Atoms - Part 5 - Particle Waves
In 1924, the French physicist Louis de Broglie reasoned that, as light has the properties of a particle under certain conditions, perhaps matter has wave properties. He showed that the wavelength l associated with a particle of mass m and velocity v would be
,
h being Planck’s constant. This was confirmed in 1927 by C. Davisson and L.H. Germer. They showed that metal foils diffracted a beam of electrons in the same way they diffracted x-rays. The wavelength of the electron beam was as de Broglie had predicted.
A wave moving in a circular path will be stable only if succesive waves follow the same path, i.e., the wavelength is a multiple of the circumference 2pr.
where n is an integer, 1, 2, 3, ... . Combining the above equation and l = h/mvgives
This is the same result that was arbitrarily imposed by Bohr, which now has an explanation. The chart below (Figure 9) represents a vibrating string. Each wave has a whole number of cycles per length of the string.
Figure 9.
If the string length in the graph represented the circumference of a circular orbit, the two ends would be connected. Only these whole number multiple waves would follow the same path in consecutive cycles. Anything other than whole numbers would result in interference.
In 1926 the Austrian physicist Erwin Schrodinger wrote the equation that describes the amplitude of the electron wave function y at any point in space.
where m is the mass of the electron, E is the total energy, V is the potential energy. The spatial dimensions are x, y, and z and y is the wave function. Solutions of this equation describe the orbitals that are occupied by electrons.
In quantum mechanics an electron is not regarded as a particle having a certain location at a certain time. Rather the probability is calculated of finding an electron at a certain position relative to the nucleus. There are many solutions for the wave function y. The probability density is given as |y|2. Solutions yield descriptions of quantized energy levels and orbital configurations.
There are three orbital quantum numbers that describe an orbital. They are called n, l and ml. The principal quantum number n determines the energy and size of the orbital. This is the number in 1s, 2s, 2p, .... The second orbital quantum number l determines the shape of the orbital. This detemines that the sublevel is type s, p, d, f, .... The third quantum number ml determines the orientation, for example, with respect to an applied magnetic field.
The value of the first quantum number n can be any positive integer (1, 2, 3, 4, ....). The value of the second quantum number l is from zero to n-1. Thus for n = 1, the value of l is 0. For n = 2, l can be 0 or 1. For n = 3, l can be 0, 1 or 2. For n = 4, l can have values of 0, 1, 2 or 3. The table below gives the values of the second quantum number l for n = 1, 2, 3, and 4.
n |
l |
| 1 | 0 |
| 2 | 0, 1 |
| 3 | 0, 1, 2 |
| 4 | 0, 1, 2, 3 |
The larger the value of l, the more complex the shape of the orbital. The various types of sublevels that correspond to the values of the second quantum number l are referred to as s, p, d and f orbitals.
| quantum number l | orbital type |
| 0 | s |
| 1 | p |
| 2 | d |
| 3 | f |
Each type has a characteristic spatial configuration or shape.
The value of n determines the principal energy level which includes one or more sublevels described by the values of l. A sublevel can be described by combining the first and second quantum numbers n and l . If n=1 and l=0, the sublevel is called 1s. Combined principal and sublevel names are 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, etc.
The 1s orbital shows the highest probability of finding the electron to be at the nucleus and decreasing with distance from the nucleus. A plot of the square of the wave function is on the left. On the right is a visual interpretation of the probability density.
Figure 10.
The 2s orbital has a node between an inner and outer sphere. The plot is of the square of the wave function. On the right is a probability density drawing of a 2s orbital.
Figure 11.
Below are the two radial probability curves (square of wave functions) for n=2.
Figure 12.
Below are the three radial probability plots for n=3.
Figure 13.
Below is a rough model illustrating how the principal levels and sublevels build outward from the nucleus. Only the s orbitals are actually spherical.
Figure 14.