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/4πε

_{0}r = E ψ

where

(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 m

_{l}.

For each combination of n,l, and m

_{l}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 m

_{l}depends only on n and may be written as

En = - mZ²e

^{4}/8 ε

_{0}²h²n²

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