application of all this stuff

chem further
represents charge on a nucleus
is charge on electron
coulomb’s law

where , represents the charges of the objects, represents the distance
and is the electric constant

if the charges are and

force between a proton and an electron

potential energy of the electron

negative because it is bound to the proton

the schrodinger equation cannot be solved in cartesian coordinates, as the variables cannot be separated

SE for hydrogen atom

change of variables spherical polar coordinates

can separate variables

solution:

3 quantum numbers will be generated instead of 1 for particle in a box - solutions to the SE

principle main shell
orbital AM subshell
magnetic quantum number which orbital

radial

where is the Bohr radius (the most probable value of )

e.g. for hydrogen 1s,

note, square modulus of wavefunction is the probability density, the wavefunction by itself doesn’t really have much of a physical representation

e.g. for hydrogen 3s,

notice the coefficient of the denominator in the exponential term
notice that the wavefunction has a quadratic, so we expect it to touch 2 twice

radial nodes are when the wavefunction touch 0, when the wavefunction changes sign

angular

complex wavefunctions

e.g. 1s

this means that electrons in the 1s orbital of the hydrogen atom has an equal chance of being found at any angle

e.g. p2

no dependence means nothing to do on rotation about

so almost like a volume of revolution around the axis

only works for because otherwise, there would be an imaginary term
taking linear combinations of complex gives real functions

1s

energy levels