clearscience: We introduced the Schrödinger Wave Equation, and now we have substituted in the operator H for an electron in a harmonic oscillator situation. The energy eigenvalues are shown, with a valid solution for every whole number value of n. The graph shows the wave function (Ψ) plotted for various eigenvalues—the lowest is in red. If you would like to go through the solution yourself, this very nice website describes it in detail, and we have borrowed their plot of the solution. There are three important things to note: The electron cannot have 0 energy. An eigenvalue of 0 gets added to 1/2 and gives a value of some energy. The wave function give us probable places to find the electron, with higher probability when the wave is far above or below the zero line for the solution. These are the only solutions: if the electron is in state n = 0 and absorbs a photon, it jumps instantly to n = 1. There is no in between. The places where the wave function crosses the zero line are places it cannot be found. So: it can go from one place to another along the wave function, but without passing through the spots of zero probability. That’s weird. It is.

clearscience:

We introduced the Schrödinger Wave Equation, and now we have substituted in the operator H for an electron in a harmonic oscillator situation. The energy eigenvalues are shown, with a valid solution for every whole number value of n.

The graph shows the wave function (Ψ) plotted for various eigenvalues—the lowest is in red. If you would like to go through the solution yourself, this very nice website describes it in detail, and we have borrowed their plot of the solution. There are three important things to note:

  1. The electron cannot have 0 energy. An eigenvalue of 0 gets added to 1/2 and gives a value of some energy.
  2. The wave function give us probable places to find the electron, with higher probability when the wave is far above or below the zero line for the solution. These are the only solutions: if the electron is in state n = 0 and absorbs a photon, it jumps instantly to n = 1. There is no in between.
  3. The places where the wave function crosses the zero line are places it cannot be found. So: it can go from one place to another along the wave function, but without passing through the spots of zero probability. That’s weird. It is.