Derivation of the Normalization Constant for the Hydrogenic Radial Wave Equation

 

The normalization constant in the texts by Goswami, Liboff, and Robinett does not give the proper normalization conditions. The authors give the constant

When their constant is used in the radial wave equation , the sum of the probability values does not equal one!! Therefore, it cannot be correct and one must trudge forward to find the true constant that gives the correct normalization conditions.

Handling the Laguerre polynomials becomes the most important aspect of doing so. The solutions of the these polynomials must satisfy the following normalization conditions and are giver as1:

Eq 1.

In looking at the radial wave function in terms of the variable r , it becomes evident that the direction one needs to take in order to find the correct normalization constant is to in some way compare these two equations. The radial wave equation in terms of r is as follows1:

Eq 2.

Although it may not be obvious at the moment, there exists a similarity between equations one and two. With a little manipulation, the resemblance between the two becomes clearer. The first step involves writing out the entire normalization expression for , and then performing a change of variable from r to r. Here is how it is done:

Eq 3.

Now make a change of variable to r knowing . Fortunately, Z and ao are both equal to 1 the way we have defined them. Thus, we make the change of variable

Eq 4.

Finally, equations one and four are starting to look alike. The next key to solving this mystery lies in the a term. We want a and n from equation one to equal 2l+1 and n-l-1 respectively. Therefore, substituting these values for a and n into equation one yields:

Eq 5.

To further the resemblance, a r2 term is needed in equation five. It can be obtained from the z2l+1 term! Here is the result:

Eq 6.

The only catch keeping us from an exact match of the left-hand side of equations four and six appears to be the z2l-1 term. This step puzzled our minds the most, until we looked at the expression for the expectation value ! According to Goswami, Liboff, and Robinett, the solution for the expectation value of is . This expression’s usefulness comes in handy only when we write out the full integral form of it.

Eq 7. ; Z=1 and a0 =1

Now, we happen to have an expression for from equation four, so substitute that in now.

Eq 8.

In looking at equation 8 above, the term stands out. In order to be able to compare this equation to equation six, we need a term that resembles , but we only have . If we move the term into the term, however, the desired result is achieved.

Now add a term to both sides:

Eq 9.

This is the coveted form that we have sought after! It looks VERY similar to equation six above! It is clear that z in equation six is equivalent to in equation nine. Two more steps are required to finalize everything before we can make the substitution of equation six into equation nine. Several terms need to be added to both sides before they are truly equal equal.

Eq 10.

Using the relation , we can finally substitute the right hand side of equation six into equation ten resulting in:

At last we have derived an expression for the constant that gives us the proper normalization conditions. This expression represents what I use in my Radial Probability applet.

 

References:

  1. Goswami, Amit. Quantum Mechanics: Second Edition. WCB Publishers: Boston. 1997.
  2. Liboff, Richard L. Introductory Quantum Mechanics: Second Edition. Addison-Wesley Publishers: New York. 1992.
  3. Robinett, Richard W. Quantum Mechanics. Oxford University Press: New York. 1997.

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