Some principles of quantum mechanics

Key points: Electrons can behave as particles or as waves; solution of the Schrödinger equation gives wavefunctions, which describe the location and properties of electrons in atoms. The probability of find ing an electron at a given location is proportional to the square of the wavefunction. Wavefunctions generally have regions of positive and negative amplitude, and may undergo constructive or destructive interference with one another.

In 1924, Louis de Broglie suggested that because electromagnetic radiation could be considered to consist of particles called photons yet at the same time exhibit wave-like properties, such as interference and diffraction, then the same might be true of electrons. This dual nature is called wave–particle duality. An immediate consequence of duality is that it is impossible to know the linear momentum (the product of mass and velocity) and the location of an electron (and any particle) simultaneously. This restriction is the content of Heisenberg’s uncertainty principle, that the product of the uncertainty in momentum and the uncertainty in position cannot be less than a quantity of the order of Planck’s constant

(specifically, 1/2 h^–, where h^-=h/2π).

Schrödinger formulated an equation that took account of wave–particle duality and ac counted for the motion of electrons in atoms. To do so, he introduced the wavefunction, (psi), a mathematical function of the position coordinates x, y, and z which describes the behaviour of an electron. The Schrödinger equation, of which the wavefunction is a solution, for an electron free to move in one dimension is

where me is the mass of an electron, V is the potential energy of the electron, and E is its total energy. The Schrödinger equation is a second-order differential equation that can be solved exactly for a number of simple systems (such as a hydrogen atom) and can be solved numerically for many more complex systems (such as many-electron atoms and molecules). However, we shall need only qualitative aspects of its solutions. The generalization of eqn 1.2 to three dimensions is straightforward, but we do not need its explicit form.

One crucial feature of eqn 1.2 and its analogues in three dimensions is that physically acceptable solutions exist only for certain values of E. Therefore, the quantization of energy, the fact that an electron can possess only certain discrete energies in an atom, follows naturally from the Schrödinger equation, in addition to the imposition of certain requirements (‘boundary conditions’) that restrict the number of acceptable solutions.

A wavefunction contains all the dynamical information possible about the electron, in cluding where it is and what it is doing. Specifically, the probability of finding an electron at a given location is proportional to the square of the wavefunction at that point, 2. Ac-cording to this interpretation, there is a high probability of finding the electron where 2 is large, and the electron will not be found where 2 is zero (Fig. 1.4). The quantity 2 is called the probability density of the electron. It is a ‘density’ in the sense that the product of 2 and the infinitesimal volume element dπ= d_xd_yd_z (where π is tau) is proportional to the probability of finding the electron in that volume. The probability is equal to 2d if the wavefunction is ‘normalized’. A normalized wavefunction is one that is scaled so that the total probability of finding the electron somewhere is 1.

Like other waves, wavefunctions in general have regions of positive and negative am plitude, or sign. The sign of the wavefunction is of crucial importance when two wave functions spread into the same region of space and interact. Then a positive region of one wavefunction may add to a positive region of the other wavefunction to give a region of enhanced amplitude. This enhancement is called constructive interference (Fig. 1.5a). It means that, where the two wavefunctions spread into the same region of space, such as occurs when two atoms are close together, there may be a significantly enhanced probability of finding the electrons in that region. Conversely, a positive region of one wavefunction may be cancelled by a negative region of the second wavefunction (Fig. 1.5b). This destructive interference between wavefunctions reduces the probability that an electron will be found in that region. As we shall see, the interference of wavefunctions is of great importance in the explanation of chemical bonding. To help keep track of the relative signs of different regions of a wavefunction in illustrations, we label regions of opposite sign with dark and light shading (sometimes white in the place of light shading).

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