Delta Definition and 1000 Threads

In Spivak's calculus book he provides a proof for: Theorem: If f is continuous on [a, b] and f(a) < 0 < f(b), then there is some number x in [a, b] such that f(x) = 0. In the proof he explicitly says, "...A has a least upper bound \alpha and that a < \alpha < b. We now wish to show that...

Hi All, I am trying to understand more rigorously why the Fourier transform of a constant functions equals the Dirac delta distribution. I found somewhere this is justified by imposing the self-adjointness, so that under a duality pairing <..,..> and indicating with F(f) the transform of...

I was wondering which are the properties of functions defined in such a way that ∫dx f(y-x) g(x-z) = δ(y-z) where δ is Dirac delta and therefore g is a kind of inverse function of f (I see this integral as the continuous limit of the product of a matrix by its inverse, in which case the...

It is easy to integrate the delta function of real variable. But when the argument of the delta function is complex, I get stuck. For example, how to calculate the integral below, where u is a complex constant: \int_{ - \infty }^{ + \infty } {f\left( x \right)\delta \left( {ux}...

Homework Statement if |x-3| < ε/7 and 0 < x ≤ 7 prove that |x^2 - 9| < ε Homework Equations The Attempt at a Solution So ths is what I did so far. |x+3|*|x-3| < ε (factored out the |x^2 - 9|) |x+3|*|x-3| < |x+3|* ε/7 < ε (used the fact that |x-3| < ε/7) |x+3|* ε/7 *7 <...

In the Principles of Quantum Mechanics, Dirac derives an identity involving his delta function: xδ(x)=0. From this he concludes that if we have an equation A=B and we want to divide both sides by x, we can take care of the possibility of dividing by zero by writing A/x = B/x + Cδ(x), because...

x''+2x'+x=t+delta(t) x(0)=0 x'(0)=1 The textbook, "Elementary differential equations" by Edwards and Penney, gives the answer as -2+t+2exp(-t)+3t exp(-t) It is clearly wrong, as in this case x'(0)=2, not x'(0)=1.

Homework Statement We have to give the total charge, dipol and quadrupol moments of a charge constellation, but I seem to be falling at the first hurdle. Q = \frac{1}{4\pi \epsilon_{0}} \int_{vol} \rho(\vec{r}) d^{3}\vec{r} whereby the charge density of the group of particles is...

Homework Statement Prove, using the formal definition of limits: If http://rogercortesi.com/eqn/tempimagedir/eqn4201.png and c>0, then [PLAIN]http://rogercortesi.com/eqn/tempimagedir/eqn4201.png (add the constant beside f(x) here, I couldn't get the equation generator to cooperate)...

Homework Statement Prove, using the formal definition of limits: If lim (x->inf) g(x) = inf and g(x) leq f(x) for x->a, then lim (x->a) f(x)=inf. leq = less than or equal to. Homework Equations The Attempt at a Solution Honestly, I'm not even sure where to start on this one. Anyone bored...

Hi! I'm having some difficulties understanding WHY sign function's derivative actually is dirac's delta function? Or more specifically why the derivative equals one at zero and NOT infinite, as the sign function's "actual" derivative at zero equals infinite? Atleast it would make sense. Thanks...

Homework Statement Determine the energy uncertainty \Delta E = \sqrt{<E^2> - <E>^2} for a particle described by a wave function \Psi (x) = c_1 \psi (x)_1 + c_2 \psi (x)_2 where \psi_1 and \psi_2 are different (orthonormal) energy eigenstates with eigenvalues E_1 and E_2. Homework...

Homework Statement A particle moves in one dimension in the delta function potential V= αδ(x). (where that is an 'alpha' ... not 'a') An initial wave function is given \Psi = A(a^2-x^2) for x between -a and a and Psi=0 anywhere else What is the probability that an energy measurement will...

Homework Statement THIS IS THE QUESTION V (x) = \sqrt{((h-bar ^{2})V_{0})/2m} [\delta(x-a)+ \delta(x+a)] -How do I find R and T? -Under what condition is there resonant transmission? 2. The attempt at a solution ok. I got these answers. Are these correct? Someone please...

I'm sick of still not getting this. I bombed the epsilon delta part of my mid term. A site where it gives many problems on epsilon delta and solutions would be amazing.

I have V (x) = \sqrt{((h-bar ^{2})V_{0})/2m} [\delta(x-a)+ \delta(x+a)] How do I find R and T? Under what condition is there resonant transmission?

