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 2.5, Problem 2.24P
To determine
The validity of uncertainty principle for wavefunction of equation 2.132.
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Students have asked these similar questions
that de/dx = 0 (x).
**Problem 2.25 Check the uncertainty principle for the wave function in
Equation 2.129. Hint: Calculating (p2) is tricky, because the derivative of has
a step discontinuity at x = 0. Use the result in Problem 2.24(b). Partial answer:
(p²) = (ma/h)².
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?
Problem 3.
A pendulum is formed by suspending a mass m from the
ceiling, using a spring of unstretched length lo and spring constant k.
3.1. Using r and 0 as generalized coordinates, show that
1
L =
= 5m (i² + r²0?) + mgr cos 0 –
z* (r – lo)²
3.2. Write down the explicit equations of motion for your generalized coordinates.
Chapter 2 Solutions
Introduction To Quantum Mechanics
Ch. 2.1 - Prob. 2.1PCh. 2.1 - Prob. 2.2PCh. 2.2 - Prob. 2.3PCh. 2.2 - Prob. 2.4PCh. 2.2 - Prob. 2.5PCh. 2.2 - Prob. 2.6PCh. 2.2 - Prob. 2.7PCh. 2.2 - Prob. 2.8PCh. 2.2 - Prob. 2.9PCh. 2.3 - Prob. 2.10P
Ch. 2.3 - Prob. 2.11PCh. 2.3 - Prob. 2.12PCh. 2.3 - Prob. 2.13PCh. 2.3 - Prob. 2.14PCh. 2.3 - Prob. 2.15PCh. 2.3 - Prob. 2.16PCh. 2.4 - Prob. 2.17PCh. 2.4 - Prob. 2.18PCh. 2.4 - Prob. 2.19PCh. 2.4 - Prob. 2.20PCh. 2.4 - Prob. 2.21PCh. 2.5 - Prob. 2.22PCh. 2.5 - Prob. 2.23PCh. 2.5 - Prob. 2.24PCh. 2.5 - Prob. 2.25PCh. 2.5 - Prob. 2.26PCh. 2.5 - Prob. 2.27PCh. 2.5 - Prob. 2.28PCh. 2.6 - Prob. 2.29PCh. 2.6 - Prob. 2.30PCh. 2.6 - Prob. 2.31PCh. 2.6 - Prob. 2.32PCh. 2.6 - Prob. 2.34PCh. 2.6 - Prob. 2.35PCh. 2 - Prob. 2.36PCh. 2 - Prob. 2.37PCh. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - Prob. 2.44PCh. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - Prob. 2.47PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - Prob. 2.54PCh. 2 - Prob. 2.58PCh. 2 - Prob. 2.63PCh. 2 - Prob. 2.64P
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Similar questions
- Problem 2.2 Show that E must exceed the minimum value of V (x), for every normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.5 in the form d² 2m [V(x) - E]; dx² if E < Vmin, then and its second derivative always have the same sign-argue that such a function cannot be normalized. h² d² 2m dx² + Vy = Ev. (2.5)arrow_forward2.03 Given f(x) = +1, 2 + sin(Tx) that is defined over [1, 6] with a step (h= 1). Using the N.G.F. function differences Interpolation. The first :derivative of P2(s) at x=3 is None of them 32 O 12 O 25 Oarrow_forwardProblem 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.arrow_forward
- Answer the following about an observable that is represented by the operator  = wo (3² + 3²). ħ (4) Is it possible to write a complete set of basis states that are simultaneously eigenstates of the operator Ĵ and Â? If so, explain how you know. If not, write an uncertainty principle for the observables A and J.arrow_forwardDemonstrate that e#ikz are solutions to both H and p, (momentum) for a free particle. Do you expect a difference for a bound particle where V (z) + 0?arrow_forwardDetermine the transmission coefficient for a rectangular barrier (same as Equation 2.127, only with +Vo in the region -a Vo (note that the wave function inside the barrier is different in the three cases). Partial answer: For Earrow_forwardProblem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: А, if -a < х <а, otherwise, (x, 0) = where A and a are positive real constants. 50 Chap. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine (k) (Equation 2.86). (c) Comment on the behavior of (k) for very small and very large values of a. How does this relate to the uncertainty principle? *Problem 2.22 A free particle has the initial wave function (x, 0) = Ae ax where A and a are constants (a is real and positive). (a) Normalize (x, 0). (b) Find V(x, t). Hint: Integrals of the form e-(ax?+bx) dx can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note that (ax? + bx) = y? – (b²/4a). Answer: 1/4 e-ax?/[1+(2ihat/m)] 2a Y (x, t) = VI+ (2iħat/m) (c) Find |4(x, t)2. Express your answer in terms of the quantity w Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again for some very large t.…arrow_forward2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)arrow_forwardProblem 7. 1. Calculate the energy of a particle subject to the potential V(x) = Vo + câ?/2 if the particle is in the third excited state. 2. Calculate the energy eigenvalues for a particle moving in the potential V(x) = câ2/2+ bx. %3!arrow_forwardConsider the half oscillator" in which a particle of mass m is restricted to the region x > 0 by the potential energy U(x) = 00 for a O where k is the spring constant. What are the energies of the ground state and fırst excited state? Explain your reasoning. Give the energies in terms of the oscillator frequency wo = Vk/m. Formulas.pdf (Click here-->)arrow_forwardProblem 2.34:- Show that E must be greater than minimum value of Ve for every normalizeable solution to time independent Schrodinger wave equation.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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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