Normalize Hydrogen Wavefunction

In summary, the problem is asking to integrate the wave function of Hydrogen in three dimensions using spherical coordinates to prove that it is normalized. However, there are some errors in the integration process, such as squaring instead of doubling the term in the exponential and missing a factor of 4πr^2. The normalization constant also has units of 1/volume, so multiplying it by dr alone will not give a unitless result.
  • #1
Nacho Verdugo
15
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Moved from a technical forum, so homework template missing
I'm trying to prove that the wave function of Hydrogen for the fundamental state is normalized:

$$ \Psi_{1s}(r)=\frac{1}{\sqrt{\pi a^3}}e^{-\frac{r}{a}} $$

What I tried is this:

$$ I= \int_{-\infty}^{\infty} | \Psi^2(x) | dx = 1$$

$$ \int_{-\infty}^{\infty} \frac{1}{\pi a^3}e^{-\frac{r^2}{a^2}} dr $$

Which lead me to two differents results:

a) First, when integrating the next integral

$$ \frac{e^{1/a^2}}{\pi a^3} \int_{-\infty}^{\infty} e^{-r^2}dr$$

and using polar coordinates to integrate ## \int_{-\infty}^{\infty} e^{-r^2}dr ##, I finally obtain:

$$ \frac{e^{1/a^2}}{\pi a^3} \sqrt{\pi} $$

and I don't see a way that this expression equals to one.

b) for my second attempt, I used the next property

$$ \int_{-\infty}^{\infty} e^{-bx^2} dx = \sqrt{\frac{\pi}{b}} $$

So, considering ## b=\frac{1}{a^2} ##, I obtain

$$ I=\frac{a\sqrt{\pi}}{\pi a^2} $$

which as well, doesn't look like one.

did i miss something?
 
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  • #2
Does your hydrogen live in 1 dimension or in 3 ?
In other words: what are you integrating over ?
 
  • #3
The first issue is that you are squaring the term in the expontial, but you just need to double it. The second issue as Bvu alludes to is that you are missing a factor of 4 pi r^2.
 
  • #4
BvU said:
Does your hydrogen live in 1 dimension or in 3 ?
In other words: what are you integrating over ?
1 dimension!
 
  • #5
Nacho Verdugo said:
1 dimension!
You need to revise integration in three dimensions using spherical coordinates.
 
  • #6
PeroK said:
You need to revise integration in three dimensions using spherical coordinates.
I thought about that, but the statement of the problem I'm trying to solve just says to integrate in one dimesion. I'll try again.
 
  • #7
Nacho Verdugo said:
I thought about that, but the statement of the problem I'm trying to solve just says to integrate in one dimesion. I'll try again.
A normalised function of one Cartesian variable ##x## is different from a normalised function of the spherical variable ##r##.

You have a function of one variable here, sure, but it is not a Cartesian variable.
 
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  • #8
Nacho Verdugo said:
I thought about that, but the statement of the problem I'm trying to solve just says to integrate in one dimesion. I'll try again.
A hydrogen atom lives in three dimensions, so you have to integrate across three-dimensional space. It turns out because of spherical symmetry, the integration over the angles is trivial, leaving you with only one variable to integrate over, which is probably what the problem statement was referring to.

Also, note that the normalization constant squared has units of 1/volume. The only way the integral is going to produce a unitless result is to multiply ##\psi^2## by a quantity with units of volume. ##dr## by itself isn't going to cut it.
 

FAQ: Normalize Hydrogen Wavefunction

What is the purpose of normalizing a hydrogen wavefunction?

The purpose of normalizing a hydrogen wavefunction is to ensure that the probability of finding an electron in all space is equal to 1. This allows for accurate calculations and predictions of the electron's position and energy in the hydrogen atom.

How is a hydrogen wavefunction normalized?

A hydrogen wavefunction is normalized by dividing the wavefunction by the square root of the integral of the square of the wavefunction over all space. This ensures that the total probability of finding the electron in all space is equal to 1.

Why is it important to normalize a hydrogen wavefunction?

Normalizing a hydrogen wavefunction is important because it allows for accurate calculations and predictions of the electron's position and energy in the hydrogen atom. It also ensures that the probability of finding the electron in all space is equal to 1, which is necessary for the wavefunction to represent a physically meaningful state.

Can a hydrogen wavefunction be normalized to a value other than 1?

No, a hydrogen wavefunction must be normalized to a value of 1 in order to accurately represent the electron's position and energy in the hydrogen atom. Normalizing to any other value would result in incorrect calculations and predictions.

What happens if a hydrogen wavefunction is not normalized?

If a hydrogen wavefunction is not normalized, the calculated probabilities of finding the electron in different regions of space may not add up to 1. This can lead to inaccurate predictions and calculations of the electron's position and energy in the hydrogen atom.

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