Fourier transformations: Say that f(x) = -1 for 0 x 1 and is zero everywhere else. Which g(α) are correct? There may be more than one.
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Fourier transformations:
Say that f(x) = -1 for 0 x 1 and is zero everywhere else. Which g(α) are correct? There may be more than one.
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- Consider a periodic signal x(t) with period T defined as follows: T x(t) = (5t, -< t <0 (10, 0I am confused with part (e). I don't understand the steps. How is 1.427 obtained? How is the 20t moved to the left side of the equation, since it is inside the COS() function? I am just not understanding the math. Can you step it through with an explanation at each individual step?For the function g(t) = 1 / 3t − 2 determine the average rate of change between the values t = 0 and t = a +1Consider a thin hoop of mass (1.420 ± 0.001) kg and radius (0.250 ± 0.002) m. The moment of inertia for a thin hoop rotating about an axis going through its center is MR2 . Calculate the moment of inertia of this hoop and its uncertainty using error propagation rules.4) [swHW] It turns out that any function that has a finite number of finite-magnitude discontinuities and a finite number of extrema (maximums and minimums) over a finite interval can be represented exactly with an infinite series of cosine and sine functions called a Fourier Series. The conditions that the function must meet, called the "Dirichlet conditions", are not very restrictive, so most functions you will encounter in physics will have an associated Fourier Series. To give you a sense of how this is possible consider a very simple f(x) = x function on the interval -πThe Fourier transform of a function f (x) is defined as: f (w) = , f(x)e-iax dx -iwx dx Similarly, the inverse Fourier transform of a function f (x) whose Fourier transform is known is found as: 1 f(x) = L f (@) e-iwx dw So, find the Fourier transform f (w) for a> 0 of the function given below, and using this result, calculate the inverse Fourier transform to verify the form f (x) given to you: for x > 0 for x < 0 -ах f (x) = }Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.Show that the minimum cnergy of a simple harmonic oscillator is Fw/2 if ArAp = h/2, where (Ap)² = ((p - (p))?). %3DA real wave function is defined on the half-axis: [0≤x≤00) as y(x) = A(x/xo)e-x/xo where xo is a given constant with the dimension of length. a) Plot this function in the dimensionless variables and find the constant A. b) Present the normalized wave function in the dimensional variables. Hint: introduce the dimensionless variables = x/xo and Y(5) = Y(5)/A.Use the commutator results, [â, ô] = iħ, [x²,p] = 2iħâ, and [ÂÂ, Ĉ] = Â[Â, Ĉ] + [‚ Ĉ]B, to find the commutators given below. (a) [x, p²] (b) [x³, p] (c) [x², p²]A particle experiences a potential energy given by U(x) = (x² - 3)e-x² (in SI units). (a) Make a sketch of U(x), including numerical values at the minima and maxima. (b) What is the maximum energy the particle could have and yet be bound? (c) What is the maximum energy the particle could have and yet be bound for a considerable length of time? (d) Is it possible for a particle to have an energy greater than that in part (c) and still be "bound" for some period of time? Explain. Responses