Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 2.2, Problem 2.5P
(a)
To determine
The normalization of
(b)
To determine
The value of
(c)
To determine
The value of
(d)
To determine
The value of
(e)
To determine
The energy of the particle, the probability of getting each of them, the expected value of
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Find the normalization constant A [in Equation Ψ(x, y, z) = A sin(k1x)sin(k2y)sin(k3z) ] for the first excited state of a particle trapped in a cubical potential well with sides L. Does it matter which of the three degenerate excited states you consider?
The wave function for the first excited state y, for the simple harmonic oscillator is y, = Axe (ax-/2), Normalize the wave function to find the value of the constant A. (Use the following as
necessary: a)
A%3D
Determine (x), (x-), and y (x2) - (x)². (Use the following as necessary: a)
(x)
(x?)
V (x?) - (x)? :
Need Help?
Read It
For a particle in a 1-dimensional infinitely deep box of length L, the normalized wave function or the 1st excited state can be written as:
Ψ2(x) = {1/i(2L)1/2} ( eibx -e-ibx), where b = 2π/L.
Give the full expression that you need to solve to determine the probalibity of finding the particle in the 1st third of the box. Simplify as much as possible but do not solve any integrals.
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- 3n (2x –), find Ynormalized, the normalized wave function for a 1-dimensional particle- Given 4 = cos in-a-box where the box boundaries are at x=0 and x=2. The potential energy is zero when 0arrow_forwardSolve the 3-dimensional harmonic oscillator for which V(r) = 1/2 mω2(x2 + y2 + z2), by the separation of variables in Cartesian coordinates. Assume that the 1-D oscillator has eigenfunctions ψn(x) that have corresponding energy eigenvalues En = (n+1/2)ħω. What is the degeneracy of the 1st excited state of the oscillator?arrow_forwardSolve the “particle in a box” problem to find Ψ(x, t) if Ψ(x, 0) = 1 on (0, π). What is En? The function of interest here which you should plot is |Ψ(x, t)|2.arrow_forwardProblem 2. Consider the double delta-function potential V(x) = a[8(x + a) + 8(x − a)], where a and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a = ħ²/ma and for a = ħ²/4ma, and sketch the wave functions.arrow_forwardFor an electron in a one-dimensional box of width L (x lies between 0 and L), (a)Write down its wavefunction and the allowed energy. (b)If the electron is in a superposition of the ground state and second excited state, write down the wavefunction and compute the probability of finding the electron at 1/6 ?. (Don’t forget to normalize it!)arrow_forwardA quantum mechanical particle is confined to a one-dimensional infinite potential well described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere. The normalised eigenfunctions for a particle moving in this potential are: Yn(x) = √ 2 Nπ sin -X L L where n = 1, 2, 3, .. a) Write down the expression for the corresponding probability density function. Sketch the shape of this function for a particle in the ground state (n = 1). b) Annotate your sketch to show the probability density function for a classical particle moving at constant speed in the well. Give a short justification for the shape of your sketch. c) Briefly describe, with the aid of a sketch or otherwise, the way in which the quantum and the classical probability density functions are consistent with the correspondence principle for large values of n.arrow_forwardThe odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ x ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc I have got the expectation value of momentum for ⟨p⟩ and ⟨p 2⟩ for n = 2 (see figures) By direct substitution, show that the wavefunction in the figure satisfies the timedependent Schrodinger equation (provided that En takes the value derived in figure).arrow_forwardConsider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)arrow_forwardConsider a state function that is real, i.e., such that y (x) = y* (x). Show that (p) Under what conditions on p (x), would the function o (p) turn out to be real, and if so, what is (x) worth? = 0. What happens in that case with (p2) and with (x) ?;arrow_forwardThe normalised wavefunction for an electron in an infinite 1D potential well of length 80 pm can be written:ψ=(0.587 ψ2)+(0.277 i ψ7)+(g ψ6). As the individual wavefunctions are orthonormal, use your knowledge to work out |g|, and hence find the expectation value for the energy of the particle, in eV.arrow_forwardAn atom in an infinite potential well of width L=207 pm (from x=0 to x=207 pm) is in the normalised superposition state we considered in class, i.e. ψ=(ψ1+ψ2)/21/2, what is the expectation of the position of the particle, in pm, at time 0.224 (in units of the ground state period, T1)? Hint: use symmetry, a product of sines is the sum of two cosines, and that the integral of x cos(a x) is x sin(a x)/a+cos(a x)/a2. Remember your answer is (hopefully!) between 0 and 207. The part of the answer that is a time-dependent function should have amplitude 16 L/(3π)2.arrow_forwardCalculate the expectation value of x2 in the state described by ψ = e -bx, where b is a ħ constant. In this system x ranges from 0 to ∞.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning