An Introduction to Thermal Physics
An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter A.4, Problem 21P
To determine

The quantum numbers ( n, l and m ) for all the independent states of a hydrogen atom with definite E , |L|2 and Lz up to n = 3 .

To check: The numbers of independent states for level n is equal to n2 .

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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 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.
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.
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