What is Approximation: Definition and 763 Discussions

If: ##\hat{H} \psi (x) = E \psi (x)## where E is the eigenvalue of the *disturbed* eigenfunction ##\psi (x)## and ##E_n## are the eigenvalues of the *undisturbed* Hamiltonian ##\hat{H_0}## and the *disturbed* Hamiltonian is of the form: ##\hat{H} = \hat{H_0} +{\epsilon} \hat{V}...

hi everyone , i don't understand these steps for Taylor Expansion , it has used for state space equations the equations are the approximations for sin and cos the equation for Taylor series is ( i don't understand at all ) please help me if you can

I'm stucked in a passage of Particle Physics (Martin B., Shaw G.) in page 41 regarding neutrino oscillations. Having defined E_i and E_j as the energies of the eigenstates \nu_i and \nu_j, we have: E_i - E_j = \sqrt{m^2_i - p^2} - \sqrt{m^2_j - p^2} \approx \frac{m^2_i - m^2_j}{2p} It...

I'm trying to understand why the WKB approximation doesn't seem to work in the following case. Suppose you have a particle of mass ##m## in a potential ##V(x)=q m\cos(2mx/\hbar)##, where ##q\ll 1##. Consider then the stationary solution with energy ##E=m/2##. The Schroedinger equation is then...

In a paper that I'm reading, the authors write:- N_e \approx \frac{3}{4} (e^{-y}+y)-1.04 ------------ (4.31) Now, an analytic approximation can be obtained by using the expansion with respect to the inverse number of "e-foldings" (N_e is the number of "e-foldings"). For instance, eq...

Homework Statement A dipole is placed next to a sphere (see image), at a large distance what is E proportional to? 3. The Attempt at a Solution or lack thereof I'm having trouble figuring out what's happening in any variations of these. How does the dipole affect the sphere's charge...

Homework Statement Find an N so that ##∑^{\infty}_{n=1}\frac{log(n)}{n^2}## is between ##∑^{N}_{n=1}\frac{log (n)}{n^2}## and ##∑^{N}_{n=1}\frac{log(n)}{n^2}+0.005.## Homework Equations Definite integration The Attempt at a Solution I began by taking a definite integral...

I know that at early stage (around 1999), GW implementation uses Matusbara frequency to help calculating self energy, and then apply analytic continuation to change it to real frequency for subsequent calculation. I don't know whether this scheme has been superceded by other implementations for...

Hi. Please help me prove the approximation formula below given in my book. This is not homework question. Thanks.

Homework Statement The restoring force of a pendulum is F_\theta=-mg\sin\theta and is approximated to F_\theta=-mg\theta. The period is T=2\pi\sqrt{\frac{L}{g}}, but can be expressed as the infinite series: T=2\pi\sqrt{\frac{L}{g}}\left(...

Hi all I was trying to understand the Born-Bethe approximation related to cross sections for atomic and molecular collisions. All the stuffs that i got are explaining in complicated way which am not able to follow. Can anyone explain in simple terms what the theory explains? It will be of...

Hello, I've come across equations where people use the approximation \int_0^1 \exp(f(x))\, dx \approx \exp \left( \int_0^1 f(x)\, dx\right) I can see that this is correct if f(x) is small, one just uses exp(x) = 1+x+... However, it appears that this approximation has a broader validity...

Homework Statement Using the Eikonal approximation (1) Determine the expression for the total scattering cross section of a particle in a potential V(r) (2) Using this result, compute the total scattered cross section for the following potential. V(r)= \begin{cases} V_0, \text{for } r < a \\...

I'm looking for a way to write down an analytic approximation for the following integral: \int_0^\infty \frac{k \sin(kr)}{\sqrt{1+v^2(k-k_F)^2}}dk Let's assume that v kF >> 1, so that the the oscillating piece at large k doesn't contribute much uncertainty. Ideas? Thus far, Mathematica has...

Homework Statement If we have the following data T = [296 301 306 309 320 333 341 349 353]; R = [143.1 116.3 98.5 88.9 62.5 43.7 35.1 29.2 27.2]; (where T = Temperature (K) and R = Reistance (Ω) and each temperature value corresponds to the resistor value at the same position) Homework...

