All you need to know about: The Electron (Part 1)
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| Electron waves for hydrogen atom with n=10, l=2 and m=0 Source: https://www.falstad.com/qmatom/ |
"All that glitters may not be gold, but at least it contains free electrons."-John Desmond Bernal
The matter around us is composed of particles. These particles, on a macroscopic scale are divisible. Take for instance a cup of hot tea. This tea is made from various ingridients which can be divided; tea leaves are a special component of our household tea, which are further made of small grains of leaves which are composed of molecules. These molecules are nothing but a large grouping of atoms, and atoms make up everything in the observable universe. It was earlier belived that atoms are indestructible and indivisible, meaning that one cannot further divide atoms and that these are the true fundamental building blocks of matter. But the experiments in the 19th and 20th century went further in answering what an atom is made up of. Scientists found out 3 fundamental particles which gives rise to an atom; Proton, Neutron and Electrons. It is true that neutrons and protons are further divisible, however in this article we will restrict ourselves to electrons.
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Let's have a formal introduction of electron. Everything we observe in the nature with naked eyes can be cateogorized into three types: Positively charged objects, Negatively charged objects and Neutral objects. The neutral objects are just a composition of large number of positive and negative charges which cancel each other. Similarly, as discussed above, an atom is made up of protons (positively charged particles), neutrons (neutral) and electrons (negatively charged particles), and thus an atom is always neutral because in order to attain stability the nature makes sure that you always have equal number of electrons and protons in an atom under normal conditions. Electrons are these little tiny particles which orbit the nucleus (nucleus is made up of protons and neutrons) as shown in Figure 1. The mass of an electron is 9.1 x 10-31 kg. This is approximately 1837 times lighter than a proton (1.67 x 10-27 kg) which itself is a billion million million times lighter than a grain of a sand (1 x 10-6 kg). Just for the record, you need approximately 60 million grains of sand to reach somewhere close to the mass of one human body. Did you know that 99.99% our Universe is made up of elements like Hydrogen (one electron-one proton atom) (75%) and Helium (two electrons two protons and two neutrons atom) (24.9%9)? A current esmitate says that there are about 1080 atoms in the observable universe. With that in consideration, there are approximately more than 1.25 x 1080 eletrons in the observable universe.
These electrons are the ones responsible for the flow of electric current in our circuits. The electrons are also responsible for the emission of light from the hot objects. When we heat any material, we are just providing an external source of energy for the atom. An electron never likes to be bound to an orbit so as soon as this electron receives energy, it jumps to a higher orbit. But it cannot stay there for longer duration and thus it goes back to its original position by emitting light. The light is such an interesting topic, it deserves an entire big series of posts of its own. The point I want to bring here is that many of the physicial processes we see around happens due to presence of electrons.
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| Figure 2: 3D Electron Orbitals (electron clouds). |
The above description of an atom where the electrons are revolving around a central nucleus is pretty crude and yields satisfactory results for simple physics and chemistry calculations. However, there is a big flaw which cannot be ignored. If we assign the electrons an orbit and consider them like point particles we are violating the Heisenberg's Uncertainty Principle which says that "one cannot determine a particles momentum and position with high precision". This is like a reminder to us, we cannot simply consider electrons as tiny little dots at all times. This means our electrons behave like particles at one moment and like particles some other time. This also states that instead of well defined orbits, we should be having electron clouds and regions in space corresponding to high and low electron probability density sites as shown in Figure 2. In this figure the nucleus is assumed to be at the origin and the balloon shaped clouds, which are called as orbitals, denote the electron clouds where the probability of finding an electron is high. This picture of atom is crucial in understanding chemistry as it deals with bonding of orbitals (i.e. overlapping of orbitals) and formation of compounds.
