Fundamentals of Physics, Binder Ready Version
10th Edition
ISBN: 9781118230640
Author: Halliday, David; Resnick, Robert; Walker, Jearl
Publisher: WILEY
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Chapter 39, Problem 35P
To determine
To find:
a) The probability density
b) Radial probability density
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For the hydrogen atom in its ground state, calculate (a) the probability density c2(r) and (b) the radial probability density P(r) for r = a, where a is the Bohr radius.
Calculate the radial probability density P(r) for the hydrogen atom in its ground state at (a) r= 0, (b) r = a, and (c) r = 2a, where a is the Bohr radius.
(a) How much energy is required to cause an electron in hydrogen to move from the n = 2 state to the n = 5 state?
in J(b) Suppose the atom gains this energy through collisions among hydrogen atoms at a high temperature. At what temperature would the average atomic kinetic energy 3/2 * kBT be great enough to excite the electron? Here kB is Boltzmann's constant.
in K
Chapter 39 Solutions
Fundamentals of Physics, Binder Ready Version
Ch. 39 - Prob. 1QCh. 39 - Prob. 2QCh. 39 - Prob. 3QCh. 39 - Prob. 4QCh. 39 - Prob. 5QCh. 39 - Prob. 6QCh. 39 - Prob. 7QCh. 39 - Prob. 8QCh. 39 - Prob. 9QCh. 39 - Prob. 10Q
Ch. 39 - Prob. 11QCh. 39 - Prob. 12QCh. 39 - Prob. 13QCh. 39 - Prob. 14QCh. 39 - Prob. 15QCh. 39 - Prob. 1PCh. 39 - Prob. 2PCh. 39 - Prob. 3PCh. 39 - Prob. 4PCh. 39 - Prob. 5PCh. 39 - Prob. 6PCh. 39 - Prob. 7PCh. 39 - Prob. 8PCh. 39 - Prob. 9PCh. 39 - Prob. 10PCh. 39 - Prob. 11PCh. 39 - Prob. 12PCh. 39 - Prob. 13PCh. 39 - Prob. 14PCh. 39 - Prob. 15PCh. 39 - Prob. 16PCh. 39 - Prob. 17PCh. 39 - Prob. 18PCh. 39 - Prob. 19PCh. 39 - Prob. 20PCh. 39 - Prob. 21PCh. 39 - Prob. 22PCh. 39 - Prob. 23PCh. 39 - Prob. 24PCh. 39 - Prob. 25PCh. 39 - Prob. 26PCh. 39 - Prob. 27PCh. 39 - Prob. 28PCh. 39 - Prob. 29PCh. 39 - Prob. 30PCh. 39 - Prob. 31PCh. 39 - Prob. 32PCh. 39 - Prob. 33PCh. 39 - Prob. 34PCh. 39 - Prob. 35PCh. 39 - Prob. 36PCh. 39 - Prob. 37PCh. 39 - Prob. 38PCh. 39 - Prob. 39PCh. 39 - Prob. 40PCh. 39 - Prob. 41PCh. 39 - Prob. 42PCh. 39 - Prob. 43PCh. 39 - Prob. 44PCh. 39 - Prob. 45PCh. 39 - Prob. 46PCh. 39 - Prob. 47PCh. 39 - Prob. 48PCh. 39 - Prob. 49PCh. 39 - Prob. 50PCh. 39 - Prob. 51PCh. 39 - Prob. 52PCh. 39 - Prob. 53PCh. 39 - Prob. 54PCh. 39 - Prob. 55PCh. 39 - Prob. 56PCh. 39 - Prob. 57PCh. 39 - Prob. 58PCh. 39 - Prob. 59PCh. 39 - Prob. 60PCh. 39 - Prob. 61PCh. 39 - Prob. 62PCh. 39 - Prob. 63PCh. 39 - Prob. 64PCh. 39 - A diatomic gas molcculc consistsof two atoms of...Ch. 39 - Prob. 66PCh. 39 - Prob. 67PCh. 39 - Prob. 68PCh. 39 - Prob. 69PCh. 39 - Prob. 70PCh. 39 - An old model of a hydrogen atom has the charge e...Ch. 39 - Prob. 72PCh. 39 - Prob. 73P
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- (a) How much energy is required to cause an electron in hydrogen to move from the n = 2 state to the n = 5 state?in J(b) Suppose the atom gains this energy through collisions among hydrogen atoms at a high temperature. At what temperature would the average atomic kinetic energy 3/2 * kBT be great enough to excite the electron? Here kB is Boltzmann's constant. in Karrow_forwardCompute the intrinsic line-width (Δλ) of the Lyman α line (corresponding to the n=2 to n=1) transition for the Hydrogen atom. You may assume that the electron remains in the excited state for a time of the order of 10^−8s. The line-width may be computed using:ΔE=(hc/λ^2)Δλarrow_forwardThe radial Hamiltonian of an isotropic oscillator ((1 = 0) is ħ² d (r² 2/1 ) + 1/ / mw² p² 2mr² dr dr Estimate the ground state energy level of the atom using variational method with the trial function & = e ararrow_forward
