# Search results

1. ### Quantization axis

What is quantisation axis? In many books authors just say that we choose that z is quantization axis.
2. ### Delta function

In Dirac definition ##\delta(x)## is ##\infty## when ##x=0##, and ##0## when ##x\neq 0##. My question is when I have some ##\alpha \delta(x)## could I interpretate this like function which have value ##\alpha## in point ##x=0##?
3. ### Antiferromagnet more subblatices

In ordinary definition antiferromagnet lattice has to sublattices, one with spins up, and one with of spin down in ##T=0##. Why in some cases people discuss situations with four or even more subblatices? Do you have explanation for this? Some references maybe?
4. ### Classical and quantum Heisenberg model

In quantum Heisenberg model \hat{H}=-J\sum_{\vec{n},\vec{m}}\hat{\vec{S}}_{\vec{n}}\cdot \hat{\vec{S}}_{\vec{m}} ##J## can be obtained from dispersion experiment. For large spin ##S## even classical Heisenberg model is good for description of Curie temperature for example. Is that with the same...
5. ### Monte Carlo step

What is refered as one Monte Carlo step. In all books, papers people is written that was performed ##5 \cdot 10^{6}## MCS on all system sizes. Or the time is measured in MCS. But what is refered as one MCS? For example in MC simulation of Ising model what is a one MCS?
6. ### Translation operator

e^{\alpha\frac{d}{dx}}=1+\alpha\frac{d}{dx}+\frac{\alpha^2}{2!}\frac{d^2}{dx^2}+...=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^n}{dx^n} Why this is translational operator? ##e^{\alpha\frac{d}{dx}}f(x)=f(x+\alpha)##
7. ### Derivative question

If ##f=f(x)## why then is ##df=\frac{df}{dx}dx##?
8. ### Linear operator

Linear operator A is defined as A(C_1f(x)+C_2g(x))=C_1Af(x)+C_2Ag(x) Question. Is A=5 a linear operator? I know that this is just number but it satisfy relation 5(C_1f(x)+C_2g(x))=C_15f(x)+C_25g(x) but it is also scalar. Is function ##A=x## linear operator? It also satisfy...
9. ### Divergence question

I see identity in one mathematical book div \vec{A}(r)=\frac{\partial \vec{A}}{\partial r} \cdot grad r How? From which equation?
10. ### Quantum mechanics probability

In the one moment you accept something in the other you're not sure why? Why probability density in quantum mechanics is defined as ##|\psi|^2## and no just ##|\psi|## if we know that ##|\psi|## is also positive quantity.
11. ### Infinite potential well

In one dimensional problem of infinite square potential well wave function is ##\phi_n(x)=\sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L}## and energy is ##E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}##. Questions: What condition implies that motion is one dimensional. Did wave function describes motion of...
12. ### Taylor series

Why in Taylor series we have some factoriel ##!## factor. f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+... Why we have that ##\frac{1}{n!}## factor?
13. ### Hypergeometric function problem

Homework Statement Calculate _2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x) Homework Equations _2F_1(a,b,c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n (a)_n=a(a+1)...(a+n-1) The Attempt at a Solution (\frac{1}{2})_n=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-1}{2}...
14. ### Hypergeometric function

Hypergeometric function is defined by: _2F_1(a,b,c,x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n where ##(a)_n=a(a+1)...(a+n-1)##... I'm confused about this notation in case, for example, ##_2F_1(-n,b,b,1-x)##. Is that _2F_1(-n,b,b,1-x)=\sum^{\infty}_{n=0}\frac{(-n)_n}{n!}(1-x)^n or...
15. ### Infinite sum question

Happy new year. All the best. I have one question. Is it true? \sum^{\infty}_{k=0}a_kx^k=\sum^n_{k=0}a_{n-k}x^{n-k} I saw in one book relation \sum^{\infty}_{k=0}\frac{(2k)!}{2^{2k}(k!)^2}(2xt-t^2)^k=\sum^{n}_{k=0}\frac{(2(n-k))!}{2^{2(n-k)}((n-k)!)^2}(2xt-t^2)^{n-k} Can you give me some...
16. ### Bessel function summation

