6.5 Quantum Mechanics

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The paradox described by Heisenberg's uncertainty principle and wavelike nature of subatomic particles (electrons) made it impossible to use equations of classical physics to describe the motion of electrons in atoms.

In 1926, an Australian physicist Erwin Schrodinger developed wave mechanics, a mathematical technique that describes the relationship between the motion of a particle that exhibits wavelike properties and its allowed energies.

WAVEFUNCTIONS

A wavefunction (Ψ) relates to the location of an electron at a given point in space to the amplitude of its wave, which corresponds to its energy. Thus, each wavefunction is associated with a particular energy, E.

Properties:

A wave function uses three variables to describe the position of an electron.

The magnitude of the wavefunction at a particular point in space is proportional to the amplitude of the wave at that point.

Many wavefunctions are complex functions, indicating they contain sqrt (-1), represented as "i". Therefore, the amplitude has no physical significance; the sign of wavefunction corresponds to wave's phase.

The square of the wavefunction at a given point is proportional to finding an electron at that point, which leads to a distribution of probabilities in space.

Ψ^2 is always a real quantity and is proportional to the probability of finding an electron at a given point. The probability is given by a product of Ψ and its complex conjugate Ψ*, in which all terms that contain "i" are replaced by "-i"

Describing the electron distribution as a standing wave leads to sets of quantum numbers that are characteristic of each wavefunction.

Each wavefunction is associated with a particular energy.

Quantum numbers

Schrodinger's approach uses three quantum numbers (n, l m^l)

The Principle Quantum Numbers (n)

Tells average relative distance of electron from a nucleus

n = 1, 2, 3, 4...

A n increases, average distance of an electron from nucleus increases with it. A negatively charged electron that is closer to the positively charged nucleus is attracted to the nucleus more strongly than an electron that is farther out in space. Therefore electrons with higher values of "n" are easier to remove from an atom. All wavefunctions that have the same value of "n" are said to constitute a principal shell because those electrons have similar average distances from the nucleus.

The Azimuthal Quantum Number (l)

Value of "l" describes the shape of the region of space occupied by the electron. The allowed values of "l" depend on values of "n" and range from 0 to n-1: l = 0, 1, 2, ... , n-1

For example, if n=1, "l" can be only 0; if n=2, "l" can be 0 or 1. For a given atom, all wavefunctions that have the same values of both "n" and "l" form a subshell. Electrons of the same subshell usually have the same shape, but at oriented differently.

The Magnetic Quantum Number (m^l)

Describes the orientation of the region in space occupied by an electron with respect to an applied magnetic field. Values of m^l depend on values of "l" (can range from -l to l)

m^l = -l, -l +1, ..., 0, ..., l-1, l