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 1.11P
(a)
To determine
The expression for speed of the particle in terms of total energy and potential.
(b)
To determine
The probability of finding the particle in a range
(c)
To determine
The expectation value of
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Students have asked these similar questions
Problem 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?
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.
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.
Chapter 1 Solutions
Introduction To Quantum Mechanics
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Similar questions
- 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).arrow_forwardCASE 2 Let three equations of the model take these forms: p = 1 1 -3U + dn 3 (р — п) - 3- dt 4 dU (т — р) dt a. Find p(t), T(t), and U(t) b. Are the time path convergent? Fluctuating? explainarrow_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
- Question related to Quantum Mechanics : Problem 2.56 For your reference , problem 2.55 shown belowarrow_forwardProblem 1.8 A particle confined in an infinite square well between x = 0 and x = L is prepared with wave function de) = { 1/VI if 0 < x < L otherwise 1. Does the particle have a well-defined energy? 2. What is the probability to find the particle in the n'th bound state n (see Eq. 1.71) that has a well-defined energy En = (Tħn/L)²/2m? 3. What is the average energy of the particle? Explain the the requirement that physical wave functions don't have discontinuities, in view of your answer. Problem 1.9 Consider a particle in the n'th bound state ,n(x) (see Eq. 1.71) of an infinite square well between x = 0 and x = L. 1. Use a symmetry argument to prove that (x) = L/2 where (x) is the average position computed from the results of many identical experi- ments. 1.7. PROBLEMS 35arrow_forward1.1 Illustrate with annotations a barrier potential defined by 0 if - oarrow_forward1.16. Establish thermodynamically the formulae v (7)= = S and v (R), V = N. Express the pressure P of an ideal classical gas in terms of the variables and 7, and verify the μl above formulae.arrow_forwardAssuming a one-dimensional collision, apply the conservation of energy theorem to the following system:In the system in the initial state, cart A is launched at the speed (vi ± delta vi) towards cart B, which is stationary.In the final state system, the two carts stick together and move together.The masses of the carts are known, as well as their uncertainty.Obtain a model for vf (the final speed of the carts) and its uncertainty based on known parameters only. Consider a collision between cart A, moving at speed (vi ± delta vi), and cart B, immobile. The masses of the carts are known, as well as their uncertainty. Friction is neglected. Using the conservation of energy theorem, program cells to predict the speed of sliders A and B after the collision as well as its uncertainty. Then test your model with the following values: mA=(0.47±0.05) kg mB=(0.47±0.06) kg vi A=(1.9±0.02) m/sarrow_forwardParticle in a Box 12. According to the correspondence principle, quantum theory should give the same results as classical physics in the limit of large quantum numbers. Show that as n ∞, the prob- ability of finding the trapped particle of Sec. 5.8 between x and x + Ax is A/L and so is independent of x, which is the classical expectation. [A+ CO]arrow_forward(3.8) This question introduces a rather efficient method for calculating the mean and variance of probability distributions. We define the moment generating function M(t) for a random variable x by M(t) = (etx). Show that this definition implies that (x) = M(n) (0), (3.51) (3.52) where M(n) (t) mean (x) = d" M/dt" and further that the M (¹) (0) and the variance σ = = M(2)(0) [M(¹) (0)] 2. Hence show that: - (a) for a single Bernoulli trial, = M(t) pe 1-p; (3.53) (b) for the binomial distribution, M(t) = (pe +1 - p)"; (3.54) (c) for the Poisson distribution, M(t) = em(et-1); (3.55) (d) for the exponential distribution, λ M(t) (3.56) Hence derive the mean and variance in each case and show that they agree with the results derived earlier.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_forwardA particle of mass in moving in one dimension is confined to the region 0 < 1 < L by an infinite square well potential. In addition, the particle experiences a delta function potential of strengtlh A located at the center of the well (Fig. 1.11). The Schrödinger equation which describes this system is, within the well, + A8 (x – L/2) v (x) == Ep(x), 0 < x < L. !! 2m VIx) L/2 Fig. 1.11 Find a transcendental equation for the energy eigenvalues E in terms of the mass m, the potential strength A, and the size L of the system.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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