hydrogen atom

• Schrödinger equation


  Equation used to model atomic systems
• Hydrogen atom


  • Electron is trapped in an electrostatic potential well created by the positively charged nucleus

  
• Bohr model


  1. Developed to explain the discrete spectral lines seen in the spectrum of atomic hydrogen

  2. Electron orbits the nucleus with quantised angular momentum

  3. Energy of the electron is quantised

  4. Discrete spectral lines occur when there is a transition between energy levels

  
• Bohr model predicted the correct energy behaviour for atomic hydrogen, but did not work for other atomic systems

  
• Schrödinger equation


  Used to model atomic systems that the Bohr model could not
• Applying Schrödinger equation to hydrogen atom

  1. Assume nucleus is massive and stationary compared to electron

  2. Consider motion of electron only

  3. System displays spherical symmetry, so solve in spherical coordinates

  4. Separate wave function into radial and angular solutions

  
• Principal quantum number (n)


  Comes from the radial solution to the Schrödinger equation, takes integer values 1, 2, 3, ...
• Orbital quantum number (l)


  Comes from the angular solution, associated with the electron's angular momentum, takes integer values 0, 1, 2, ..., n-1
• Orbital magnetic quantum number (ml)

  Comes from the angular solution, describes how the atom behaves in a magnetic field, takes integer values -l, -l+1, ..., l-1, l

  
• For each value of l there are 2l+1 possible values of ml
• The solution to the Schrödinger equation for hydrogen yields the same energy as predicted by the Bohr model

  
• The electron's energy is quantised
• A photon is emitted when an electron transitions from a high energy level to a lower energy level
• A photon is absorbed and forces an electron to move from low energy level to a higher energy level

  
• Ground state for any electron in an atom

  n=1, l=0, ml=0

  
• The probability that an electron is found in any small volume dV at a distance r from the centre of the atom is |ψ|^2 dV

  
• Radial probability density


  The quantity (r^2)e^(-2r/a0) where a0 is the Bohr radius
• The radius at which the radial probability is largest corresponds to the Bohr radius a0
• Hydrogen n=2 wave functions and states

  • Multiple wave functions possible as l and ml can take values other than 0
  • Radial probability distributions and volume probability densities have different shapes

  
• Shell


  All states having the same principal quantum number n
• Subshell


  All states having the same principal quantum number n and orbital quantum number l
• Atomic shell notation uses letters K, L, M, ... to identify the shell based on n
• Atomic subshell notation uses letters s, p, d, f, g, ... to identify the subshell based on l
• The allowed combinations of n, l, ml quantum numbers are defined by the value of n
• Each allowed combination of n, l, ml quantum numbers is a state that can hold an electron
• For each n, the number of states is set by the total number of possible values of ml, which is n^2
• The actual number of states is half the n^2 value
• Quantum numbers


  n, l, ml, ms
• Electrons


  • Electrons with the same value of n are in the same shell
  • Electrons with the same value of l are in the same sub-shell
• Allowed quantum numbers


  • n
  • l
  • ml
• For each n the number of states is set by the total number of possible values of ml (=n^2)
• This is not quite true
• Atoms


  • Atoms are stable. Aside from radio-active decay they do not change into other forms
  • Atoms combine with each other. They join together to make molecules, solids, amorphous materials
  • Atoms are put together systematically. There are 6 complete periods and a 7th incomplete period. The number of atoms in the complete periods are 2, 8, 8, 18, 18, 32
  • Atoms like Li, Na, K have 1 electron in the s sub-shell and are easy to ionise, remove an electron
  • Atoms like He, Ne, Ar, have completely full shell and so are difficult to ionise
• Electrons make transitions between energy levels

  The energy needed E = hf = Ehigh - Elow

  
• Atoms


  • Atoms have Angular Momentum and Magnetism
• Orbital angular momentum


  • L = √l(l+1)ℏ, l = 0, 1, ..., n-1
  • Lz = mlℏ, ml = 0, ±1, ±2, ..., ±l
  • cos(θ) = Lz/L
• Orbital magnetic dipole moment

  • μorb = (e/2m0)L
  • μz = -μBml
• Spin angular momentum


  • S = √s(s+1)ℏ, s = 1/2
  • Sz = msℏ, ms = ±1/2
• Experimental evidence for spin came from the Stern-Gerlach experiment
• Electrons


  • An electron is described by the quantum numbers n, l, ml, and ms
  • Each sub shell can hold 2(2l+1) electrons
  • In a given shell there are 2n^2 electrons