Finding the Normalization Constant for the Hydrogen Radial Wave Function

In summary, the normalization constant for the radial wave function for Hydrogen can be found by applying separation of variables to the Schrödinger equation and solving for R(r). The resulting equation for R(r) is then used to derive the normalization constant, which can be normalized using the properties of Laguerre polynomials. Alternatively, more information is needed for a simpler solution.
  • #1
Patroclus
2
0
1. Find the normalization constant for the radial wave function for Hydrogen.

I'm told that C = 1/(24a^5)^1/2
But how do I get that?


2.
n=2, l=1
R(2)(1)=Cr^(-r/2a。)
the integral from 0 to infinity of (x^4 * e^-"alpha"x) = 24 / alpha^5


3. I honestly don't know where to start
 
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  • #2
Depends what your starting point is.
If you're going from the Schrödinger equation, you can apply separation of variables:
[tex]\psi(r, \theta, \phi) = R(r) Y_{\ell m}(\theta, \phi)[/tex]
and plug this into the time independent wave equation
[tex]\left( - \frac{\hbar^2}{2m} \nabla^2 + V(r) \right) \psi(r, \theta,\phi) = E \psi(r, \theta, \phi)[/tex]
and derive the equation for R(r).
Then if you solve it and impose appropriate boundary conditions, you will find
[tex] R_{nl} (r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \frac{2 Z r}{n a_{\mu}} \right )[/tex]
which you can normalise using the properties of the Laguerre polynomials.

If you were looking for something simpler, please give us more information :-)
 

What is a normalization constant?

A normalization constant is a mathematical factor used to scale a probability distribution so that the total probability of all possible outcomes sum to 1.

Why is a normalization constant important in science?

A normalization constant is important in science because it allows for the comparison and interpretation of data from different experiments or models. It also ensures that the probability of all possible outcomes is accurately represented.

How is a normalization constant calculated?

A normalization constant is calculated by dividing each individual probability value by the sum of all probabilities. This ensures that the total probability of all possible outcomes is equal to 1.

What happens if a normalization constant is not used?

If a normalization constant is not used, the probability distribution may not accurately reflect the true probabilities of the outcomes. This can lead to incorrect interpretations and conclusions.

Can a normalization constant be greater than 1?

No, a normalization constant cannot be greater than 1. It is a scaling factor that is used to ensure the total probability of all outcomes is equal to 1, so it must be less than or equal to 1.

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