Introduction

For a scientist, knowing that matter acts as a wave is helpful only if one knows something about that wave. In the year 1926 Erwin Schrödinger introduced a mathematical equation whereby, if one knows the potential energy acting on an object, one can compute the wave function for that object. Heisenberg had previously introduced a numerical formalism for performing quantum mechanics calculations, without explicitly including the concept of waves. It was later shown that, although the approaches of Schrödinger and Heisenberg looked very different, they made exactly the similar predictions. In practice, the Schrödinger formalism is more helpful for explaining the problem being studied, and the Heisenberg methodology permits for more straightforward computation. Accordingly, a mixture of the 2 approaches is typically utilized in modern quantum chemistry. Once we know the wave function of the atom or molecule under learn, we can compute the properties of that atom or molecule.

Definitions of Wave Function

The wave function is a mathematical function describing the wave. For example,  y(x)  =  A   sin(kx)  might  be  the  wave-function  for  a  one- dimensional  wave,  which  exists  along  the  x-axis.  Matter waves are three-dimensional; the relevant wave function depends on the x, y, and z coordinates of the system being studied (and sometimes on time as well). We conventionally label the wave function for a three- dimensional object as ψ(x, y, z). Since electrons are fermions, they are subject to the Pauli Exclusion Principle,  which  states  that  no  two  fermions  can  occupy  the  same quantum state at once. This is the fundamental basis of the configuration of electrons in an atom: once a state is occupied via an electron, the next electron must occupy a different quantum mechanical state.

In an atom, the stationary states of an electron's wave function (i.e. the states which are eigenstates of the Schrodinger equation [HΨ = EΨ, where H is the Hamiltonian] are termed to as orbitals, by analogy through the traditional picture of electron particles orbiting the nucleus. In general, an orbital represents the region where an electron can be expected to exist (with ~90per cent probability). Such states have 4 principal quantum numbers:  n, l, ml and ms, and by the Pauli principle no two electrons may share the similar values for all four numbers. The 2 most significant of such are n and l.

Energy Levels (n)

The 1^st quantum number n corresponds to the overall energy and therefore as well the distance from the nucleus of an orbital, therefore sets of states by the similar n are often termed to as electron shells or energy levels. Such aren't piercingly delineated zones inside the atom, but rather fuzzy-edged regions inside that an electron is likely to be found, due to the probabilistic nature of quantum mechanical wave functions.

Any discussion of the shapes of electron orbitals is essentially imprecise, since a specified electron, regardless of that orbital it occupies, can at any instant be found at any distance from the nucleus and in any direction due to the uncertainty principle. Though, the electron is much more likely to be originating in certain regions of the atom than in others. Specified this, a boundary surface can be drawn so that the electron has a high probability to be create anywhere inside the surface, and all regions outside the surface have low values. The exact placement of the surface is arbitrary, but any logically compact determination must follow a pattern specified via the behavior of ψ2, the square of the wave function. This boundary surface is what is meant when the "shape" of an orbital is mentioned.

Nature of Wave Function

The wave function ψ is a sort of amplitude function. It isn't an observable quantity but ψ ψ* is observable where ψ*is the conjugate of ψ. If the function is equal to its conjugate than ψ2 provides the probability of discovering the electron inside a specified volume. Where ψ2 is high the probability of finding the electron is high. Where ψ2 is low, the electron is rarely originated. The information enclosed in ψ2 can be likened to the information contained in the in a dart board. The pattern of holes illustrates that there is a high probability that the dart will land inside a circle enclosing many dart holes and a low probability for a circle of equal area but enclosing few dart holes. The density of dart holes provides probability distribution.

The wave function ψ must obey sure mathematical conditions, that is, ψ(x) be single valued, finite and continuous for all physically possible values of x. It must be single valued since the probability of discovering the electron at any point x must have only one value. It cannot be finite at any point since this would mean that it is fixed exactly at point.

This is inconsistent by the wave properties. The criterion of continuous wave function is useful in the selection of physically feasible solutions for the wave function.

Wave Properties of Matter

The thought that particles could exhibit wave properties just as wave could exhibit particle properties got serious attention in the year 1920s when Louis de Broglie suggested in his  doctoral thesis that particles in motion possessed a wavelength an could hence act as waves. De Broglie obtained an expression for the wavelength of a particle via drawing an analogy to the behaviour of photons (Porlie, 1987). The association between the energy of the photon and its wavelength is specified as

εγ  = hc/λ

Where  h  is  Planck's  constant,  c  the  velocity  of  light  and  λ  the wavelength the photon. The energy of the a particle and its momentum (p) are related by

εγ  = pc

The relation between the momentum and the wavelength of a photon is

p = h/ λ

For a particle of mass m, the momentum is given as

p = mv 

According to de Broglie the momentum of the photon and that of the particle should be equal. That means the wavelength of the particle should be

λ = h/p = h/mv

The equation symbolizes a completely dissimilar view of the behaviour of particles such as the electron, which had been thought to possess only corpuscular properties. The wave nature of particles and the quantization of angular momentum provide calculation procedures that are intimately related. Though, different difficulties need different approaches. If we consider a easy system of a particle moving in a circular orbit (path) about a fixed point, we can illustrate the relation between the wave and angular momentum procedure.

