3The eigenstates y y of a quantum system are associated, respectively, to the1' 2'eigenvalues ε1, 2, 3 of the observable E. The system is prepared in such a way that itsstate is described by the wavefunction:Ψ=1-2Assuming the w; are individually normalised, determine the normalisation factor N. Entera number into the empty box below.N=Calculate the probability of obtaining ε in a measurement of E. Enter a number, specifiedto two significant figures, in the box below.probability of obtaining &2 =

Answered: 3 The eigenstates y y of a quantum…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -
Normalize the following wavefunction and solve for the coefficient A. Assume that the quantum particle is in free-space, meaning that it is free to move from x € [-, ∞]. Show all work. a. Assume: the particle is free to move from x € [-0, 00] b. Wavefunction: 4(x) = A/Bxe¬ßx²
Below is a figure that depicts the potential energy of an electron (a finite square well), as well as the energies associated with the first two wave-functions. a) Sketch the first two stationary wavefunctions (solutions to the Schrődinger equation) for an electron trapped in this fashion. Pay attention to detail! Use the two dashed lines as x-axes. U(x) E2 E1 b) If the potential energy were an infinite square well (not finite well as shown above), what would the energy of the first two allowed energy levels be (i.e., E1 and E2). Write the expressions in terms of constants and a (the width of the wellI) and then evaluate numerically for a = 6.0*1010 m. [If you don't remember the formula, you can derive it by using E = h²k²/2m, together with the condition on À = 2a/n.] c) Let's say I adjust the width of the well, a, such that E1 = 3.5 ev. In that case, calculate the wavelength (in nanometers) of a photon that would be emitted in the electron's transition from E2 to E1. [Remember: hc =…
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Y:0. 4 %YO N Derive an equation which is represent the time-dependent Schrödinger equation. * 1 Add file This is a required question If Å is the position operator, and P, is the linear momentum operator, solve the following commutative, [X² ,P.] =?. 1 Add file If the operator Â = 3 in , and the wave function Y = 8 e-4ix , find the eigenvalue of (Â¥),Is ¥ an eigen function of the operator Â? dx 1 Add file Page 4 of 5 Вack Next Never submit passwords through Google Forms. العربية الإنجليزية
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1.The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ ? ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc a) Sketch the potential of this system , including in your sketch the positions of the lowest three energy levels . Indicate in your sketch the form of the wavefunction for a particle in each of these energy levels , and state which of the wavefunctions you have drawn could be decirbed by the Ψn written above (see figure) . b) Calculate the expectation value of momentum , ⟨p⟩ for a particle with n=2 c) Calculate the expectation value of momentum squared ⟨p 2⟩ , for a particle with n = 2 . Hint : you may use the mathematical identiy sin2 x = 1/2 (1 − cos 2x) without proof .

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