Electron has a wave character as well as a particle character. The wave function of the electron ψ(r,t ) is obtained by solving Schrodinger’s wave equation. The probability of finding an electron is high where | ψ(r,t )|² is greater. Not only the information about the electron’s position but information about all the properties including energy etc. that we calculated using the Bohr’s postulates are contained in the wave function of ψ(r,t).
Quantum Mechanics of the Hydrogen Atom
The wave function of the electron ψ(r,t) is obtained from the Schrodinger’s equation
-(h²/8π²m) [∂²ψ /∂x² + ∂²ψ /∂y² + ∂²ψ/∂z²] - Ze²ψ/4πε0r = E ψ
(x.y,z ) refers to a point with the nucleus as the origin and r is the distance of this point from the nucleus.
E refers to the energy.
Z is the number of protons.
There are infinite number of functions ψ(r,t) which satisfy the equations.
These functions may be characterized by three parameters n,l, and ml.
For each combination of n,l, and ml there is an associated unique value of E of the atom of the ion.
The energy of the wave function of characterized by n,l, and ml depends only on n and may be written as
En = - mZ²e4/8 ε0²h²n²