamiltonian is bounded from below. This means that there is a state of minimal energy (called the ground state or vacuum state). This is needed for the third law of thermodynamics to hold.

— Wikipedia on Unitarity (physics)

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2019.05.23 Thursday (c) All rights reserved by ACHK

Multiverse

A physics statement is meaningful only if it is with respect to an observer. So the many-world theory is meaningless.

— Me@2018-08-31 12:55:54 PM

— Me@2019-05-11 09:41:55 PM

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Answer me the following yes/no question:

In your multi-universe theory, is it possible, at least in principle, for an observer in one universe to interact with any of the other universes?

If no, then it is equivalent to say that those other universes do not exist.

If yes, then those other universes are not “other” universes at all, but actually just other parts of the same universe.

— Me@2019-05-11 09:43:40 PM

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2019.05.11 Saturday (c) All rights reserved by ACHK

Classical probability, 7

Classical probability is macroscopic superposition.

— Me@2012.04.23

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That is not correct, except in some special senses.

— Me@2019-05-02

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That is not correct, if the “superposition” means quantum superposition.

— Me@2019-05-03 08:44:11 PM

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The difference of the classical probability and quantum probability is the difference of a mixed state and a pure superposition state.

In classical probability, the relationship between mutually exclusive possible measurement results, before measurement, is OR.

In quantum probability, if the quantum system is in quantum superposition, the relationship between mutually exclusive possible measurement results, before measurement, is neither OR nor AND.

— Me@2019-05-03 06:04:27 PM

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2019.05.03 Friday (c) All rights reserved by ACHK

Mixed states, 4

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How is quantum superposition different from mixed state?

The state

\displaystyle{|\Psi \rangle = \frac{1}{\sqrt{2}}\left(|\psi_1\rangle +|\psi_2\rangle \right)}

is a pure state. Meaning, there’s not a 50% chance the system is in the state \displaystyle{|\psi_1 \rangle } and a 50% it is in the state \displaystyle{|\psi_2 \rangle}. There is a 0% chance that the system is in either of those states, and a 100% chance the system is in the state \displaystyle{|\Psi \rangle}.

The point is that these statements are all made before I make any measurements.

— edited Jan 20 ’15 at 9:54

— Mehrdad

— answered Oct 12 ’13 at 1:42

— Andrew

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Given a state, mixed or pure, you can compute the probability distribution \displaystyle{P(\lambda_n)} for measuring eigenvalues \displaystyle{\lambda_n}, for any observable you want. The difference is the way you combine probabilities, in a quantum superposition you have complex numbers that can interfere. In a classical probability distribution things only add positively.

— Andrew Oct 12 ’13 at 14:41

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— How is quantum superposition different from mixed state?

— Physics StackExchange

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2019.04.23 Tuesday ACHK

Physical laws are low-energy approximations to reality, 1.3.1

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Symmetry breaking is important.

When there is symmetry-breaking, the system goes to a low-energy state.

Each possible low-energy state can be regarded as a new “physical world”.

One “physical world” cannot jump to another, unless through quantum tunnelling. But the probability of quantum tunnelling happening is low.

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Low-energy physics theories, such as harmonic oscillator, are often simple and beautiful.

— Professor Renbao Liu

— Me@2019-04-08 10:46:32 PM

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2019.04.09 Tuesday (c) All rights reserved by ACHK

Quantum classical logic

Mixed states, 2 | Eigenstates 4

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— This is my guess. —

If the position is indefinite, you can express it in terms of a pure quantum state[1] (of a superposition of position eigenstates);

if the quantum state is indefinite, you can express it in terms of a mixed state;

if the mixed state is indefinite, you can express it in terms of a “mixed mixed state”[2]; etc. until definite.

At that level, you can start to use classical logic.

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If you cannot get certainty, you can get certain uncertainty.

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[1]: Me@2019-03-21 11:08:59 PM: This line of not correct. The uncertainty may not be quantum uncertainty. It may be classical.

[2]: Me@2019-03-22 02:56:21 PM: This concept may be useless, because a so-called “mixed mixed state” is just another mixed state.

For example, the mixture of mixed states

\displaystyle{p |\psi_1 \rangle \langle \psi_1 | + (1-p) |\psi_2 \rangle \langle \psi_2 |}

and

\displaystyle{q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_2 |}

is

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\displaystyle{\begin{aligned}  &w \bigg[ p |\psi_1 \rangle \langle \psi_1 |+ (1-p) |\psi_2 \rangle \langle \psi_2 | \bigg] +  (1-w) \bigg[ q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_1 | \bigg] \\  &= w p |\psi_1 \rangle \langle \psi_1 | + w (1-p) |\psi_2 \rangle \langle \psi_2 | +  (1-w) q |\phi_1 \rangle \langle \phi_1 | + (1-w) (1-q) |\phi_2 \rangle \langle \phi_1 | \\  \end{aligned}}

— This is my guess. —

— Me@2012.04.15

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2019.03.22 Friday (c) All rights reserved by ACHK

Physical laws are low-energy approximations to reality, 1.2

d_2019_01_31__23_12_26_PM_

When the temperature \displaystyle{T} is higher than the critical temperature \displaystyle{T_c}, point \displaystyle{O} is a local minimum. So when a particle is trapped at \displaystyle{O}, it is in static equilibrium.

However, when the temperature is lowered, the system changes to the lowest curve in the figure shown. As we can see, at the new state, the location \displaystyle{O} is no longer a minimum. Instead, it is a maximum.

So the particle is not in static equilibrium. Instead, it is in unstable equilibrium. In other words, even if the particle is displaced just a little bit, no matter how little, it falls to a state with a lower energy.

This process can be called symmetry-breaking.

This mechanical example is an analogy for illustrating the concepts of symmetry-breaking and phase transition.

— Me@2019-03-02 04:25:23 PM

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2019.03.02 Saturday (c) All rights reserved by ACHK

The Door 1.1

The following contains spoilers on a fictional work.

In Westworld season 2, last episode, when a person/host X passed through “the door”, he got copied, almost perfectly, into a virtual world. Since the door was adjacent to a cliff, just after passing through it, the original copy (the physical body) fell off the cliff and then died.

Did X still exist after passing through the door?

Existence or non-existence of X is not a property of X itself. So in order for the question “does X exist” to be meaningful, we have to specify “with respect to whom”.

In other words, instead of “does X exist”, we should ask

With respect to the observer Y, does X exist?

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There are 3 categories of possible observers (who were observing X passing through the door):

  1. the original person (X1)
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    X_1 == X

  2. the copied person (X2) in the virtual world
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    For simplicity, assume that X2 is a perfect copy of X.

  3. other people (Y)

— Me@2019-02-09 1:09 PM

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2019.02.28 Thursday (c) All rights reserved by ACHK

Quantum decoherence 9

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This is a file from the Wikimedia Commons.

In classical scattering of target body by environmental photons, the motion of the target body will not be changed by the scattered photons on the average. In quantum scattering, the interaction between the scattered photons and the superposed target body will cause them to be entangled, thereby delocalizing the phase coherence from the target body to the whole system, rendering the interference pattern unobservable.

The decohered elements of the system no longer exhibit quantum interference between each other, as in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum-entangled with the environment. The converse is not true: not all entangled states are decohered from each other.

— Wikipedia on Quantum decoherence

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2019.02.22 Friday ACHK