From dirac, if A=B, then \frac{A}{x}=\frac{B}{x}+c\delta(x) (1) How this formula is derived? Since \frac{dlnx}{dx} = \frac{1}{x}-i\pi\delta(x) We can get \frac{A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x) \frac{B}{x} = B\frac{dlnx}{dx}+Bi\pi\delta(x) So if A=B, \frac{A}{x}=\frac{B}{x}...

Hi, I'm reading through one of my books and it's explaining how a vector is eqaul to multiplying sin\phi and \Delta\vartheta. the way it's written in the text is as followed, |\Deltai| \approx (sin\phi)\Delta\vartheta I have never understood how things like this work. Could...

Hello forum, I have a question regarding the delta function potential well. Given the following potential: V(x) = -αδ(x) for -a/2 < x < a/2 (α- positive constant) and V(x) = 0 elsewhere, how would one show that the ground state is the only eigenstate with E <0. One could of course solve the...

Let δ(x)=∞ at x = 0, and zero elsewhere. Then δ(x)(1-exp(x)) = ? It seems the above expression is zero. But isn't it zero times infinity at x = 0?

Why there is a phase shift of 30 degree (positive or negative) if we convert our winding from star to delta or vice versa?

Hello, I am trying to show that: \delta(x) = \lim_{\epsilon \to 0} \frac{\sin(\frac{x}{\epsilon})}{\pi x} Is a viable representation of the dirac delta function. More specifically, it has to satisfy: \int_{-\infty}^{\infty} \delta(x) f(x) dx = f(0) I know that the integral of...

Hi everyone, I have trouble understanding the multiple delta function. For one dimensional delta function, we have δ(\varphi(x))=\sum_{i=1}^{N}δ(x−xi)|\varphi′(xi)| where xi's (for i = 1 to N) are simple zeros of f(x) and it is known that f(x) has no zeros of multiplicitiy > 1 but...

What is the experimental evidence that a wavefunction will collapse to a dirac delta function, and not something more 'smeared' out?

Well, I understand q = mc∆T, along with q = mHv and q = mHf What I don't understand is this graph: http://dinosaurtheory.com/phase_change.jpg Well, I mean, I understand THAT graph. Here's what I don't understand: Today in chemistry, we received a very similar graph, but the X-axis was...

Need help proving lim(x)->(a) sqrt(x)=sqrt(a) using epsilon delta definition. Homework Statement Prove that the limit of \sqrt{x} is \sqrt{a} as x approaches a if a>0 Homework Equations in words By the epsilon delta definition we know that the distance between f(x) and the limit...

Homework Statement This is my first delt/epsilon proof ever, so please understand if I seem ignorant. e=epsilon d = delta Let f(x) = 1/x for x>0 If e is any positive quantity, find a positive number d, which is such that: if 0 < |x-2| < d, then |f(x) - 1/2| < e Homework...

In the equation for determining the coefficients of eigenfunctions of a continuous spectrum operator, I have trouble understanding the origin of the Dirac delta. a_f = INTEGRAL a_g ( INTEGRAL F_f F_g ) dq dg a is the coefficient, F = F(q) is an eigenfunction. From this it is shown that...

first of all i don't know anything about this epsilon and delta method.explain this a bit. secondly i have been given a problem involving this method: f(x)=x^2 given: limit x-->2 [x^2] = 4 a) what is the value of x' such that f(x')= 4 + .01? find \delta=x'-2 b) what is the value of...

Homework Statement I'll post it as an image since the notation will be tricky to type out. It's problem 4. http://img29.imageshack.us/img29/1228/307hw3.jpg Homework Equations Not sure this really applies hereThe Attempt at a Solution This is for a physics course but as you can see it's...

Homework Statement Suppose |f(x)-5|<0.1 when 0<x<5. Find all values \delta>0 such that |f(x)-5|<0.1 whenever 0<|x-2|<\delta Homework Equations The Attempt at a Solution I know that 0<|x-2|<\delta 2-\delta<x<2+\delta \delta=2 but how does this part of the equation help me find...

Homework Statement Integrate exp[abs(x)+3]*delta(x-2) dx, -1, 1 2. The attempt at a solution f(x)=exp[abs(x)+3]*delta(x-2) f(2)=148.4 Integrate exp[abs(x)+3]*delta(x-2) dx, -1, 1 = 0 [b]3. Why is the answer 0?

I'm having trouble understanding the proof/solution below (please see photo, I also wrote out the problem below). I highlighted the part of my problem in red (in the picture attached). Basically I'm not sure what identity they use to get the Kronecker delta after differentiating or whether they...

