Modern Physics
Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Chapter 7, Problem 11CQ
To determine

To Explain:The reason that hydrogen’s electron has only one certain energy but for some energies, electrons can still be in different states.

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. Find the average (expectation) value of 1/r in the 1s state of hydrogen. Is the result equal to the inverse of the average value of r ? Note that the general expression is given by,
Find expectation value of position and for an electron in the ground state of hydrogen across the radial wave function. Express your answers in terms of the Bohr radius a.
A simple illustration of the variation method is provided by the hydrogen atom in the 1s state. Let us assume a form of the trial wave function ψ=e–br where b is a constant. For hydrogen atom, V=-e2/r, so the Hamiltonian operator is Ĥ=-h2/8π2m V2- e2/r.The energy depends on r for the 1s state of the hydrogen atom so the angular portion of the Laplacian can be omitted and replaced by the factor 4π after integration. Therefore, the radial portion of V2=1/r2 (∂/∂r) r2 ∂/∂r.  Solve the energy minimum of the hydrogen atom, E = -2π2 me4/h2.

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