Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8, Problem 8.20P
To determine
The upper bound on the energy of the first excited state of the hydrogen atom.
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Chapter 8 Solutions
Introduction To Quantum Mechanics
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- 3arrow_forward8.8 Calculate by direct integration the expectation values (r) and (1/r) of the radial position for the ground state of hydrogen. Compare your results to the quoted expressions in Eq. (8.89) and discuss your results. Did you expect that (1/r) # 1/(r)? Use your result for (1/r) to find the expectation value of the kinetic energy of the ground state of hydrogen and discuss your result. 8.9 Calculate by direct integration the expectation value of the radial position for each of thearrow_forwardO Consider the kinetic energy matrix elements between Hydrogen states (n' = 4, l', m'| |P|²| m -|n = 3, l, m), = for all the allowed l', m', l, m values. What kind of operator is the the kinetic energy (scalar or vector)? Use this to determine the following. For what choices of the four quantum numbers (l', m', l, m) can the matrix elements be nonzero (e.g. (l', m', l, m) (0, 0, 0, 0),...)? Which of these nonzero values can be related to each other (i.e. if you knew one of them, you could predict the other)? In this sense, how many independent nonzero matrix elements are there? (Note: there is no need to calculate any of these matrix elements.)arrow_forward
- a. Write down the energy eigenfunctions for a particle in an infinitely deep one- dimensional square well extending from z = -L/2 to z = +L/2 b. Check that they are eigenfunctions of parity operator (that maps z → −z) corresponding to the eigenvalue (-1)n-1, where n labels the energy level.arrow_forwardFor Problem 8.37, how do I find <1/r> within the integral? I think that the exponent function inside of P(r) is actually troublesome to finding what I need to find; however, I am not certain of what's really the correct procedure here.arrow_forwardTake as a a trial function for the ground state of the hydrogen atom (a) e−kr, (b) e−kr^2 and use the variational principle to find the optimal value of k in each case. Identify the better wavefunction. The only part of the laplacian that need be considered is the part that involves radial derivatives.arrow_forward
- Solve 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_forwardhelp with modern physics questionarrow_forwardFor Problem 8.16, how do I prove the relations and give the correct expressions?arrow_forward
- PROBLEM 1. Calculate the normalized wave function and the energy level of the ground state (1 = 0) for a particle in the infinite spherical potential well of radius R for which U(r) = 0 at r R. HINT: Reduce the spherically-symmetric SE to a ld form using the substi- tution (r) = x(r)/r.arrow_forwardConsider the spherical harmonic having l = 2, me = -2. (a) Show that it is an eigenfunction of the operator , for the projection of the angular momentum onto the axis of quantization (the z-axis). What is the eigenvalue? (b) What is the magnitude of the angular momentum for this state in units of h?arrow_forwardConsider an infinite well, width L from x=-L/2 to x=+L/2. Now consider a trial wave-function for this potential, V(x) = 0 inside the well and infinite outside, that is of the form (z) = Az. Normalize this wave-function. Find , .arrow_forward
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