As revealed in equation, the wavelength of the wave is associated to the mass and velocity of the particle by the expression

λ = h/mv

As the particle shifts in a wavelike manner there is the possibility of both constructive and destructive interferences, in that the waves either cancel out themselves or compliment themselves to become a bigger wave through bigger intensity (Porlie, 1987).

If there is constructive interference the relation between the radius of the orbit and the wavelength is

2πr = nλ 

Where n is an integer which tells the number of waves that fit in the orbit.

mvr = nh/2π

The LHS of equation (3.2.8) is the angular momentum Iω. By implication, the wave-fitting   process provides angular momentum restriction.

Iω = nh/2π (n= 1, 2, 3, 4,...)

Instance

What is the wavelength of a 100eV electron?

Initially, the speed of the electron must be computed from the kinetic energy. The relation between the kinetic energy and speed is given as ? = (½)mv2. Therefore, speed is given as

v = (2 ε/m)½

= [(2x100eVx1.602x10^-19JeV)/(99.110x10^-31kgx1Jkg^-1m^-2s²)] ½

= 5.930x106 ms^-1

On substituting into equation (3.2.6) we obtain

λ = h/mv

= (6.626x10^-34Js)(109nm m^-1)/(9.110x10^-31kg)(5.930x106 ms^-1)(1Jkg^-1m^-2s²)

= 0.12 nm

This effect implies that a 100eV has a wavelength that is inside the range of atomic size.

Application of De Broglie Conjecture

The confirmation of the de Broglie conjecture via experiment made it possible to be applied in dissimilar electron diffraction of crystalline solids in the similar way as X-rays. The diffraction patterns attained through electron are extremely similar by those of X-rays. Consequently, the techniques of electron diffraction and neutron diffraction have turned out to be very important tools in the study of molecular structures. This observation implies that the ideas of wave and particle blend mutually at the atomic level. That is, particles exhibit the properties of waves and waves exhibit those of particles (Porlie, 1987). Similar calculations have led to the consequences exposed in Table.

Table:  The de Broglie Wavelength of some Typical Particles

Heisenberg's uncertainty principle

Since the electron is described as a wave then its position at a given time cannot be specified with complete accuracy because it isn't localized in space. Using electromagnetic radiation of short wavelength we can find out the position and momentum of an electron. The position of the electron cannot be determined more closely than ±λ, where λ is the wavelength of the incident photons. The uncertainty in the position of the electron, Δx, is therefore approximately equal to λ. It has been established above that the momentum of photon of wavelength λ is equal to h/λ. Therefore, the uncertainty in momentum of the electron, ?p = h/λ. The product of the uncertainty in the momentum and position of the electron is

(Δp)(Δx) ≈ (h/ λ)( λ) ≈ h

This is a simplified statement of Heisenberg's uncertainty principle which indicates that it is impossible to simultaneously measure both the momentum and the position of a particle so that the product of the 2 uncertainties is less than h/4π. This is mathematically symbolized as

(Δp)(Δx) ≥ h/4π

The implication of the Heisenberg's uncertainty principle is that any effort to minimise the uncertainty in any of the two parameters will definitely lead to increase in the uncertainty of the other such that the relation in equation is maintained (Porlie, 1987).

Instance

An electron travels through the speed of 3x106 m s^-1.What is the minimum uncertainty in its momentum if se assume that its position is calculated inside 10 per cent of its atomic radius.  Do the similar computation for a 0.03kg ball travelling at a speed of 25 ms^-1. Assume that the uncertainty in position of the ball is equal to the wavelength light of 600 nm.

From uncertainty principle the minimum uncertainty in the momentum, (Δp), is

(Δp) = h/4πΔx

= (6.63x10^-34Js)/(4π)(0.01 nm)(10^-9 m nm^-1)(1 Jkg^-1m^-2s²)

= 5 x 10^-24 kg m s^-1

The momentum of the electron is p = mv = (9.1 x10^-31)(3x106  m s^-1) = 2.73x 10^-24 kg ms^-1

The minimum uncertainty is about twice as large as the momentum itself. Such a large uncertainty makes it difficult to predict the position of the particle at a later time.

For the ball, the momentum is p = (0.03kg)(25 ms^-1) = 0.75 kg m s^-1. The uncertainty in the momentum of the ball is (Δp) = h/4πΔx

= (6.63x10^-34Js)/(4π)(600 nm)(10^-9 m nm^-1)(1 Jkg^-1m^-2s²)

 = 5.01 x10^-39kg m s^-1.  

This uncertainty in momentum is approximately negligible compared to the value of the momentum as to be entirely undetectable. The dissimilarity between the 2 cases illustrates why macroscopic objects perform differently from microscopic ones. This is why the orbits of macroscopic objects as the planets and moon can be computed to elevated accuracy and the occurrences of eclipses are predictable even into the far future. On the other hand, as a consequence of wave nature of microscopic objects, it is not possible to make these predictions about their motion (Porlie, 1987).

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