The Schrodinger Wave Equation definition

The Schrodinger Wave Equation - Atomic structure and the periodic table

Chemistry Explain provide notes about Atomic structure and the periodic table today we discuss “The Schrodinger Wave Equation” Inorganic Chemistry

For a standing wave (such as a vibrating string) of wavelength λ, the whose amplitude at any point along x may be described by a function f (x). it can be shown that:

If an electron is considered as a wave which moves in only one dimension then:

An electron may move in three directions x. y and z so this becomes

Using the · symbol ∇ instead of the three partial differentials, this is shortened to

The de Broglie relationship states that

(where h is Planck's constant, m is the mass of an electron and v its velocity); hence:

or

However, the total energy of the system Eis made up of the kinetic energy K plus the potential energy V

Substituting for v² in equation (1.5) gives the well-known form of the Schrodinger equation

Acceptable solutions to the wave equation, that are solutions which are physically possible, must have certain properties:
1. ♆ the must be continuous.
2. ♆ the must be fi.nite.
3. ♆ the must be single-valued.
4. The probability of finding the electron over all the space from plus infinity to minus infinity must be equal to one.
The probability of finding an electron at a point x, y, z is ♆², so

Several wave functions called ♆₁. ♆₂ , ♆₃ . will satisfy these conditions to the wave equation, and each of these has a corresponding energy E₁, E₂ , E₃ . Each of these wave functions ♆₁ ♆₂ , etc. is called an orbital, by analogy with the orbits in the Bohr theory. In a hydrogen atom, the single electron normally occupies the lowest bf the energy levels E₁• This is called the ground state. The corresponding wave function ♆₁ describes the orbital, which is the volume in space where there is a high probability of finding the electron.
For a given type of atom, there are a number of solutions to the wave equation which are acceptable, and each orbital may be described uniquely by a set of three quantum numbers, n, I, and m. (These are t_he same quantum numbers - principal, subsidiary and magnetic - as were used in the Bohr theory).
The subsidiary quantum number I describe the shape of the orbital occupied by the electron. l may have values 0, 1, 2 or 3. When I= 0, the orbital is spherical and is called an s orbital; when I = l, the orbital is dumb-bell shaped and is called a p orbital; when I= 2, the orbital is double dumb-bell shaped and is called a d orbital; and when I = 3 a more complicated f orbital is formed (see Figure 1.6). the letters s, p, d and i come from the spectroscopic terms sharp, principal, diffuse and

.fundamental, which were used to describe the lines in the atomic spectra.

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Examination of a list of all the allowed solutions to the wave equation shows that the orbitals fall into groups. In the first group of solutions the value of the wave function ♆ and
hence the probability of finding the electron ♆² depends only on the distance r from the nucleus and is the same in all directions.

♆ = f(r)

This leads to a spherical orbital and occurs when the subsidiary quantum number I is zero. These are called s orbitals. When I = 0, the magnetic quantum number m = 0, so there is only one such orbital for each value of n.
In the second group of solutions to the wave equation, ♆ depends both on the distance from the nucleus, and on the direction in space (x, y or z). Orbitals of this kind occur when the subsidiary quantum number I = 1. These are called p orbitals and there are three possible values of the magnetic quantum number (m = -1, 0and+1). There are therefore three orbitals that are identical in energy, sh.ape, and size, which differ only in their direction in space. These three solutions to the wave equation may be written

Orbitals that are identical in energy are termed degenerate, and thus three degenerate p orbitals occur for each of the values of n = 2, 3, 4 ... The third group of solutions to the wave equation depends on the

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distance from the nucleus r and also on two directions in space, for example

This group of orbitals has I = 2, and these are called d orbitals. there are five solutions corresponding tom = -2, -1, 0, + 1 and +2, and these are all equal in energy. Thus five degenerated orbitals occur for each of the values of n = 3, 4, 5 .... A further set of solutions occurs when I = 3, and- these are called f orbitals. There are seven values of tn: -3, -2, -l, O", +l, +2 and +3, and seven degenerate f orbitals are formed when n = 4, 5, 6 ....