Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 7, Problem 7.49P
To determine
Prove the relation
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Problem 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?
Problem #1
(Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and
derive the differential form that reveals A as a potential:
dA < -SdT – pdV [Eqn 5.20]
Problem 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!
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
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- The Hamilton function for a point particle moving in a central potential is given by p? H + a|x|". 2m Consider the vector A = p x L+ ma where L is the angular momentum of the particle. (a) Calculate the Poisson bracket {H, Ar}, where A is the k-th component of the vector A. (b) Determine the value of the exponent n for which the vector A becomes a conserved quantity.arrow_forwardDerive Eq.(7.13) by using set of equations (7.2 a,b,c and d).arrow_forwardFor Problem 8.37, how do I find <1/r> within the integral? I think that the exponent function inside of P(r) is actually troublesome to finding what I need to find; however, I am not certain of what's really the correct procedure here.arrow_forward
- 8.8 ** Two masses m, and m, move in a plane and interact by a potential energy U(r) = Įkr?, Write down their Lagrangian in terms of the CM and relative positions R and r, and find the equations of motion for the coordinates X, Y and x, y. Describe the motion and find the frequency of the relative motion.arrow_forwardWrite the matrices which produce a rotation θ about the x axis, or that rotation combined with a reflection through the (y,z) plane. [Compare (7.18) and (7.19) for rotation about the z axis.]arrow_forwardA molecule is made up of three identical atoms at the corncrs of an equilateral triangle as showi in Fig. 7.12. We consider its ion to be nade by adding one clectron with some amplitude on cach site. Suppose the matrix element of the Hamiltonian for the electron on two adjacent sites i, j is (i|H|j) = -a for i # j. 1 o' O? Fig. 7.12 (a) Calculate the energy splittings. (b) Suppose an electric field in the z direction is applied, so that tlhe potential energy for the electron on top is lowered by b witli b < Ja|. Now calculate the levels. (c) Suppose the electron is in the ground state. Sucdeuly the field is rotated by 120° and points toward site 2. Calculate the probability for the electron to remain in the ground state.arrow_forward
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