Probably the most startling revelation of Quantum Theory was the postulate that all particles have a waviness to them. Initially, when this was proposed for photons, physicist were more open to accepting it – since light was already clearly a wavy phenomenon to begin with. However, when De-Broglie applied it to all matter – including cherished solid notions of atoms, the result was a bit harder to digest.

Firmly established since then, we no longer worry about how it is possible for matter to be wavy – we just accept that it is.

Using the Waviness of Electrons to Predict Atomic Sizes

This is the first cool application of the wave like nature of particles.

Step 1-  Estimate the electron’s velocity using wavelike nature of matter

The velocity obviously depends on the charges that cause the electron to move Ze^2 where Z is the number of protons (the atomic number). That part is classical. If we want to accommodate the quantum nature of the electron, planck’s constant, \hbar, needs to figure in the equation somewhere. The only combination of Ze^2 and \hbar that has the dimensions of velocity is :

(1)   \begin{equation*} v \sim \frac{Ze^2}{\hbar} - \text{Estimate of electron velocity} \end{equation*}

Another way to relate the velocity to the atomic constants is to realize that the Potential Energy (\frac{Ze^2}{a}) is of the order of the kinetic energy mv^2.

(2)   \begin{equation*} mv^2 \sim \frac{Ze^2}{a} \end{equation*}

\frac{Ze^2}{ma} \sim \frac{Z^2e^4}{\hbar^2}

Step 2 – This also helps us get an estimate of the atomic radius a:

(3)   \begin{equation*} a \sim \frac{\hbar^2}{mZe^2} - \text {Estimate of atomic radius} \end{equation*}

So – here we have the first appearance of Quantum Theory (\hbar) in an otherwise mundane calculation – that of the size on an atomic orbit (orbital radius).

Using waviness (the wave like nature of matter) to calculate the stationary states

If electrons are really waves, let us think about the electron bound in a Hydrogen atom. It is subject to a potential energy, V (we say the particle is in a potential well). The size of the box (well), a represents the range of the potential that binds the electron. So, we have the following assumption:

Assumption 1 (completely based on the wave like nature of the particle): Since the particle must ‘fit’ in the box, the ‘wave’ must fit in the box. The dimension a of the box that contains the particle, must contain an integral number of wavelengths  (actually ‘half-wave-lengths). So – we have the equation below – the energy level L(n) is proportional to the wavelength of the ‘wave’ aspect of the particle.

The formal proof of the above assumption comes from solving the SE:

(4)   \begin{equation*} \psi\prime\prime = -k^2\psi \end{equation*}

where k^2 = 2(E - V)

For V < E (classical region), the equation has an oscillatory solution. Back to our assumption – energy levels, L(n), are proportional to the wavelength of the particle.

(5)   \begin{equation*} L(n) \sim n \frac{\lambda}{2} \end{equation*}

where n is the integral number of wavelengths  – and \lambda is the wavelength of the particle (this is where our particle’s waviness enters the picture).

De-Broglie didn’t just postulate that all matter behaves like waves, he was kind enough to provide a mathematical relationship between the matter-like nature (momentum of the particle) and the wave-like nature (wavelength) of particles.

(6)   \begin{equation*} p = \frac {\hbar}{\lambda} \end{equation*}

So – energy levels are proportional to the wavelength, the wavelength is inversely proportional to the momentum. All we need is way to relate the momentum to the energy of the particle (E-V) – and we should have our estimate. From \frac{p^2}{2m} = E-V, we have p \sim \sqrt(E-V).

Last Trick for our estimate

With the equation for p \sim \sqrt(E-V), we are almost there. Now – if we just choose V=0 as our origin (physically, this means taking the top of the well as our origin), we have an equation for p.

p \sim \sqrt(E).

Hence, our estimate for the wavelength becomes:

(7)   \begin{equation*} \lambda \sim \frac{1}{\sqrt(E)} \end{equation*}

And our estimate for our energy levels, L(n), which are just proportional to the wavelength above:

(8)   \begin{equation*} L(n) = \frac{n \lambda}{2} \sim \frac{2\pi n}{\sqrt(E)} \end{equation*}

L(n) should just be the width of the well = L

So – we have our final equation  L = \frac{2\pi n}{\sqrt(E)}  Or

(9)   \begin{equation*} E_n = \frac {\pi^2 n^2}{2 L^2} \text{Estimate of stationary states (energy levels) of a particle in a box} \end{equation*}

Summary – Matter Waves, Wave Like Nature of Matter

One does not need to solve the Schrodinger Wave Equation simply to get estimates of atomic values such as the electron velocity or atomic radius. We calculated the velocity of an electron in an atom as well as the atomic radius – with just dimensional analysis. The energy levels for a bound electron could also be calculated without solving the Schrodinger equation. These energy levels (for a particle in a box) are proportional to the quantum number n as well as inversely proportional to the width L of the well. This is in agreement with Bohr’s correspondence principle in the limit of large n. We used a combination of dimensional arguments (in the case of the electron velocity and atomic radius) and proportionality arguments (in the case of the stationary states). All of these results relied on the wave like nature of matter – and provided results that agree with conventional derivations (that DID not bring waviness into the derivation).

Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.

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Anuj Varma – who has written posts on Anuj Varma, Hands-On Technology Architect, Clean Air Activist.

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