Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 3.3, Problem 3.10P
To determine
The ground state wavefunction of infinite square well is an eigen function of momentum or not. In the case it is an eigen function, the value of momentum.
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Students have asked these similar questions
Problem 2.15 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)mw²a², where a is the amplitude. So the "classically allowed region" for an
oscillator of energy E extends from -√2E/mw² to +√2E/mw². Look in a math
table under "Normal Distribution" or "Error Function" for the numerical value of
the integral.
An electron in a region of length L is described by the normalized wavefunction ψ(x) = (2/L)1/2sin(2πx/L) in the range x = 0 to x = L; outside this range the wavefunction is zero. Evaluate ⟨x⟩. The necessary integrals will be found in the Resource section.
1.4 Show that the distance of closest approach d, in Rutherford scattering leading to an
angle of deflection 0, is given by
d=(1+ cosec 0,2).
where p is detined in Fig. 1.6.
(Use the conservation of energy and angular momentum.]
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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- 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.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_forwardAnswer the following with detailed and clear solution. 31. Substitute the function ψ (x, t) = e-2πiEt/h ψ (x) into the time-dependent Schrodinger equation and determine the eigenvalue.arrow_forward
- I have an electron that I want to put in a rigid box. How small do I need to make the box so that the speed of my electron in its ground state inside the box will be equal to the speed of light? [Don’t worry about relativity—do the calculation with our usual non-relativistic equation(s).] Include a sketch of U(x) and ?(x).arrow_forwardWrite your complete justification of the answer on your paper. (To accept your answer, you should write it as ( x= , y= ) just no-digit below the comma and without unit): Three particles are placed in the xy plane. A 40-g particle is located at (3, 4) m, and a 50-g particle is positioned at (-2, –6) m. Where must a 20-g particle be placed so that the center of mass of this three- particle system is located at the origin?arrow_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_forward
- Bit confused you say with is the definition of a complex conjugate but all I've ever seen is |X|^2=(X*)(X). Can you provide maybe a reference or proof of this?arrow_forwardProblem 2.8 A particle of mass m in the infinite square well (of width a) starts out in the left half of the well, and is (at t = 0) equally likely to be found at any point in that region. (a) What is its initial wave function, (x, 0)? (Assume it is real. Don't forget to normalize it.) (b) What is the probability that a measurement of the energy would yield the value л²ħ²/2ma²?arrow_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_forward
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