In this problem, you will find the average distance r from the nucleus, which we

Question

In this problem, you will find the average distance r from the nucleus, which we assume to be a point, for an electron in a hydrogen atom's ground state. The average value of a variable is often called the expectation value of that variable and denoted by angle brackets: r. To calculate the average of the collection of numbers {2,3,3,5,6}, you sum the five numbers and divide by 5: 2+3+3+5+65=12+23+15+165= 152+253+155+156. Each of the fractions in the last expression corresponds to the probability of picking the number by which it is multiplied if you choose one element from the collection at random. From this, you can see that computing the average of a collection of values is equivalent to multiplying each value in the collection by the probability of choosing that value if you pick one number from the collection at random. This idea of summing a set of numbers, with each multiplied by the probability of finding it, generalizes to integrals. Thus, to find the expectation value of the variable r, you would use the formula rP(r)dr, where P(r) is a probability density function, where for an infinitesimally small linear element dr, the value P(r)dr is the probability that a measurement of the electron position would yield a value in the interval [r,r+dr]. Since ||2dV is the probability of finding the particle in the volume element dV, you can rewrite the integral as rP(r)dr=r||2dV. In spherical coordinates, this becomes 0020r||2r2sin()dddr. In this problem, we are interested specifically in hydrogen. Since the wave function for hydrogen has the form =R(r)()(), you may separate the r-dependent part of the function, obtaining 0r3(R(r))2dr020(())2(())2sin()dd. It is conventional to normalize the angular and radial portions of the wave function seperately, with the result that the expression in brackets above evaluates to 1. So, to find the expectation (average) value of r, you simply need to evaluate r=0r3(R(r))2dr, where R(r) is the normalized radial wave function.

Part A

Find the expectation value r of r for an electron in the ground state of hydrogen. The normalized radial wave function for such an electron is 2a3/2er/a, where a is the Bohr radius.

Express your answer in terms of a

Explanation / Answer

expectation value r of r for an electron in the ground state of hydrogen

= 3*a/2

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