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Quantum Number Hydrogen

Quantummechanics began with a daring hypothesis by Louis de Broglie, if light has a dualistic wave/particle nature, why not matter? His reasoning led to the prediction that a particle of mass m and velocity u would exhibit wave like properties with wavelength,

wavelength

Thusfor an electron beam with u = 1.46 × 106 m/s, above equation gives l = 5.0 Å, comparable to the ionic spacing in a crystal. In 1920, it was found that a crystal would indeed diffract an electron beam, and de Broglie's crazy idea was proven! Once it was established that electrons could behave like waves, physicists began looking for ways of dealing with electrons using their wave like properties, and in 1926, Schrödinger found the solution. Schrödinger's wave equation doesnot attempt to calculate the trajectory of an electron nor does it makeany assumptions about planetary orbits of electrons about nuclei. Rather it computes a wave function, the square of which is the probability density of finding an electron at some point in space.


quantum mechanics of hydrogen atom:


WhenQuantum mechanics applied to the hydrogen atom, the results are similar to those of Bohr, with the following equation for the allowed energies:

Rydbergformula

where m is the reduced mass, m = memn/(me + mn) » me (me and mn, are the masses of the electron and the nucleus), e is the electronic charge, Z is the nuclear charge in units of |e| (Z = 1 for H, 2 for He+), e0 = 8.854 ×10-12 C2J-1m-1 is the permittivity of free space, h is Planck's constant, and n isa positive integer. The energy of the photon emitted from an excited Hydrogen  atom is the difference between allowed energies:


Energychangewhenanelectronjumpsfromoneleveltoanother

The integer n in Eq. (2), called the principal quantum number,determines the energy levels in an one electron, atom or ion and largely determines the average distance of the electron from the nucleus. A complete description of the Hydrogen atom requires two additional quantum numbers: The angular momentum quantum number, l, defines the shape of the electronic distribution, called an orbital. l, may have any integral value between 0 and n - 1. Chemists often use a letter to represent the numerical value of l:


l =

0

1

2

3

4

Letter:

s

p

d

f

g

The magnetic quantum number, m, describes the orientation of orbitals in space. m may have any integral value between -l and +l.

Thus for l = 0, m = 0, for l = 1, m = ± 1, 0; in general there are 2l + 1 values of m.

The shapes of the s, p, and d orbitals are shown in Figure


 shapesofs,panddorbitals


The quantum numbers of the Hydrogen atom:


  • The principal quantum number, n = 1, 2,3, . . .

  • The angular momentum quantum number, l = 0, 1, . . . n – 1

  • The magnetic quantum number, m = 0, ±1, . . . ±l