Homework Statement What is easiest way to summate \sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}] where ##i## is imaginary unit. Homework Equations The Attempt at a Solution I don't need to write explicit Bessel function so in sum could stay C_1J_(x)+C_2J_2(x)+... Well I see that...
17. ### Power series identity

Homework Statement Show e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Homework Equations J_k(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{(n+k)!n!}(\frac{x}{2})^{2n+k} The Attempt at a Solution Power series product (\sum^{\infty}_{n=0}a_n)\cdot (\sum^{\infty}_{n=0}...
18. ### Potential series method

Why sometimes we search solution of power series in the way: y(x)=\sum^{\infty}_{n=0}a_nx^n and sometimes y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}???
19. ### Archived Optics - magnification of a converging lens

Optics -- magnification of a converging lens Homework Statement Dioptry of converging lens is $D=3$. What is magnification ##u##? Homework Equations The Attempt at a Solution ##\frac{1}{f}=D## - dioptry. \frac{1}{f}=\frac{1}{p}+\frac{1}{l} u=\frac{l}{p} l=up...
20. ### Coefficients derivative

Why we always write equation in form y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write: m(x)y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write coefficient ##m(x)## for example?
21. ### Can you explain me why this is also isomorphism?

Homomorphism is defined by ##f(x*y)=f(x)\cdot f(y)##. One interesting example of this is logarithm function ##log(xy)=\log x+\log y##. Can you explain me why this is also isomorphism?
22. ### Inverse Laplace transform

Homework Statement Find inverse Laplace transform \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}] Homework Equations The Attempt at a Solution I try with theorem \mathcal{L}[f(t)*g(t)]=F(s)G(s) So this is some multiple of \mathcal{L}[\sin at*\sin at] So \mathcal...
23. ### Partial fraction

Homework Statement How to get partial fraction decomposition for \frac{1}{(x^2+a^2)(x^2+p^2)} Homework Equations The Attempt at a Solution I tried with \frac{1}{(x+ia)(x-ia)(x+ip)(x-ip)}=\frac{A}{x+ia}+\frac{B}{x-ia}+\frac{C}{x-ip}+\frac{D}{x+ip} and get the result at the end of the...
24. ### Poisson kernel

Why Poisson kernel is significant in mathematics? Poisson kernel is ##P_r(\theta)=\frac{1-r^2}{1-2rcos\theta+r^2}##. http://www.math.umn.edu/~olver/pd_/gf.pdf [Broken] page 218, picture 6.15. If we have some function for example ##e^x,sinx,cosx## what we get if we multiply that function with...
25. ### Fredholm integral equation

Is there any way to solve Fredholm integral equation without using Fourier transform. \varphi(t)=f(t)+\lambda\int^b_aK(t,s)\varphi(s)ds?
26. ### Laplace transform limits?

How we get relation \lim_{t\to 0}f(t)=\lim_{p\to \infty}pF(p)? Where ##\mathcal{L}\{f\}=F##.
27. ### Laplace transform converge

##\mathcal{L}\{f(t)\}=F(s)## \mathcal{L}\{e^{at}\}=\frac{1}{s-a},Re(s)>a \mathcal{L}\{\sin (at)\}=\frac{a}{s^2+a^2}, \quad Re(s)>0 \mathcal{L}\{\cos (at)\}=\frac{s}{s^2+a^2},Re(s)>0 If we look at Euler identity ##e^{ix}=\cos x+i\sin x##, how to get difference converge intervals...
28. ### Eigenvalue question

If ##\hat{A}\vec{X}=\lambda\vec{X}## then ##\hat{A}^{-1}\vec{X}=\frac{1}{\lambda}\vec{X}## And what if ##\lambda=0##?
29. ### Laplace transform question

\mathcal{L}\{f(t)*g(t)\}=F(s)G(s) Is there some relation between F(s)*G(s) and f(t)g(t)? ##*## is convolution.
30. ### Angle between spins

If ##|\alpha>## is spin up, and ##|\beta>## is spin down. Then if angle between those spins and some other up and down spin is ##\theta##, then |\alpha'>=\cos \frac{\theta}{2}|\alpha>+\sin \frac{\theta}{2}|\beta> |\beta'>=\sin \frac{\theta}{2}|\alpha>-\cos \frac{\theta}{2}|\beta> Why?