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I was reading about the one-dimensional tight-binding Hamiltonian (TBH) $$H=E_0\sum\limits_{n}|n\rangle\langle n|-t\sum\limits_{n}\Big(|n\rangle\langle n+1|+|n+1\rangle\langle n|\Big)\tag{1}$$ where $E_0$ and $t$ denote the on-site energy and the hopping parameter, repectively. The Hamiltonian of...
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Is it possible to define quantum mechanics in real vector spaces instead of complex vector spaces and what would the dimensionality be of such a vector space?
Can anyone referee me such a treatment of a 2 state system, say the spin system.
While the calculations seem fine, I can't make sense of the solution to this problem. How can the uncertainty principle be used to tell us what the shortest pulse is? Wouldn't using it tell us the error in the pulse lengths? And the pulse lengths then could be arguable any size we want it to be.
if we have position operator x and momentum operator p as follows:
$$ \hat x = x \space ; \space \hat p = - i \hbar {∂ \over ∂x} \space $$
How do we show the commutator is:
$$ [\hat x, \hat p] = i \hbar$$
I am stuck at:
$$ [\hat x, \hat p] = x \space (-i \hbar {∂ \over ∂x}) - (-i \hbar {∂ \o...
"Everything we call real is made of things that cannot be regarded as real."
And also can someone say what is the source of this quote.
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I should note that this is an integration algorithm and therefore intermediate steps DO appear to be unjustified. The purpose of this question is to justify or reject this algorithm as always giving correct solutions.
Suppose we have some piecewise continuous function $f$ and that it can be writ...
The busy beaver game consists of designing a halting, binary-alphabet Turing machine which writes the most 1s on the tape, using only a limited set of states. The rules for the 2-state game are as follows:
the machine must have two states in addition to the halting state, and
the tape starts with 0s only.
As the player, you should conceive each state aiming for the maximum output of 1s on the tape while making sure the machine will halt eventually.
The nth busy beaver, BB-n or simply "busy beaver" is the Turing machine that wins the n-state Busy Beaver Game. That is, it attains the maximum number...
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt. These numbers are formed from a construction due to Gregory Chaitin.
Although there are infinitely many halting probabilities, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction instead of Chaitin's constant when not referring to...
In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces
[
0
,
ω
]
{\displaystyle [0,\omega ]}
and
[
0
,
ω
1
]
{\displaystyle [0,\omega _{1}]}
, where
ω
{\displaystyle \omega }
is the first infinite...
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Suppose that there exists functions $f$ and $g$ defined on the real numbers and differentiable everywhere. If their derivatives $f'$ and $g'$ are linearly independent on some nonzero interval then are $f$ and $g$ linearly independent on the same interval? Similarly if $f$ and $g$ are linearly ind...
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Q: Does observation require understanding to collapse the wave function in the double slit experiment?
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Can a particle, having zero kinetic energy but with rest mass $m_0$ ($\sim E_0$) > V (V is potential energy), penetrate the potential wall V ?
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