Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 6.5, Problem 6.17P
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To show that
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The first four Hermite polynomials of the quantum oscillator areH0 = 1, H1 = 2x, H2 = 4x2 − 2, H3 = 8x3 − 12x.
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Chapter 6 Solutions
Introduction To Quantum Mechanics
Ch. 6.1 - Prob. 6.1PCh. 6.2 - Prob. 6.2PCh. 6.2 - Prob. 6.3PCh. 6.2 - Prob. 6.4PCh. 6.2 - Prob. 6.5PCh. 6.2 - Prob. 6.7PCh. 6.4 - Prob. 6.8PCh. 6.4 - Prob. 6.9PCh. 6.4 - Prob. 6.10PCh. 6.4 - Prob. 6.11P
Ch. 6.4 - Prob. 6.12PCh. 6.4 - Prob. 6.13PCh. 6.5 - Prob. 6.14PCh. 6.5 - Prob. 6.15PCh. 6.5 - Prob. 6.16PCh. 6.5 - Prob. 6.17PCh. 6.6 - Prob. 6.18PCh. 6.6 - Prob. 6.19PCh. 6.7 - Prob. 6.20PCh. 6.7 - Prob. 6.21PCh. 6.7 - Prob. 6.22PCh. 6.7 - Prob. 6.23PCh. 6.7 - Prob. 6.25PCh. 6.8 - Prob. 6.26PCh. 6.8 - Prob. 6.27PCh. 6.8 - Prob. 6.28PCh. 6.8 - Prob. 6.30PCh. 6 - Prob. 6.31PCh. 6 - Prob. 6.32PCh. 6 - Prob. 6.34PCh. 6 - Prob. 6.35PCh. 6 - Prob. 6.36PCh. 6 - Prob. 6.37P
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- 1) Explain why any physically realizable solution to the Schrodinger’s equation must be square-integrable. 2)The Chebyshev polynomials of the first kind can be otained from the recurrence relation, Tn+1(x) = 2xTn(x) −Tn−1(x) with T0(x) = 1 and T1(x) = x.a) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (1 − x2)− 12 inthe closed interval [−1, 1]. b) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form anorthonormal set in the closed interval [−1, 1].arrow_forwardFind expectation value of position and for an electron in the ground state of hydrogen across the radial wave function. Express your answers in terms of the Bohr radius a.arrow_forwardin quantum mechanics ; calculate the eigenvalue of these operators L2 , Lz when l equal to 6 ?arrow_forward
- The radial wave function of a quantum state of Hydrogen is given by R(r)= (1/[4(2π)^{1/2}])a^{-3/2}( 2 - r/a ) exp(-r/2a), where a is the Bohr radius.(a) Determine the radial probability density P(r) associated with the quantum state in question. (b) Show that the function P(r) you determined in part (a) is properly normalized.arrow_forwardA simple illustration of the variation method is provided by the hydrogen atom in the 1s state. Let us assume a form of the trial wave function ψ=e–br where b is a constant. For hydrogen atom, V=-e2/r, so the Hamiltonian operator is Ĥ=-h2/8π2m V2- e2/r.The energy depends on r for the 1s state of the hydrogen atom so the angular portion of the Laplacian can be omitted and replaced by the factor 4π after integration. Therefore, the radial portion of V2=1/r2 (∂/∂r) r2 ∂/∂r. Solve the energy minimum of the hydrogen atom, E = -2π2 me4/h2.arrow_forwardConsider an electron in a 2D harmonic trap with force constants kxx = 232 N/m and kyy = 517 N/m. List the energies of the lowest 10 eigenfunctions.arrow_forward
- Show that ψ2 and ψ3 for the one-dimensional particle in a box are orthogonal.arrow_forwardShow transcribed data (d) Find (r) and (r2) for an electron in a circular orbit of hydrogen with arbitrary prin- cipal quantum number n (corresponds to l = n - 1 and any allowed m). (e) Compute the RMS uncertainty ✓(r) – (r)2 in terms of r for the electron in part (d). Note that the fractional spread in r decreases with increasing n (in this sense the system "begins to look classical” for large n). How much more volume does a hydrogen atom in the n = 100 state occupy compared to the hydrogen atom in the ground state. (Hint - you might want to look at Griffiths 4.55, or 4.15 in the second edition)arrow_forwardFind the real component if the complex number a + bi is raised to m if a = 7.4, b = 4, and m = 5.arrow_forward
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