All you need to know about: The Electron (part2)

"The task is not to see what has never been seen before, but to think what has never been thought before about what you see everyday."

 -Erwin Schrodinger

Continuing from the last post, we were obsessed with a bound electron; i.e. all we were talking about was an electron trapped in the potential of an atom. However this is not always true as electrons can also exist as free particles. Consider an electron moving along the positive X-axis having a momentum \(\vec p\) and energy \(E\). The wavefunction of such an electron is given by:

\(\psi (x,t) = e^{ik x - i\omega t}\)

where \(\kappa\) is the propagation vector given by \(\kappa = \frac{2\pi}{\lambda}\) and \(\omega\) is given by \(\omega = \frac{E}{\hbar}\). This is just a general equation for travelling waves. One special thing to be noticed here is that there is no normalization factor. Carefully trying our hands at math, we can generate a localized wave-packet which logically makes more sense and is given by the following equation:

\(\Psi (x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k) e^{i(kx-\omega(k)t)}dk\)

That is we take a superposition of plane waves to create a localized wave. The Gaussian plot for this type of wave packet is given in figure 1. 

Figure 1: A probability distribution of a Gaussian wave-packet, i.e. a wave representation of an electron (represented in red) and a particle representation of an electron (in blue; just for intuition)

Thus it is very clear that a wave representation of an electron is pretty crazy. You don't have a definite boundary. This particular fact is useful in many quantum mechanical phenomena such as quantum tunneling, interference, etc. Instead of thinking about the subatomic world as a giant 3 dimensional billard game, we have to visualize it as a constant interaction and scattering of waves in 3 dimensional world.

There is one more important aspect regarding electron which can never be neglected is the spin of the electron. Spin is as important as charge and there is a whole different field dedicated to the study of spin of particles which is called as Spintronics. So what is a spin? The angular momentum of a particle (in this case an electron) can be divided into two types; one is orbital angular momentum and other is spin angular momentum. The orbital angular is the kind of momentum we have in classical mechanics, like that of a particle moving in a closed orbit. Spin of a particle is a purely quantum and an intrinsic property which is independent of other properties. To get the intuition (not entirely correct but it still helps), consider a top spinning about its axis. Theoretically speaking there are two ways a top can spin i.e. clockwaise and counter-clockwise. The angular-momentum of this spinning top (which follows from right hand thumb rule) is thus given as a positive (upward direction) or negative (downward direction). In somewhat a similar way, an electron has 2 spins, we call it an up-spin and a down-spin electron. So where does this unique quantum property occur in our wavefunction? 

Like mentioned above, spin of a particle is itself a type of angular momentum which is itself an observable. Observables are nothing but some special operators in quantum mechanics which gives a physical meaning to our observations (examples of observables also include velocity, momentum, etc.). So this means we can express the spin angular momentum of any particle in the form of a matrix \(\hat S\). Now in quantum mechanics, an observable basically follows an eigenvalue equation and thus in this case we have:

\(\hat S \psi = \sqrt{s(s+1)}\hbar \psi\)

that is, \(\sqrt{s(s+1)}\hbar\) is the eigenvalue of spin angular momentum \(\hat S\). For an elementary particle such as an electron, the value of \(s = \frac{1}{2}\) and this is an intrinsic value. So here it becomes \(\hat S = \frac{\sqrt{3}}{2}\hbar\). So far so good, however there's a little problem. In sophisticated experiments, it is very difficult to extract information about spin of a particle directly. That is the reason why we measure the spin of any particle with respect to a fixed axis (mostly it is the z-axis). So this brings us to the following eigenvalue equation:

\(\hat S_z \psi= m_s\hbar \psi\)

where \(m_s\) takes the integer step values from \(-s,-s+1,-s+2,...,s-2,s-1,s\). Again substituting the value of \(s=\frac{1}{2}\) we finally get two values of spin of an electron with respect to z-axis which is \(m_s = +\frac{1}{2}\) and  \(m_s = -\frac{1}{2}\). The following figure explains the spin quantization of electron spin. 

Figure 2: The spin angular momentum, denoted by \(\hat S\), has a magnitude of \(\frac{\sqrt{3}}{2}\hbar\) for particles having intrinsic spin \(s = \frac{1}{2}\) (for example electron). The z-component of this vector \(\hat S_z\) has eigenvalues \(\pm \frac{\hbar}{2}\); this is what determines whether the electron is in spin "up" or spin "down" state.

Now what effect does spin has on an electron? The angular momentum (both the orbital and the spin) is a major candidate responsible for magnetic field in an atom, which deserves a different post for itself. For an electron trapped in a potential well, the complete wavefunction is now given by:

\(\Psi(r,\theta,\phi;m_s) = \psi(r,\theta,\phi) \cdot \chi(m_s)\)

where \(\psi\) is a spatial function of \(r\),\(\theta\) and \(\phi\), and \(\chi\) is purely a spin dependent wavefunction such that \(m_s\) is a quantum number taking the values \(\pm\frac{1}{2}\). The four quantum numbers \((n, l, m_l, m_s)\) play a very significant role in determining the state of an electron. The particles which carry a half integral spin are called as fermions and the ones with an integral spins are called as bosons. Clearly electron is a fermion (\(s=\frac{1}{2}\)) and hence it must obey Pauli's exclusion principle which states that "No two electron can occupy the same quantum state". What this tells, in simple words, is that no two electrons can have the same four quantum numbers \((n, l, m_l, m_s)\). If in an atom, the ground state (\(n=1,l=0,m_l=0,m_s=\pm\frac{1}{2}\)) can only accomodate two electrons, one with a positive \(\frac{1}{2}\) spin and other with a negative \(\frac{1}{2}\) spin.

To conclude this part, we have seen how an electron behaves like a wave. Whether or not in a potential well, an electron exhibits a smooth wave-like behaviour in a 3 dimensional space carrying two very important characteristics; charge and spin. We have also seen what information spin carries and how the four quantum numbers make sure that an electron follow the Pauli's exclusion principle. There are still a lot of tricks this little fellow has up its sleeves and this is exactly what we will be covering in the next part.