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Chapter 39: Q55P (page 1217)

The radial probability density for the ground state of the hydrogen atom is a maximum when r = a , where is the Bohr radius. Show that the average value of r, defined as

ravg=鈭�P(r)rdr,

has the value 1.5a. In this expression for ravg , each value of (P)r is weighted with the value of r at which it occurs. Note that the average value of is greater than the value of r for which (P)r is a maximum.

Short Answer

Expert verified

ravg=1.5a

Step by step solution

01

Identification of the given data

The given data is listed below as-

The radial probability density is maximum when r = a

02

The radial probability function

Theradial probability functionfor the ground state is given by-

P(r)=(4r2a3)e-2r/a

Here, r is the radius of Hydrogen atom.

03

To Show that the average value of r, defined as ravg=∫P(r)r dr , has the value 1.5 a   

The average value of is defined as:

ravg=鈭�0鈭�rP(r)dr鈥︹赌�(1)

Now, Pr=4r2a3e-2/a

Substitute the value of P(r) in equation (1)

ravg=鈭�0鈭�4r3a3e-2r/adr

Now, let x=2ra

dx=2dra

ravg=鈭�0鈭�ax34a3e-xadx

Solving the above equation,

ravg=a4鈭�0鈭�x3e-xdx

Now, ravgwill become ravg.=a(3!)4

Therefore, it is shown that ravg=1.5a.

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Most popular questions from this chapter

The two-dimensional, infinite corral of Fig. 39-31 is square, with edge length L = 150 pm. A square probe is centered at xy coordinates (0.200L,0.800L)and has an x width of 5.00 pm and a y width of 5.00 pm . What is the probability of detection if the electron is in the E1.3energy state?

A muon of charge -eand massm=207me(wheremeis the mass of an electron) orbits the nucleus of a singly ionized helium atom (He+). Assuming that the Bohr model of the hydrogen atom can be applied to this muon鈥揾elium system, verify that the energy levels of the system are given by

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If you double the width of a one-dimensional infinite potential well, (a) is the energy of the ground state of the trapped electron multiplied by4,2,12,14 or some other number? (b) Are the energies of the higher energy states multiplied by this factor or by some other factor, depending on their quantum number?

Consider a conduction electron in a cubical crystal of a conducting material. Such an electron is free to move throughout the volume of the crystal but cannot escape to the outside. It is trapped in a three-dimensional infinite well. The electron can move in three dimensions so that its total energy is given by

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