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ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Quantum Mechanics > Quantum Theory > Trapped Particles > H Atom Wavefunctions

H Atom Wavefunctions

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

Introduction

The solution to the hydrogen atom Schrodinger Equation leads to a wavefunction with a dependence on three quantum numbers n, \(l\) and \(m_l\).

  • n is the principle quantum number in which the energy depends \(E_n = \frac{-e^2}{8 \pi \epsilon_0 (a_0n^2}\).  n has values of 1, 2, 3,...
  • \(l\) is the angular momentum quantum number in which the angular momentum depends \(|L|=\hbar[l(l+1)]^{1/2}\).  \(l\) has values of 0, 1, 2, ..., n-1
  • \(m_l\) is the magnetic quantum number in which the z component of the angular momentum depends \(L_z = m_l\hbar\).  \(m_l\) has values of \(-l\), \(-l+1\), ...,-2, -1, 0, 1, 2, ..., \(l\)

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 \(l\) .  \(l\)  tends to be the quantum number that is associated with the shape of the orbital.  The angular portion of the wavefunction depends on \(m_l\).  \(m_l\) attributes to the degeneracy of the orbitals.  There are \(2l+1\) possible values for \(m_l\), therefore, there are \(2l+1\) 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 \(l\)  nodes.  Therefore, there are \(n-l-1\) radial nodes.  The \(l\)  quantum number is commonly expressed by a letter when describing the shape.

  • \(l=0\) is denoted as an s orbital.  The s orbitals are spherically symmetric (no \(\theta\) or \(\phi\) dependence) and have no angular nodes.  s orbitals can be visualized as spheres which have increasing radii with increasing values of n.
  • \(l=1\) 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.
  • \(l=2\) 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.
  • \(l\)  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?

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Last Modified
09:39, 2 Oct 2013

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