Quantum Mechanical Hydrogen Atom

The complete specification of an electron's state in the hydrogen atom according to quantum mechanics requires four quantum numbers:

• The principal quantum number is n = 1, 2, 3, ...
• The orbital quantum number gives the magnitude of the orbital angular momentum as

  Notation: = 0,1,2, ..., n-1 means that can have any value from 0 to n-1.
  Example: For n=6, can take the values 0 or 1 or 2 or 3 or 4 or 5.

  [Please note the difference between the correct quantum mechanical expression for the angular momentum and the expression obtained from the Bohr model, L = n. In the Bohr model, there is only one quantum number, the main quantum number n.]

• The third quantum number is called the magnetic quantum number and specifies the component of the orbital angular momentum along some direction usually taken as the Z-axis

  The Z axis can be located by placing the hydrogen atom in a magnetic field. Because m_l is constrained to integral values, the orbital angular momentum takes only specific angles with respect to the Z axis. The case for = 2 is shown above.

• The fourth quantum number, spin, has no classical analog. The spin angular momentum is

  The Z component is given by S_Z = m_s and m_s = -s and +s. For the electron, m_s = + or -.

In the hydrogen atom, the energy depends only on n. For other atoms, the energy levels depend on n, , m_l, and m_s.

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