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 3.5, Problem 3.20P
To determine
Test energy – time uncertainty principle for the wave function in Problem 2.5 and the observable x, by calculating
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1 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).
Problem 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.)
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 E
Chapter 3 Solutions
Introduction To Quantum Mechanics
Ch. 3.1 - Prob. 3.1PCh. 3.1 - Prob. 3.2PCh. 3.2 - Prob. 3.3PCh. 3.2 - Prob. 3.4PCh. 3.2 - Prob. 3.5PCh. 3.2 - Prob. 3.6PCh. 3.3 - Prob. 3.7PCh. 3.3 - Prob. 3.8PCh. 3.3 - Prob. 3.9PCh. 3.3 - Prob. 3.10P
Ch. 3.4 - Prob. 3.11PCh. 3.4 - Prob. 3.12PCh. 3.4 - Prob. 3.13PCh. 3.5 - Prob. 3.14PCh. 3.5 - Prob. 3.15PCh. 3.5 - Prob. 3.16PCh. 3.5 - Prob. 3.17PCh. 3.5 - Prob. 3.18PCh. 3.5 - Prob. 3.19PCh. 3.5 - Prob. 3.20PCh. 3.5 - Prob. 3.21PCh. 3.5 - Prob. 3.22PCh. 3.6 - Prob. 3.23PCh. 3.6 - Prob. 3.24PCh. 3.6 - Prob. 3.25PCh. 3.6 - Prob. 3.26PCh. 3.6 - Prob. 3.27PCh. 3.6 - Prob. 3.28PCh. 3.6 - Prob. 3.29PCh. 3.6 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Prob. 3.32PCh. 3 - Prob. 3.33PCh. 3 - Prob. 3.34PCh. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - Prob. 3.40PCh. 3 - Prob. 3.41PCh. 3 - Prob. 3.42PCh. 3 - Prob. 3.43PCh. 3 - Prob. 3.44PCh. 3 - Prob. 3.45PCh. 3 - Prob. 3.47PCh. 3 - Prob. 3.48P
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Similar questions
- Determine 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_forwardDivergence theorem. (a) Use the divergence theorem to prove, v = -478 (7) (2.1) (b) [Problem 1.64, Griffiths] In case you're not persuaded with (a), try replacing r by (r² + e²)2 and watch what happens when ɛ → 0. Specifically, let 1 -V². 4л 1 D(r, ɛ) (2.2) p2 + g2 By taking note of the defining conditions of 8°(7) [(1) at r = 0, its value goes to infinity, (2) for all r + 0, its value is 0, and (3) the integral over all space is 1], demonstrate that 2.2 goes to 8*(F) as ɛ → 0.arrow_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_forwardProblem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d?/dx², with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).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_forwardProblem 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?arrow_forwardProblem 2.7 A particle in the infinite square well has the initial wave function JAx, У (х, 0) — 0< x < a/2, a/2 < x < a. А (а — х), (a) Sketch ¥ (x, 0), and determine the constant A. (b) Find ¥(x, t). (c) What is the probability that a measurement of the energy would yield the value E1? (d) Find the expectation value of the energy, using Equation 2.21.21arrow_forwardProblem 3.30 Derive the transformation from the position-space wave function to the “energy-space” wave function using the technique of Example 3.9. Assume that the energy spectrum is discrete, and the potential is time-independent.arrow_forwardProblem 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_forwardProblem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?arrow_forwardHow would I be able to sketch the graph in problem 7.36?arrow_forwardSolve the time-independent Schrödinger equation with appropriate boundary conditions for an infinite square well centered at the origin [V (x) = 0, for -a/2 < x < +a/2; V (x) = 00 otherwise]. Check that your allowed energies are consistent with mine (Equation 2.23), and confirm that your y's can be obtained from mine (Equation 2.24) by the substitution x x - a/2.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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