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Chapter 36: Problem 34

Adapt part (b) of Example 36.1 to find the probability that an electron in the hydrogen ground state will be found beyond two Bohr radii.

Short Answer

Expert verified
The probability that an electron is found beyond two Bohr radii can be calculated by integrating the probability density function from two Bohr radii to infinity. This involves the wave function for the hydrogen ground state and some known integral calculations. The result is a decimal or fraction that represents the probability.

Step by step solution

01

Understand the wave function for the hydrogen atom

The wave function for an electron in the ground state of the hydrogen atom is given by \( \Psi (r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0} \), where \( a_0 \) is the Bohr radius. The wave function describes the quantum state of the electron and the probability density associated with this state is given by \( |\Psi (r)|^2 \). This provides a measure of the probability that the electron is at a particular location.
02

Calculate the probability density

In order to get the radial probability distribution function, the square of the wave function, \( |\Psi (r)|^2 \), needs to be multiplied by \( 4\pi r^2 \), which gives the volume of the spherical shell at radius \( r \). This gives \( P(r) = 4\pi r^2 |\Psi (r)|^2 \), and it provides the probability that an electron is found in the shell between \( r \) and \( r+dr \).
03

Integrate the probability density

To find the probability that the electron is beyond two Bohr radii, integrate the probability density from two Bohr radii (2\( a_0 \)) to infinity. This gives the integral \( P_{>2a_0} = \int_{2a_0}^鈭� P(r) dr \). This integral calculation will tell you the probability that the electron is beyond two Bohr radii.
04

Evaluate the Integral

This is a well-known integral and is found in integral tables and textbooks. Though a derived solution usually results in a fraction or decimal that represents the desired probability.

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Most popular questions from this chapter

Repeat Exercise 25 for the case where you know only that the principal quantum number is \(3 ;\) that is, \(l\) might have any of its possible values.

What are the possible \(j\) values for a hydrogen atom in the \(3 D\) state?

Find (a) the energy and (b) the magnitude of the orbital angular momentum for an electron in the \(5 d\) state of hydrogen.

Suppose you put five electrons into an infinite square well of width \(L .\) Find an expression for the minimum energy of this system, consistent with the exclusion principle.

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