Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 11.5, Problem 11.20P
To determine
The sum of the squares of the coefficient.
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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?
The dynamics of a particle moving one-dimensionally in a potential V (x)
is governed by the Hamiltonian Ho = p²/2m + V (x), where p =
is the momentuin operator. Let E, n =
of Ho. Now consider a new Hamiltonian H
given parameter. Given A, m and E, find the eigenvalues of H.
-ih d/dx
1, 2, 3, ... , be the eigenvalues
Ho + Ap/m, where A is a
%3|
poblem 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) = -
Chapter 11 Solutions
Introduction To Quantum Mechanics
Ch. 11.1 - Prob. 11.1PCh. 11.1 - Prob. 11.2PCh. 11.1 - Prob. 11.3PCh. 11.1 - Prob. 11.4PCh. 11.1 - Prob. 11.5PCh. 11.1 - Prob. 11.6PCh. 11.1 - Prob. 11.7PCh. 11.1 - Prob. 11.8PCh. 11.1 - Prob. 11.9PCh. 11.3 - Prob. 11.10P
Ch. 11.3 - Prob. 11.11PCh. 11.3 - Prob. 11.12PCh. 11.3 - Prob. 11.13PCh. 11.3 - Prob. 11.14PCh. 11.3 - Prob. 11.15PCh. 11.3 - Prob. 11.16PCh. 11.4 - Prob. 11.17PCh. 11.5 - Prob. 11.18PCh. 11.5 - Prob. 11.19PCh. 11.5 - Prob. 11.20PCh. 11.5 - Prob. 11.21PCh. 11.5 - Prob. 11.22PCh. 11 - Prob. 11.23PCh. 11 - Prob. 11.24PCh. 11 - Prob. 11.25PCh. 11 - Prob. 11.26PCh. 11 - Prob. 11.27PCh. 11 - Prob. 11.28PCh. 11 - Prob. 11.29PCh. 11 - Prob. 11.30PCh. 11 - Prob. 11.31PCh. 11 - Prob. 11.33PCh. 11 - Prob. 11.35PCh. 11 - Prob. 11.36PCh. 11 - Prob. 11.37P
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- 问题6 The Hamiltonian operator Ĥ for the harmonic oscillator is given by h d? + uw? â2, where u is the reduced mass and w = 2Tv is the angular 2µ da? frequency. Is the function f(x) = x exp- iwa? an eigenfunction of that Hamiltonian? 2h O No. Yes, and the eigenvalue is hw. Yes, and the eigenvalue is hw. 3ħw Yes, and the eigenvalue is 4arrow_forwardVerify that the two eigenvectors in (11.8) are perpendicular, and that C in (11.10) satisfies the condition (7.9) for an orthogonal matrix.arrow_forwardProblem #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]arrow_forward
- Consider a spin-1 particle with Hamiltonian Ĥ = AS² + B(S² − S²). Assume B < A, treat the second term as a perturbation, and calculate the unperturbed energies and their first-order corrections using perturbation theory. Be careful of the degen- eracy. Compare your perturbative results with the exact eigenvalues.arrow_forward(a) Derive the following general relation for the first order correction to the energy, E, in time-independent perturbation theory where the 0 are the eigen-functions of the unperturbed Hamiltonian Ho and H' is the time independent perturbation. (b) A particle moves in a potential given by Vo sin(ax/a) for 0 < x < a V : %D otherwise where Vo is a small constant. Treat this as a perturbation for the case of a particle in an infinitely deep square well of width a and calculate the change in energy of the lowest energy state to first order in Vo.arrow_forwardObtain the solution for Eq. (12.62) for the forced harmonic oscillator using Laplace transforms.arrow_forward
- Show that a gaussian psi (x) = e ^(-ax^2) can be an eigenfunction of H(hat) for harmonic oscillator 1. Compute T(hat)*psi 2. Compute Vhat* psi - assume V operator is 1/2w^2x^2 3. Write out Hbar*psi and identify terms so Hber*psi=E*psi is true 4. From cancellation find a 5. insert back a to Schrodinger eq above and find Earrow_forwardThe 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_forwardSpin/Field Hamiltonian Consider a spin-1/2 particle with a magnetic moment u = -e/mS placed in a uniform magnetic field aligned along the z axis. (a) Write the Hamiltonian for this system in matrix form. (b) Verify by explicit matrix calculation that the Hamiltonian does not commute with the spin operators in the r and y directions. Comment on how this affects the expectation values of these operators.arrow_forward
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