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| Figure 3: An electron (Orange) trapped in a potential well in a one-dimensional case. |
The probability of finding an electron in a given region of space is determined by the Schrodinger's equation. We consider an example of a bound state electron, i.e. an electron trapped in a potential well of some finite width 'L' and depth 'V' (Figure 3). By the term 'trapped' we mean that the only way an electron can get out of the well is by providing an external energy such that the perturbation effects will allow the electron to achieve energy greater than the well and then it can escape. The wavefunction of such a bound electron is given by the eigenvalue equation:
\(\hat H\psi = E\psi\)
Where \(\hat H\) is the Hamiltonian of the system given by
\(\hat H = -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi \)
is the total energy of the system. The first and the second term of the equation correspond to the Kinetic Energy and the Potential Energy of the electron respectively. The E is called as the energy eigenvalue and this tells us about the energy of the electron present in the system. The equation for E is given by:
\(E=\frac{n^2\pi^2\hbar^2}{2mL^2}\)
Here n is an integer, m is the mass of electron, \(\hbar\) is the reduced Planck's constant and L is the width of the well. The quantity that interests us the most is the wavefunction \(\psi\) which is given by:
\(\psi = \sqrt{\frac{2}{L}}sin(\frac{n\pi x}{L})\)
When we plot this function we realise that the electron is like a standing wave inside a bound potential. The figure 3 plots the wavefunction only for the ground state electron i.e. for n=1 however we can similarly generate higher order wavefunctions. The region of peaks (in this example, at the centre of the well) is the region of maximum probability. One can find the maximum occurrence for an electron is at the centre of the well. An important thing to note here is that the example we just encountered had to do with a single electron in 1-dimensional world. Since we live in a 3-dimensional world our Schrodinger's equation will take the form:
\(\frac{1}{R}\frac{d}{dr}(r^2\frac{dR}{dr}) + \frac{1}{Ysin\theta}\frac{\partial}{\partial \theta}(sin\theta\frac{\partial Y}{\partial \theta}) + \frac{1}{Ysin^2\theta}\frac{\partial^2 Y}{\partial \phi^2} + \frac{2\mu r^2}{\hbar^2}(E+\frac{Ze^2}{4\pi\epsilon_o r}) = 0\)
Where
\(\psi(r,\theta,\phi) = R(r)\cdot Y(\theta,\phi)\)
The 3 dimensional wavefunction for the electron (i.e. the solution of the above equation \(\psi\)) in terms of spherical coordinates is then finally given by:
\(\psi(r,\theta,\phi) = NR_{n,l}(r)P_l^m(cos\theta)e^{im\phi}\)
Here N is just the normalization constant which is a contribution of the radial, polar and azimuthal equations. \(P_l^m(cos\theta)\) is the Legendre Polynomial which provides a correlation between \(l\) and \(m\) and \(R_{n,l}(r)\) is the solution of the radial part of the differential equation given by:
\(R_{n,l}(r) = R_\infty (r)b_0exp(\frac{\mu Ze^2r}{2\pi\epsilon_o\hbar^2n})\)
The term \(b_0\) contains the \(l\)-dependence and \(n\) is the a set of integers. The terms \(n\), \(l\) and \(m\) are called as quantum numbers and it is used in defining an atomic structure and determination of the energy or other properties exclusive to an atom. We will touch upon this concept in some later post. Below is a snapshot of a simulation for a hydrogen electron in 3 dimensions. I highly recommend everyone to try this hands on experience of how does the electron cloud shape changes for the different values of quantum numbers.
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| Figure 4: Hydrogen atom for the state n=3, l=1 and m=0. Source: http://www.falstad.com/qmatom/ |
So in this article we have built an intuition as to what exactly an electron is. From the fundamental indivisible particle, we have seen how we now define an electron like a wave. We have also discussed the Schrodinger's equation for a 1 dimensional and a 3 dimensional case and realized that an electron shows a probabilistic nature. Of course there are a lot of things to discuss about the electron, however, we are not going to over burden anyone with more knowledge in this post. So stay tuned for the part 2 in the upcoming series.
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