Questions tagged [quantum-calculus]

Quantum calculus encompasses $q$-calculus and $h$-calculus, and is a notion of "calculus without limits". Do not confuse with the (quantum-mechanics) tag. For questions on Schrodinger's equation and solutions, use (quantum-mechanices), (pde), (fourier-analysis), and/or (calculus) as appropriate.

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q-differentiability in $R_q$

Wokring in q-calculus where everything is defined on the set $$R_q =\{\pm q^k,k \in \mathbb{Z} \} \cup \{ 0\}$$ In which they define the q-derivative (or q-difference operator) as $$D_qf(x)=\frac{f(x)−...
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1st order q-differential equations

Can we solve 1st order q-differential equations using the usual methods of 1st order differential equations? For example, can we use integration factor method to solve this q-differential equations? $$...
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Combinatorics - Q Calculus Pascal's Identity Proof

I have been trying to get started on this simple combinatorics proof. This has led me to start a proof by induction using pascal's identity from https://en.wikipedia.org/wiki/...
MathMan2021's user avatar
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Quantum Calculus Prove Q Binomial Coefficient Analogue

Parts a and b are trivial so those are of no concern. My conclusion for part c is that a proper q analog is the q binomial coefficient, counting the number of subspaces of dimension k in a vector ...
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Bound for deformed binomial coefficients

Let $\mu$ be a positive real number smaller than $1$. Define $${k\choose{r}}_{\mu}=\frac{\prod_{i=1}^{k}(1-\mu^{2i})}{\prod_{i=1}^{k-r} (1-\mu^{2i}) \prod_{i=1}^{r} (1-\mu^{2i})}$$ Then I need to show ...
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Why is $q$ sometimes a complex number, but other times a prime power?

In the fields of representation theory and quantum algebra, we often start with some $\mathbf{C}$-algebra and study it's quantization as an algebra over $\mathbf{C}(q)$, using the algebra structure to ...
Mike Pierce's user avatar
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Studying quantized algebras, what motivates the choice of base ring?

In the fields of representation theory and quantum algebra, we often start with, for example, some $\mathbf{C}$-algebra $A$ and study a quantization of $A$ by adjoining an indeterminate $q$, or ...
Mike Pierce's user avatar
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Studying quantized algebras, why introduce $q^{1/2}$ instead of just $q$?

In the fields of representation theory and quantum algebra, we often study quantized versions of algebraic objects by regarding them as algebras over $\mathbf{C}(q)$, or some subring of $\mathbf{C}(q)$...
Mike Pierce's user avatar
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Intuition behind this q-binomial formula counting sums of subsets

We have these transparent, motivating interpretations for binomial coefficients and their $q$-analogue. $$ \binom{n}{j} = \begin{matrix} \text{"The number of subsets of size $j$}\\ \text{of a set ...
Mike Pierce's user avatar
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q-derivative of binomial

Problem Find the q-derivative of $(a-x)_q^n$ for $n \ge 1$. Answer This is actually equation (3.11) of Kac and Cheung's Quantum Calculus $$D_q(a-x)_q^n=-[n](a-qx)_q^{n-1}.\tag{3.11}$$ My question ...
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Q-Exponential Sum Identity

We define the "standard" q-exponential as follows $$ e_q(x) = 1 + \frac{1}{1} x+ \frac{1}{(1+q)} x^2 + \frac{1}{(1+q)(1+q+q^2)}x^3 ... =$$ $$ \sum_{i = 0}^{\infty} \left[ (1-q)^ix^i \prod_{j=1}^{...
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Connection between the quantum calculus and fractional calculus developed by Riemann-Liouvelle, Caputo and others?

I wonder are there any connection between the quantum calculus and fractional calculus developed by Riemann-Liouvelle, Caputo and others?
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Calculate density matrix by a given observable

Given the value of my observable (Energy) and with a known hamiltonian, I would like to compute the value of my thermal state. That is, with a given observable defined as: $$\langle\hat{O}\rangle=tr(\...
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What is the way to change the limits of q-integration of a double integral

What is the way to change the limits of q-integration of a double integral. For exemple, what is the answer after change the order of integration of $$\int_0^1 \int_0^{x} f(x,y)\ d_qy\ d_qx$$ https:/...
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Cannot solve this integral used in quantum chemistry

I am writing computer code for an implementation of the Hartree Fock algorithm and I am stuck on a certain integral. This is a great walkthrough to get some background : HFTheory Anyway, the set-up ...
user2879934's user avatar
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Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
James Smithson's user avatar
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What does the statement "x is a diagram of classical logics" mean?

From the Wikipedia entry on quantum logic A more modern approach to the structure of quantum logic is to assume that it is a diagram – in the sense of category theory – of classical logics (see ...
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double integrals on quantum calculus

I need references or book recommendations to find properties of double integrals on quantum calculus. Especially i need analogue of Fubini's theorem on q-calculus.
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improper integrals in q-calculus

In quantum calculus is this equality possible for improper integrals? $\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
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Ejemplos de la integral de Jackson (Examples of Jackson's Integral)

Original question in Spanish La integral de Jackson está definida en el cálculo cuántico, y quisiera que alguien me ayudara a la explicación de un ejemplo de este estilo de integrales. Gracias Added ...
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Is there a gamma-like function for the q-factorial?

I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in ...
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