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H Atom Wavefunctions

    The solution to the hydrogen atom Schrodinger Equation leads to a wavefunction with a dependence on three quantum numbers.

    1. 1. Introduction
    2. 2. Visualizing Wavefunctions
    3. 3. Outside links
    4. 4. References
    5. 5. Problems
    6. 6. Contributors

    Introduction

    The solution to the hydrogen atom Schrodinger Equation leads to a wavefunction with a dependence on three quantum numbers n,

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    and
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    .

    • n is the principle quantum number in which the energy depends
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      .  n has values of 1, 2, 3,...
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       is the angular momentum quantum number in which the angular momentum depends
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      . 
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       has values of 0, 1, 2, ..., n-1
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      is the magnetic quantum number in which the z component of the angular momentum depends
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      . 
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      has values of
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      ,
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      , ...,-2, -1, 0, 1, 2, ...,
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    Visualizing Wavefunctions

    It is difficult to visualize the complete hydrogen wave function, or orbital, because the wave function depends on three variables.  For this reason, the wavefunctions are commonly separated into radial and angular portions.

    The radial portion of the wavefunction depends on both n and

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     .  
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      tends to be the quantum number that is associated with the shape of the orbital.  The angular portion of the wavefunction depends on
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    . 
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    attributes to the degeneracy of the orbitals.  There are
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    possible values for
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    , therefore, there are
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    degenerate orbitals.

    Associated with shape is the number of nodes present in a particular orbital.  There are n-1 total nodes.  These nodes are distributed between angular and radial nodes.  Angular nodes are planes separating the orbital and attribute 

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      nodes.  Therefore, there are
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    radial nodes.  The 
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      quantum number is commonly expressed by a letter when describing the shape.

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     is denoted as an s orbital.  The s orbitals are spherically symmetric (no
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    or
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     dependence) and have no angular nodes.  s orbitals can be visualized as spheres which have increasing radii with increasing values of n.

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    is denoted as a p orbital.  There are three degenerate p orbitals which are commonly viewed as three dumb bells, each lying in one of the three planes.

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    is denoted as a d orbital.  There are five degenerate d orbitals.  One of the orbitals is similar in appearance to a p orbital with a radial node around the center and an additional  loop of electron density.  Four of the orbitals consist of four lobes lying in one plane, meeting at the origin.

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      can assume a higher value, but these orbitals become very difficult to visualize.

    Outside links

    References

    1. McQuarrie, Donald A. Quantum Chemistry. 2nd ed. United States Of America: University Science Books, 2008. 321-24.

    Problems

    1. Draw a 3d orbital, labeling the radial and angular nodes.
    2. What are the possible quantum numbers for a 4p electron?

    Contributors

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    Viewing 4 of 4 comments: view all
    #1
    dlarsen says:
    Do you know the origin of the letters s,p,d,f?
    Posted 20:35, 7 Apr 2009
    #2
    Elizabeth Weidenhaft says:
    1. You might consider talking about nodes that contain the z-axis depend upon ml.
    2. Examples of how to draw the angular part such as .... a 3s and 2p1 orbital or something.
    3. Drawing some examples of the radial part and then explaining, from the equations how they were obtained would probably help someone replicate them especially if they don't have any experience in it. For example, maybe draw some s orbitals, p-orbitals and d-orbitals
    4. Providing links to answers to your questions might also help people trying to learn about your subject.
    Posted 19:41, 13 Apr 2009
    #3
    Kaleb Topp says:
    This page simplifies wavefunctions. It is clearly written and provides useful information to anyone looking to visualize the wavefunctions of the s,p,d,f orbitals. I thought the link to the visualization wavefunction calculator was beneficial. I thought you may want to place it at the end of the Visualizing Wave Function section because the user will then have the background to properly insert values into the calculator, but I can also understand why you would want it at the beginning of the page.
    1.) The following sentence was a bit confusing "m(l) attributes to the degeneracy of the orbitals." I thought that it could be changed by saying: "m(l) contributes to the degeneracy of the orbitals." An explanation of how and why m(l) contributes would be a good follow up paragraph to this statement.
    2.) One improvement could be to import a picture of the s,p,d,f orbitals to compliment your description of the orbitals. Then a visual representation of the orbital will be there to help the reader better understand your description of it.
    3.) In the introduction you mention l and m(l). I do not know if it is possible but if you could include a description of what those values mean physically it may help the reader to better understand the angular momentum. For example, you could use an electron spinning around a nucleus. edited 10:34, 14 Apr 2009
    Posted 10:33, 14 Apr 2009
    #4
    Zachary White says:
    This is a good introduction to H atom wavefunctions. I would include the Schrodinger equation at the top of your page so readers will know how Energy fits in to the equation. Maybe include a link to an outside source for information about the Schrodinger equation.

    I would include common pictures of the different types of oribitals if they can easily be found on the internet.

    Also for the section on the nodes, it would be helpful to show a graph like figure 7.2 in our book. This helps to show the nodes as they are represented by a graph. Remember a picture is worth 1000 words.

    I think this is a great start and I agree with the comments left by Kaleb, Elizabeth, and Professor Larson. With the addition of a few keys graphs this page will be in great shape. Good job!
    Posted 20:42, 15 Apr 2009
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