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Atomic Structure Part IV

 

 Some Wave Properties of Particles

 The de Broglie relation

                             

                                               is a key to the wave properties of small particles.

 

The de Broglie relation states that there must be a wave associated with the particle

shown in the diagram above. The wave must fit into the space allowed to the particle. The wave can have finite amplitudes inside the boundaries of the system; the amplitudes at the walls must be zero.

a. n = 1, λ /2 = d

 

b. n = 2, 2λ =d

 

c. n = 3, 3λ = d

 

 

The examples above show some waves that meet these conditions.

 

Mathematical expressions called wave functions can be written which give the amplitude of each wave as a function of position. Accordingly, if the square of the value of the wave function at any given point is taken, this equals the probability of finding the particle at that point. Examining a wave function similar to that of figure “a” above, the particle would be found in the middle region and never near the wall.

For small particles it is not possible to define such properties as position and momentum definitively because these properties have associated probabilities. This is the Heisenberg uncertainty principle. This principle really says that the more accurately we know a particle’s position, the less accurately we can know its momentum, and visa versa. Applied to the electron, the uncertainty principle implies that we cannot know the exact motion of the electron as it moves around the nucleus. It is therefore not appropriate to assume that the electron is moving around the nucleus in a well defined orbit, as in the Bohr model.

 

The wave function that describes the waves associated with a particle allows for the calculation of the energy of the particle. For waves that satisfy the condition that the amplitude be zero at the walls (see diagrams “a”, “b”, and “c” above), the wavelengths, λ, of the allowed waves can be set by the relationship

 

                                            

where is the distance between the walls and is an integer equal to 1, 2, 3, …..

 

By combining de Broglie’s equation with the equation above, an equation that imposes a quantum condition on the momentum, , of the particle is achieved.

 

                                             or

 Since the momentum can have only certain values, this limits the energy of the particle to:

 

                                           

 

This equation tells us that a particle confined to move in a region of any length, will have certain energies allowed to it.  Those energies depend, on the mass, , of the particle, the distance, , through which it moves, and the quantum number,

 

Since = 1, 2, 3,…, the particle can have different energy values; the lowest being for =1.

 

Also, the lowest energy available depends on the mass and the interval length to which the particle is confined. From the equation you can see that the energy varies inversely with the mass. Every time the mass is decreased by a factor of two, the energy is doubled.

The minimum energy also varies inversely with the square of the length through which the particle can move. If the length is halved then the quadruples the energy of the particle.

 

Erwin Schrödinger was the first person to successfully apply the concept of the wave nature of matter to an explanation of electronic structure.  He developed another equation called the Schrödinger wave equation which allowed him to determine the energy levels and wave properties of the hydrogen atom, other atoms and molecules. For a one partice system the equation takes the form:

 

                                           

where is the mass of the particle,  and V are its total and potential energies, respectively,  is Plank’s constant, and the first three terms are partial derivatives with respect to x, y, and z, of the wave function, , to be associated with the particle.

 

The wave properties of matter form the foundation for the theory called wave mechanics, which serves as the basis of all current theories of electronic structure. The term quantum mechanics is also used because wave mechanics predicts quantized energy levels.

 

The theory of quantum mechanics tells us that in the atom, electron waves are standing waves. The electrons can have different waveforms or wave patterns. Each of these waveforms which are called orbitals, is described in quantum mechanics by a mathematical expression called a wave function.  The wave function can be used to describe the shape of the electron wave and its energy.  Energy changes within an atom are simply the result of an electron changing from a wave pattern with one energy to a wave pattern with a different energy.

 

When an atom is in its most stable state, the ground state, the electrons of the atom have waveforms with the lowest possible energies.  The shapes of the wave patterns are important because the theory tells us that the amplitude of a wave at any particular place is related to the likelihood of finding the electron there.

 

Wave mechanics tells us that the three-dimensional waves, orbitals, can be characterized by three integer quantum numbers n, l, and ml.   

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