Homework Statement Show that ∫f'(x)dx/f(x) = ln|(f(x)|+C where f(x) is a differential function. Homework Equations First order Taylor approximation? f(x)=f(a)+f'(a)(x-a) The Attempt at a Solution Well, I'm not really sure how to approach the question. It's my Numerical...

Hey! :o I am looking at the identities of the optimal approximation. At the case where the basis consists of orthogonal unit vectors,the optimal approximation $y \in \widetilde{H} \subset H$, where $H$ an euclidean space, of $x \in H$ from $\widetilde{H}$ can be written $y=(x,e_1) e_1 +...

1. Consider a quantum well described by the potential v(x)=kx^{2} for \left|x\right|<a and v(x)=ka^{2} for \left|x\right|>a. Given a^{2}\sqrt{km}/\hbar =2, show that the well has 3 bound states and calculate the ratios between the energies and ka^{2}. You may use the standard integral...

Hey! :o I am looking at the following that is related to the existence of the optimal approximation. $H$ is an euclidean space $\widetilde{H}$ is a subspace of $H$ We suppose that $dim \widetilde{H}=n$ and $\{x_1,x_2,...,x_n\}$ is the basis of $\widetilde{H}$. Let $y \in \widetilde{H}$ be...

I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...

I'd rather not post the exact problem since it's homework, I don't think my instructor in E&M would want me posting full problems but I will just ask relevant conceptual question... Let's say we I have a long cylinder with time-defendant surface current density \vec K(t)=K_of(t). So if I want...

I have to solve the integral equation y(x)= -1+\int_0^x(y(t)-sin(t))dt by the method of successive approximation taking y_0(x)=-1. Sol: After simplification the given equation we have y(x)=-2+cos(x)+\int_0^x y(t)dt . So comparing it with y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt we have...

I have to solve the integral equation y(x)=1+2\int_0^x(t+y(t))dt by the method of successive approximation taking y_0(x)=1. Sol: After simplification the given equation we have y(x)=1+x^2+2\int_0^xy(t)dt. So comparing it with y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt we have f(x)=1+x^2...

Hello everyone, I'm working on a problem and it turns out that this equation crops up: 1 = cos^{2}(b)[1-(c-b)^{2}] where c > \pi Now I'm pretty sure you can't solve for b in closed form (at least I can't), so what I need to do is for some value of c, approximate the value of b to...

Here is the question: I have posted a link there to this thread so the OP can view my work.

I am trying to follow the reasoning of the last problem in the set linked below. I can't figure out what approximation they used in step 24. Thanks. http://www.physics.fsu.edu/courses/spring08/phy5524/sol1.pdf It looks like it might be an approximation based on a geometric series, but there is...

Homework Statement In a spherical well in which.. V= \begin{cases} 0,\text{for }0 \le r < R \\ ∞, \text{for } r > R \end{cases} the s-wave eigenstates are \phi_n(r)=\frac{A}{r}\sin\left( \frac{n\pi r}{R} \right) where A is a normalization constant. If a particle is in the ground state and...

Homework Statement Question: is the charge sum an approximation? Homework Equations E.g. consider 0.10 M Na_{2}SO_{4} solution. Charge sum appears to be 0.20 from elementary stoichiometric considerations. The Attempt at a Solution The charge sum, however, seems to be ignoring...

Triangle ABC is an equilateral triangle with side 4 cm long which is measured corrected to the nearest cm. Find the percentage error of the perimeter of triangle ABC.The Attempt at a Solution Is [(0.5 x 2 x 3) / 12] x 100% correct? the '2' here is the measurement errors of the starting pt and...

Hey! :o Given the following two-point problem: $$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$ $$y(0)=0, y'(1)=my(1)$$ where $ b \in C^1([0,1];R), f \in C([0,1];R)$ and $ m \in R$ a constant. Give a finite element method for the construction of the approximation of the solution $y$ of the...

Homework Statement Apply saddle point approximation to the following integral: I = ∫0∞xe-ax-b/√xdx a,b > 0 Recall that to derive Stirling formula from the Euler integral in class we required N >> 1. For the integral defined above, identify in terms of a and b appropriate parameter that...

Hi, I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is: u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2 or when discretized u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2 In addition to...