Homework Statement I am a bit confused with how to deal with Kronecker Delta. I need to show that P,i = ( P*δij ),j The i and j are subscripts. Homework Equations The Attempt at a Solution P,i = ( P*δij ),j = P,j*δij + P*δij,j I assumed I could get rid of P*δij,j leaving...

Apparently, when convolving, for example: [δ(ω-π) - δ(ω+π)] * (δ(ω+50π)-δ(ω-50π)) the result is δ(ω+49π)-δ(ω-51π)-δ(ω+51π)+δ(ω-49π) where δ() is the Dirac delta function, * the convolution operator and ω the frequency variable How do we get to this? Can you help me on the intuition in...

Probably a trivial question, but does Dirac delta function has (to have always) a physical dimension or is it used just as a auxiliary construct to express e.g. sudden force impulse, i.e. Force = Impulse \times \delta, where 'Impulse' carries the dimension? Any comments would be highly...

Hi, if the definition of a dirac delta (impulse) function is just a sinc function with an infinite height and 0 width, why is it that they are shown and used in Fourier analysis as having a finite height? for example g(t) = cos(2*PI*f0*t) has two impulses of height = 1/2 at f=+/-f0

Hello, I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however. Number 1. "Let f be a continuous function on the real numbers. Then the set {x in R ...

does anyone know why djmdkl = dkm when j = l thanks! can i interchange it to djmdkl = dkldjm ? since l = j do the "j" in dkjdjm signify summation? so i get dk1d1m + dk2d2m + dk3d3m ?

Homework Statement For some reason these are just messing me up. I need to prove: 1. \delta(y)=\delta(-y) 2.\delta^{'}(y) = -\delta^{'}(-y) 3.\delta(ay) = (1/a)\delta(y) In 2, those are supposed to be first derivatives of the delta functions Homework Equations Use an integral...

Hi, I was wondering about something a friend told me. He said that we can regularize this product in this form sgn(x)^2 \delta(x)=\frac{\delta(x)}{3} Any guess on how to derive it Thanks.

I am aware that the following operation: mathbf{M}_{ij} \delta_{ij} produces mathbf{M}_{ii} or mathbf{M}_jj However, if we have the following operation: mathbf{M}_{ij} \delta^i{}_j will the tensor M be transformed at all? Thank you for your time.

Homework Statement With the discriminate, why is delta sometimes used? Homework Equations \Delta = b2 - 4ac The Attempt at a Solution I get the logic behind what the discriminate is and how and why it works, but I don't understand why delta is used in the equation. What change is...

in QF, every dirac delta function is accompanied by 2\pi,i.e.(2\pi)\delta(p-p_0) or (2\pi)^3\delta(\vec{p}-\vec{p_0}) the intergral element in QF is \int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_P}, it comes from the integral element \int\frac{d^4p}{(2\pi)^4}(2\pi)\delta(p^2-m^2),I want to know why...

The Relation between Levi-Civita Density and Kroneckers Delta as follows \sum^{3}_{k=1} \epsilon_{mnk} \epsilon_{ijk} = \delta_{mi} \delta_{nj} - \delta_{mj} \delta_{ni} Logically we can satisfy both sides of the expression but Can anyone tell me how to derive this analytically ?

Hello again. Thank you guys. You have been great help... I have another one: Given a potential well- 2a is it's width, and in the middle - there is a delta potential: V(x)= \frac {\hbar^2} {2m} \frac {\lambda} {a} \delta(x) I am looking for the odd solution to this problem. I thought...

Homework Statement Homework Equations The Attempt at a Solution Can I write, say, f(x) \delta(x)=f(2)\delta(x)? Since \delta(x) =0 for x\neq0

let θ(x-x') be the function such that θ = 1 when x-x' > 0 and θ = 0 when x-x' < 0. Show that d/dx θ(x-x') = δ(x - x'). it is easy to show that d/dx θ(x-x') is 0 everywhere except at x = x'. To show that d/dx θ(x-x') is the dirac delta function i also need to show that the integral over the...

Homework Statement Simplify the following expressions involving the Kronecker delta in N dimensions. Where possible, write the final result without indices. C_{ns}\delta_{rn} Homework Equations The Attempt at a Solution I know Kronecker delta is symmetric but that doesn't seem to help. Is...

Could anybody explain me what's the difference between \delta\Gamma^{\rho}_{\mu\nu} and \Delta\Gamma^{\rho}_{\mu\nu}? Thank you