Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 6.4, Problem 6.12P
To determine
Show that, due to this equation
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8.8 Calculate by direct integration the expectation values (r) and (1/r) of the radial position for
the ground state of hydrogen. Compare your results to the quoted expressions in Eq. (8.89)
and discuss your results. Did you expect that (1/r) # 1/(r)? Use your result for (1/r) to
find the expectation value of the kinetic energy of the ground state of hydrogen and discuss
your result.
8.9 Calculate by direct integration the expectation value of the radial position for each of the
Prove the following commutator identity:
[AB, C] = A[B. C]+[A. C]B.
Use the angular momentum raising and lowering operators in order to evaluate the
following matrix elements: (r|ix)
(r|L|x).(r|L|x).(x|L|x).
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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- 4.7 Let (x.t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that (x.-) is the wave function for the plane wave with the momentum direction reversed. b. Let x(n) be the two-component eigenspinor of a-n with eigenvalue +1. Using the explicit form of x(n) (in terms of the polar and azimuthal angles $ and y that characterizen) verify that -io₂x() is the two-component eigenspinor with thearrow_forwardpoblem 11.13 cies wx # 0, express the angular momentum operatorl, in terms of creation and annihilation operators. Consider the limiting transition to the isotropic case. For a two-dimensional harmonic oscillator in the xy-plane with different frequen- and show that this operator becomes a constant of motion, in agreement with Section 11.6. OProve that I mn = Vn 2^n! Smn Find fow I Yes> =L[21>+i12>e -iwt Find for1 Yes> Find the time-deperden t uncert arty la Hint APe) = -arrow_forwardThe commutation relations among the angular momentum operators, Lx, Ly, Lz, and β, are distinctive and characteristic of all angular momentum systems. Define two new angular momentum operators as : Îx + ily Î_ = Îx – iÎy - Î+ 2 (a) Write the operator product, ÎÎ_, in terms of β and Î₂. Note: β = Îx² + Îy² + Îz². (b) Evaluate the commutator, [1²,1+]. (L+ is shorthand for Îx ± ily.) (c) Evaluate the commutator, [L₂,L+].arrow_forwardQUESTION 3: Abstract angular momentum operators: In this problem you may assume t commutation relations between the general angular momentum operators Ĵ, Ĵy, Ĵz. Use whenev possible the orthonormality of normalised angular momentum eigenstate |j, m) and that α = Îx±iĴy, Ĵ²|j,m) = ħ²j(j + 1)|j,m) and Ĵz|j,m) (a) Express ĴĴ_ in terms of Ĵ² and Ĵ₂. = ħmlj, m). (b) Using the result from (a) find the expectation value (j,m|εÎ_|j,m). (This is the nor squared of the state Î_|j,m).)arrow_forwardThe eigenstates of the 1² and Î₂ operators can be written in Dirac notation as Ij m) where β|j m) = j(j + 1)ħ²|j m) and Î₂|j m) = mħ|j m). Using these relationships, and assuming that all angular momentum is evaluated in units of ħ (so you can set ħ = 1), evaluate the following operations for the state |j m) = 15,2). (b) (β — ħ²)² \j m) = ([² − 1)² |j m) 2 2 2 (a) (Îx² + ΂² − Î₂²) |j m)arrow_forwardProblem 9.4 For the 2D LHO with K1 = K2 show that and [ê, ²] = 2ihxy, (ê, p}] = -2ihxy Problem 9.5 It follows from the above that [ê., Ĥ] = 0 if K1 = K2 only Work out the equivalent commutator for ê and é, with the Hamiltonian. What do these mean?arrow_forwardWhat is the value of the commutator [J, , 2]? Here J., is the y-component of the angular momentum operator of a particle, and ê is the z-component of its position operator.arrow_forwardProblem 4.25 If electron, radius [4.138] 4πεmc2 What would be the velocity of a point on the "equator" in m /s if it were a classical solid sphere with a given angular momentum of (1/2) h? (The classical electron radius, re, is obtained by assuming that the mass of the electron can be attributed to the energy stored in its electric field with the help of Einstein's formula E = mc2). Does this model make sense? (In fact, the experimentally determined radius of the electron is much smaller than re, making this problem worse).arrow_forwardFind the moments Ma, My and the mass m, of the triangular lamina defined by the vertices (0,0), (0,3) 1 and (9,3). The density function is p(æ, y) least significant digits. (xy)? . Provide an exact answer or answer accurate to at M, = %3D My = m =arrow_forwardThe spin operators Ŝ, and S; can be represented by the following 2x2 Pauli matrices: Ŝ, = } C ) and Ŝ, (6–)- The operator can be written as S" = 6 D). Which of the three operators commute with each other? In your work, show explicitly how your conclusion is true and comment on what this means for measuring certain components of the spin of an atom simultaneously. O Ŝ, and Ŝ, do commute with each other, but both do not commute with S". O None of the operators commute with each other. O §, and Ŝ, do not commute with each other, Ŝ, commutes with S“ , but Ŝ, does not. O All of the operators commute with each other. O §, and S, do not commute with each other, but both commute with S.arrow_forwardProblem 6.25 Express the expectation value of the dipole moment pe for an electron in the hydrogen state 1 4 = (211 +210) √2 in terms of a single reduced matrix element, and evaluate the expectation value. Note: this is the expectation value of a vector so you need to compute all three components. Don't forget Laporte's rule!arrow_forwardWrite down Pauli Spin matrix and find out (0,0, -0,0₂). Also discuss the result. yarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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