- Why is the following situation impossible? An experiment is performed on an atom. Measurements of the atom when it is in a particular excited state show five possible values of the z component of orbital angular momentum, ranging between 3.16 x 10-34 kg ⋅ m2/s and -3.16 x 10-34 kg ⋅ m2/s.arrow_forward(a) How much energy is required to cause an electron in hydrogen to move from the n = 1 state to the n = 3 state? eV(b) If the electrons gain this energy by collision between hydrogen atoms in a high temperature gas, find the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by 3kBT/2, where kB is the Boltzmann constant. Karrow_forwardChapter 39, Problem 043 In the ground state of the hydrogen atom, the electron has a total energy of -13.6 ev. What are (a) its kinetic energy and (b) its potential energy if the electron is a distance 4.0a from the central nucleus? Here a is the Bohr radius. (a) Number Units eV (b) Number Units eVarrow_forward
- The radial part of the Schrödinger equation for the hydrogen atom д ħ ² l ( l + 1 ) Ze² 240² or (2² (1) 2μr dr ər R(r) = ER(r) 2μr² + -R(r) – Απε has eigenvalues that depend on only the principal quantum number, n. True Falsearrow_forwardThe average value (or expected value) of r^k, where r is the distance of an electron in the state with principal quantum number n and orbital quantum number leo proton in the hydrogen atom is given by the integral below, where Pnl(r) is a radial probability density of the state with quantum number n, lek is an arbitrary power. For an electron in the ground state of the hydrogen atom. a) calculate <r>nl in terms of the Bohr radius aB b) calculate <l/r>nl in terms of aB c) calculate <U(r)>nl, where U(r) = -e^2/(4piE0r). Respond in eV units. d) Considering also that the electron is in the ground state, estimate the expected value for two kinetic energy <K> and its mean quadratic velocity v. e) Is it justifiable to disregard relativistic corrections for this system? Justify.arrow_forwardDetermine the integral | P(r) dr for the radial probability density for the ground state of the hydrogen atom 4 P(r) = - r²e-2rla a³ O 1 O-1 O 0.5arrow_forward
- (d) The following orbital belongs to the 3d subshell of the Hydrogen atom: r Y(r, 0, 0) = A(Z) θ, φ) 2 r e 3ao sin² (0) e²i зао where A and ao are constants. Using the operator for the z-component of orbital angular momentum (L₂ = -ih d/do) determine the m, for this particular orbital. (e) Consider the wavefunction, r r Y(r,0,0) = A-e 2do cos(0) do (i) Identify the radial part of this orbital function and the number of radial nodes. (ii) Identify the angular part of the orbital function and the number of angular nodes. Z (iii) Using this information and the L₂ = -ih d/do operator obtain the n, 1, and, m quantum numbers and identify the orbital.arrow_forwardThe radial wave function of a quantum state of Hydrogen is given by R(r)= (1/[4(2π)^{1/2}])a^{-3/2}( 2 - r/a ) exp(-r/2a), where a is the Bohr radius.(a) Sketch the graph of R(r) x r. For a decent sketch of this graph, take into account some values of R(r) at certain points of interest, such as r=0, 2a, 4a, and so on. Also take into account the extremes of the function R(r) and their inflection points, as well as the limit r--> infinity.arrow_forwardThe ground-state wave function of a hydrogen atom is: 1 where r is distance from the nucleus and a, is the Bohr radius (53 pm). Following the Born approximation, calculate the probability, i.e., |w?dz, that the electron will be found somewhere within a small sphere of radius, ro, 1.2 pm centred on the nucleus.arrow_forward
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