Hello! I have a nifty set of problems (or rather one problem, gradually building itself to be a great problem) that I like to collectively call "The final problem" as it is the last thing I need before I can take the exam in Numerical Methods.Information There is given a Laplace equation...

The explanation below illustrates why I think the method of successive approximations is merely a sneaky way of working with power series when you're not formally allowed to use a Taylor series expansion for a function (i.e. when it doesn't exist, as in proving the existence theorem on ode's for...

Homework Statement A coaxial cable with inner radius a and outer radius b lies on the z-axis (such that the cable's axis merges with the z-axis). its length (along z-axis) is L. at z=-L there are voltage sources that are distributed uniformly connecting the inner wire to the outer one. at...

I'll look over your other posts tomorrow but I'd like to post the 3rd problem for now in case you got the time to answer somewhere between that time. So the third and final (for now, anyway) problem. Problem III Approximate the differential equation $$\frac{d^3 u}{dx^3}=g(x)$$ on a model*...

Hi, I need help determining which of the statements are true and false. http://imgur.com/9BC1Wqf I know this involves steady state approximation, but I find that when I try it I am never able to rid of all the intermediates. Please help Thanks

Homework Statement I'm trying to learn how to apply the WKB approximation. Given the following problem: An electron, say, in the nuclear potential $$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0} \end{cases}$$ 1. What is the...

Here is another problem I should understand of differential approximation. Maybe I shouldn't have posted it before solving the first one but I'm really anxious to learn them so I can pass the class. Problem II Using the Taylor series and the undetermined coefficient method, approximate the...

So in this thread I plan to present 3 problems and my takes on them. I think I'll post each next one after I get some help and solution with the current one. Note that this is a translated problem from my native language. Task Nr. 1 Write the difference analogue (I mean in discrete form) to a...

I'm looking for a concise description of an algorithm for low-rank approximation via SVD. I've seen a number of articles referring to Lanczos method and finding eigenvalues but nothing going all the way to determining all the matrices involved in the low-rank SVD of a given matrix. Any...

Use a linear approximation to find a good approximation to \sqrt{100.4} x = 100.4 x1 = 100 y1 = 10 y - 10 = \frac{1}{20}(100.4 - 100) y = 10.20

I'm studying low rank approximation by way of SVD and I'm having trouble understanding how the result matrix has lower rank. For instance, in the link the calculation performed on page 11 resulting in the so-called low rank approximation on page 12. This matrix on page 12 doesn't appear to me to...

Homework Statement consider a capacitor with circular plates of radius a, separated by a distance d (d<<a) and V(t)=V_{0}sin(wt) a)Considering the z axis to be the capacitor axis, verify that the electric field between the plates is , in good approximation, given by \vec{E}(t)\approx E_{0}...

So \sqrt[5]{306} is a pretty good approximation for Pi (=3.14155). If you add 1/51, so that you have \sqrt[5]{306+1/51} you get 3.1415925 (last digit is 6 for actual Pi.) If you add 1/12997, \sqrt[5]{306+1/51+1/12997} you get 3.141592653587 (vs 3.141592653589 for actual Pi.) And so on. As you...

Hi all, Could someone please help me understand a small but significant step in the derivation of the wave equation for a string with stiffness. I am trying to follow the notes here: http://courses.physics.illinois.edu/phys406/Lecture_Notes/Waves/PDF_FIles/Waves_2.pdf The statement...

Hi all, I try to understand the difference which can made by using or not using NWA .. I have a process have cross section (p p > x x) ~ 10^-5 pb , where x is a paricle have mass mx = 2 TeV and dominant decay channel (x > b b~) with Gamma (x > b b~) ~ 6 * 10^2 GeV , while sigma ( p p > x x ...

Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Hi, could someone please help me with this one: I'd need to form a sequence of rational functions ##R_{n}(x)## such that ##\lim_{n \to \infty} R_{n}(x)=\theta(x)##, where ##\theta(x)## is the Heaviside step function. The functions ##R_{n}(x)## should preferably be limited in range, i.e. for some...

Homework Statement Suppose that X is the number of successes in a Binomial experiment with n trials and probability of success θ/(1+θ), where 0 ≤ θ < ∞. (a) Find the MLE of θ. (b) Use Fisher’s Theorem to find the approximate distribution of the MLE when n is large. Homework Equations...