Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8.2, Problem 8.7P
To determine
The ionization energy for the helium.
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Consider a hydrogen atom and a singly ionized helium atom. Which atom has the lower ground state energy? (a) Hydrogen (b) Helium (c) The ground state energy is the same for both. Why?
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A Hydrogen atom initially in its ground state i.e., n = 1 level, absorbs a photon and ends up in n = 4 level. (a) What must have been the frequency of the photon? Now the electron makes spontaneous emission and comes back to the ground state. (b) What are the possible frequencies of the photons emitted during this process?
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Introduction To Quantum Mechanics
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- Sketch the potential energy function of an electron in a hydrogen atom, (a) What is the value of this function at r=0 ? in the limit that r=? (b) What is unreasonable or inconsistent with the former result?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) Determine the radial probability density P(r) associated with the quantum state in question. (b) Show that the function P(r) you determined in part (a) is properly normalized.arrow_forward(b) Calculate the half width in nanometers for Doppler broadening of the 4s S 4p transition for atomic nickel at 361.939 nm (3619.39 Å) at a temperature of 20,000 K in both wavelength and frequency units. (e) Calculate the speed that an iron atom undergoing the 4s S 4p transition at 385.9911 nm (3859.911 Å) would have if the resulting line appeared at the rest wavelength for the same transition in nickel. (f) Compute the fraction of a sample of iron atoms at 10,000 K that would have the velocity calculatedin (e). (g) Create a spreadsheet to calculate the Doppler half width DlD in nanometers for the nickel and iron lines cited in (b) and (e) from 3000–10,000 K. (h) Consult the paper by Gornushkin et al. (note 10) and list the four sources of pressure broadening that they describe. Explain in detail how two of these sources originate in sample atoms.arrow_forward
- Consider hydrogen in the ground state, 100 . (a) Use the derivative to determine the radial position for which the probability density, P(r), is a maximum. (b) Use the integral concept to determine the average radial position. (This is called the expectation value of the electrons radial position.) Express your answers into terms of the Bohr radius, a0. Hint: The expectation value is the just average value, (c) Why are these values different?arrow_forwardAt time t = 0 the wave function of the hydrogen atom is: where we ignore the spin.(a) Calculate the expected value of energy for this system.(b) What is the probability of finding the system at l = 1, m = +1 as a function of time?(c) What is the probability of finding the electron around 10−10 cm from the proton, at t = 0s (canapproximate).(d) Write the time-dependent wave function: ψ (r,t)arrow_forward(a) How much energy is required to cause an electron in hydrogen to move from the n = 1 state to the n = 2 state? (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 3kBTarrow_forward
- E Suppose an electron is in the ground state of hydrogen. (a) Whatis the highest-energy photon this system can absorb without dissociating the electron from the proton? Explain. (b) What is thelowest-energy photon this system can absorb? Explainarrow_forwardCalculate the longest wavelength of the electromagnetic radiation emitted by the hydrogen atom in undergoing a transition from the n = 6 level.arrow_forwardTo a good approximation, the sun’s surface is a blackbody with a surfacetemperature of 5300 K. (a) At what wavelengthdoes the sun emit most strongly? (b) What is the total radiated powerper unit surface area?arrow_forward
- With the aid of tunable lasers, Rydberg atoms of sodium have been produced with n =100. The resulting atomic diameter would correspond in hydrogen to n = 600.(a) What would be the diameter of a hydrogen atom whose electron is in the n =600 orbit? (b) What would be the speed of the electron in that orbit? (c) How does the result in (b) compare with the speed in the n = 1 orbit?arrow_forwardA hydrogen atom is excited from its ground state to the state with n= 4. (a) How much energy must be absorbed by the atom? Consider the photon energies that can be emitted by the atom as it de-excites to the ground state in the several possible ways. (b) How many different energies are possible; what are the (c) highest, (d) second highest, (e) third highest, (f) lowest, (g) second lowest, and (h) third lowest energies?arrow_forward
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