# Search results for query: *

1. ### A Ground state of the one-dimensional spin-1/2 Ising model

Never mind, I've solved it myself. Simply using that $$\min_{\lbrace s_n\rbrace}\mathcal{H} = \sum_n\min\left(-Js_ns_{n+1}\right).$$

9. ### Solve this functional equation

The function ##f(x) = \mathrm{max}(x,0)## (i.e. the positive part) is also a solution to Cauchy as well and though it is continuous it isn't differentiable at ##x=0## and thus not analytic. Check for yourself if also satisfy the original functional equation.
10. ### Solve this functional equation

I believe that ##f(x) = ax## for ##a\in\{0,1\}## are the entire family of continuous solutions (consisting of elementary functions) to the original functional equation. As any continuous solution also has to be a solution to the Cauchy's functional equation. However there exist non-continuous...
11. ### Solve this functional equation

True, however that just imply that either ##a=1## or ##a=0##.
12. ### Solve this functional equation

From ##f(x^2)=xf(x)## conclude that ##f## is an odd function i.e. ##f(-x)=-f(x)##. If we take ##(x,y) = (-x,y)## then we get that $$f(-xf(y)+f(-x)) + yf(y) = f(-x) + yf(y-x)$$ or, by using the odd property of ##f##, that $$-f(xf(y)+f(x)) + yf(y) = -f(x) + yf(y-x).$$ Now, add this equation...
13. ### Solve this functional equation

No, you are right. What I wrote in 19 wasn't correct.
14. ### Solve this functional equation

That is also tremendous progress. This allow you to simplify the functional equation to $$xf(y) + yf(y) = yf(x+y)$$
15. ### Find a function satisfying these conditions: f(x)f(f(x)) = 1 and f(2020) = 2019

Sure! However call ##f(x) = 2020##. Then, by an identical argument, we instead get that ##f(2020) = 1/2020##, which clearly contradicts the condition ##f(2020) = 2019##. I agree. If the function exist then its graph is intersting. It clearly isn't monotone.
16. ### Find a function satisfying these conditions: f(x)f(f(x)) = 1 and f(2020) = 2019

Not sure if this is going to be of any help. But doesn't the condition ##f(2020)=2019## seem to be contradictory? I mean, let ##y=f(x)## for some arbitrary chosen ##x\in\mathbb{R}##. Then $$yf(y) = 1,$$ according to the functional equation. Or equivalently that $$f(x) = \frac{1}{x}.$$ This...

Your PDE is linear, you should therefore try to look for a solution by separation of variables; i.e. assume that the solution is of the form ##T(t,x) = u(t)v(x)## and derive a space-independent equation for ##u(t)## and a time-independent equation for ##v(x)##. Remember that ##v(x)## inherit...
18. ### Orthogonality Relationship for Legendre Polynomials

Legendre polynomials are orthogonal but not orthonormal over the interval ##[-1,1]##. Thus, you shouldn't expect your orthonormal basis to be identical to the Legendre polynomials. NB. If you are trying to construct and orthonormal set ##\{p_0,p_1,p_2\}## of polynomials over the interval...

I agree.
25. ### Help solving this equation please: y^2-2ln(y)=x^2

It is possible to express ##y## in terms of ##x##. From your calculations you have that $$y^{-2}e^{y^2} = e^{x^2},$$ this it great. Rearrange the equation to look like $$-y^2e^{-y^2} = -e^{-x^2}.$$ This is an equation of the form ##we^w = z## and can thus be solved using the Lambert ##W##...
26. ### I Solving an ODE with Legendre Polynomials

Begin by deviding both side of $$\frac{d}{d\theta}\bigg(\sin\theta\frac{d\Theta}{d\theta}\bigg) = - l(l+1)\sin\theta\,\Theta$$ by ##\sin\theta## to get $$\frac{1}{\sin\theta}\frac{d}{d\theta}\bigg(\sin\theta\frac{d\Theta}{d\theta}\bigg) + l(l+1)\Theta = 0.$$ Now, introduce the new variable...
27. ### Chemistry Quantum Chemistry - Particle in a box

Apply l'Hospital's rule to deduce that \begin{align*} \lim_{m\rightarrow\infty}\frac{m\pi \pm 2}{4m\pi} &= \frac{1}{4} \end{align*} and conclude that the probability for finding the particle in the leftmost quarter of the box is ##1/4## at high energies (i.e. large ##m##).
28. ### How to get the Uncertainty of a percentage change

Have you tried propagation of error?
29. ### I Assigning a value for integrating a divergent oscillatory function to infinity

Sure! However, as you probably know, an antiderivative to ##f(x) = x\sin(x)## is ##F(x) = \sin(x) - x\cos(x)##. Thus, the original improper integral is \begin{align} I &= \int_0^\infty x\sin(x)\,dx \\ &= \lim_{a\rightarrow\infty}\Big[\sin(x) - x\cos(x)\big]. \end{align} This limit doesn't exist...
30. ### Perturbation theory for solving a second-order ODE

How sure are you that the equation reads $$\ddot{\xi} = -b\xi + \cos(\omega t)(a-c\xi^2)$$ and not instead $$\dot{\xi} = -b\xi + \cos(\omega t)(a-c\xi^2).$$ I only ask because the latter is an Ricatti equation and thus exactly solvable. If you only have calculated the pertubation series upto...
31. ### Why is my simulation of projectile trajectory with air resistance wrong?

Congratulation! So what is the ideal angle when you include air resistance?
32. ### I Does this ODE have any real solutions?

There are an infinite number of trivial solutions to this ODE. Supose ##f:\Omega\rightarrow\mathbb{C}## is a holomorphic function, then ##\big(y(x), z(x)\big) = \big(f(x),\pm f(x)\big)## are trivial solutions to your original ODE $$(y^\prime)^2 - (z^\prime)^2 + 2m^2(y^2 - z^2) = 0.$$ So to...
33. ### Probability density function in classical mechanics

In the case of a simple harmonic oscillator (SHO); your intuition is right, it is more likely to encounter it near its endpoints of oscillation. If we choose our coordinate frame such that the origin is at the equilibrium point for the SHO and starts the oscillation at the displacement ##A##...
34. ### Internal Energy of a Mole of Particles each with 3 Energy Levels

No, this is generally not true for indistinguishable particles as the following minimal example will demonstrate. Consider a system consisting of two particles, each of which can be in one out of two possible one-particle-states with the energies ##E_1 = 0## and ##E_2 = \epsilon## respectively...
35. ### Internal Energy of a Mole of Particles each with 3 Energy Levels

In general, $$Z=z^N$$ is the partition function for a system consiting of ##N## distinguishable and noninteracting particles with single particle partition function ##z## (prove this from first principle, good exercise). Thus, in your case where the single particle partition function is given...