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 1, Problem 1.12P
(a)
To determine
The classical probability distribution for harmonic oscillator in terms of momentum.
(b)
To determine
The expectation value of
(c)
To determine
The product of classical uncertainties in position and momentum and the limiting case when
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Check out a sample textbook solutionStudents have asked these similar questions
Problem 2.11
(a) Compute (x). (p). (x²), and (p²), for the states yo (Equation 2.60) and 1 (Equation
2.63), by explicit integration. Comment: In this and other problems involving the
harmonic oscillator it simplifies matters if you introduce the variable = √mo/hx
and the constanta (m/h)¹/4
(b) Check the uncertainty principle for these states.
(c) Compute (T) and (V) for these states. (No new integration allowed!) Is their sum
what you would expect?
Problem 2.14 In the ground state of the harmonic oscillator, what is the probability (correct
to three significant digits) of finding the particle outside the classically allowed region?
Hint: Classically, the energy of an oscillator is E = (1/2) ka² = (1/2) mo²a², where a
is the amplitude. So the “classically allowed region" for an oscillator of energy E extends
from –/2E/mw² to +/2E/mo². Look in a math table under “Normal Distribution" or
"Error Function" for the numerical value of the integral, or evaluate it by computer.
Problem 1.17 A particle is represented (at time=0) by the wave function
A(a²-x²). if-a ≤ x ≤+a.
0,
otherwise.
4(x, 0) = {
(a) Determine the normalization constant A.
(b) What is the expectation value of x (at time t = 0)?
(c) What is the expectation value of p (at time t = 0)? (Note that you cannot
get it from p = md(x)/dt. Why not?)
(d) Find the expectation value of x².
(e) Find the expectation value of p².
Chapter 1 Solutions
Introduction To Quantum Mechanics
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- Problem 1.6.5. A magnetic moment µ in a magnetic field h has energy E+ = Fµh when it is parallel (antiparallel) to the field. Its lowest energy state is when it is aligned with h. probabilities for being parallel or antiparallel given by P(par)/P(antipar) = exp(-E+/T]/ exp[-E-/T] where T is the absolute temperature. Using the fact that the total probability must add up to 1, evaluate the absolute probabilities for the two orientations. Using this show that the average magnetic moment along the field h is m = µ tanh(uh/T) Sketch this as a function of temperature at fixed h. Notice that if h = 0, m vanishes since the moment points up and down with %3D However at any finite temperature, it has a nonzero %3D equal probability. Thus h is the cause of a nonzero m. Calculate the susceptibility, dm lh=0 as a function of T.arrow_forward1 W:0E *Problem 1.3 Consider the gaussian distribution p(x) = Ae¬^(x-a)² %3D where A, a, and A are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find (x), (x²), and ơ. (c) Sketch the graph of p(x).arrow_forwardProblem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?arrow_forward
- 2.2. (a) Verify explicitly the invariance of the volume element do of the phase space of a single particle under transformation from the Cartesian coordinates (x,y, z, px, Py, Pz) to the spherical polar coordinates (r,e,4,Pr.Pe,Po). (b) The foregoing result seems to contradict the intuitive notion of "equal weights for equal solid angles," because the factor sine is invisible in the expression for do. Show that if we average out any physical quantity, whose dependence on po and po comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor sin0 to appear with the subelement (de do).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_forwardFigure 1.30arrow_forward
- Problem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle." (a) Show that the chemical potential is N+ - kT ln N (b) Discuss this result in the limits N > q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?arrow_forwardProblem 3.27 Sequential measurements. An operator Ä, representing observ- able A, has two normalized eigenstates 1 and 2, with eigenvalues a1 and a2, respectively. Operator B, representing observable B, has two normalized eigen- states ø1 and ø2, with eigenvalues b1 and b2. The eigenstates are related by = (3ø1 + 402)/5, 42 = (401 – 302)/5. (a) Observable A is measured, and the value aj is obtained. What is the state of the system (immediately) after this measurement? (b) If B is now measured, what are the possible results, and what are their probabilities? (c) Right after the measurement of B, A is measured again. What is the proba- bility of getting a¡? (Note that the answer would be quite different if I had told you the outcome of the B measurement.)arrow_forwardQuestion related to Quantum Mechanics : Problem 1.16arrow_forward
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