Category: Chandra & Satya

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Electrons move around the nucleus in circular orbits just like how planets move around sun.

Is that how electrons really move?

That’s what Bohr proposed. It was a brilliant model because it correctly predicted the allowed energies of hydrogen.

But quantum mechanics revealed something even more fascinating.

Ohh… is there much more to it?

Yes, an electron is not described as a tiny particle travelling along a well-defined circular path.

Instead, quantum mechanics describes the electron by a wavefunction.

A wavefunction?

Yes …I can recollect, electron has a dual nature just like light. And de Broglie introduced matter waves.

But if electron behaves like a wave why can’t we simply draw their paths?

Because waves are not described by trajectories. They are described by wavefunctions.

But how does a wavefunction tell us where the electron is?

It doesn’t tell us a trajectory.

Instead, the wavefunction contains all the information that quantum mechanics allows us to know about the electron.

From it, we can calculate the probability of finding the electron at different locations.

Hmm….…$\psi$ represents wavefunction and $|\psi|^2$ represents its probability density. But where this wavefunction comes from?

By solving Schrodinger equation.

I could recollect I solved couple of Schrödinger equations to get different wavefunctions. Now I am able to connect back.

Exactly, Do you remember how do we solve them?

A potential landscape is given and we substitute the potential expression in either Schrödinger time dependent equation or Schrödinger time independent equation based on whether V(r) or V(r, t).

Then we solve the equation and get solutions.

Exactly… and in the case of atom, potential is V(r) and is electrostatic potential. When we solve it, we get

Notice that the Schrödinger equation doesn’t give just one solution. It gives many different mathematical functions.

The allowed solutions of the Schrödinger equation are wavefunctions.

The simplest one is $\psi_{100}(r)=constant\, e^{-r/a_0}$

This is 1s orbital. Next, we have 2s orbital:

$\psi_{200}(r)=constant\,\left( 2-\frac{r}{a_0}\right)e^{(-r/2a_0)}$

Notice that this solution has a completely different mathematical form.

Next, observe these two plots

The first two wavefunctions ($\psi_{100}\, , \psi_{200}$) depends only on the distance from the nucleus. But this one also depends on direction.

We have,

Unlike the $1s$ and $2s$ wavefunctions, this one depends on both $r\,$and $\theta$. This different mathematical form eventually gives rise to the familiar dumbbell-shaped orbital.

In general we can write:

$R_{n\,l}$ is the radial part, determined by the Coulomb potential. (radial part plot).

$Y_m^l\left(\theta\,,\phi\right)$are the spherical harmonics, which describe the angular dependence. (angular part plot)

This separation is possible because the Coulomb potential depends only on the distance from the nucleus:

So, each allowed solution of the Schrödinger equation is an atomic orbital.

Exactly.

An orbital is simply one of the allowed wavefunctions or equivalently, one of the allowed quantum states of the electron in the atom.

So we can say:

Coulomb Potential –> Schrödinger Equation –> Many Allowed Wavefunctions $\psi_{100}\,\,, \psi_{200}\,\,,\psi_{210}\,\,,\psi_{211}\cdots$ 

Wait a minute.

We’ve been calling these orbitals $1s\,,2s$ and $2p.$

Are these just convenient names given by physicists?

Or is there a reason why one is called $1s$ and another $2p$?

Excellent question.

These names are not arbitrary.

Every allowed solution of the Schrödinger equation naturally comes with a unique set of quantum numbers.

The familiar names $1s\,,2s$ and $2p\,$are simply a convenient way of representing those quantum numbers.

So quantum numbers are not assigned afterwards.

They naturally emerge while solving the Schrödinger equation?

Precisely.

They uniquely characterize each allowed quantum state.

Then I have one last question.

If every electron is identified by its quantum state…

can two electrons occupy exactly the same one?

That’s one of the most profound questions in quantum mechanics.

The answer is the Pauli Exclusion Principle.

But before we move on, remember the biggest takeaway from today’s discussion: an orbit is a path, but an orbital is a wavefunction obtained by solving the Schrödinger equation.

And that’s where our journey continues.

This is episode 3 of the series Why matter exists?

If squeezing the electron is energetically expensive, shouldn’t the electron just keep moving farther and farther away?

It would if kinetic energy is the only energy in the atom.

Imagine an electron very far from proton. Now you try to bring it closer to proton. Does proton help you or oppose you?

They are oppositely charged, so the proton helps by attracting the electron.

So the system releases energy as they come together. That is why the potential energy becomes more and more negative.

Where does the released energy come from?

From the potential energy of the electron–proton system. As the electric field does work on the electron, the system loses potential energy.

I see… if the PE were positive instead… the electron would not remain bound to the proton and there would be no atom.

But wait, bringing the electron closer also increases the uncertainty in its momentum.

Exactly, greater momentum uncertainty means greater kinetic energy.

  (click here to see how   )

Even the graphs can teach more.

Look at this graph:

What do you notice?

The kinetic energy shoots up very rapidly as the radius decreases.

How does potential energy changes with radius?

The PE decreases…

Ummm….. But, as the electron comes closer, the potential energy should increase as it is proportional to 1/r …..right?

Be careful, this is a very common mistake.

The potential energy has actually decreased.

The negative sign completely changes the interpretation.

As the electron falls toward the proton, the electron–proton system releases energy and moves to a lower-energy state. That is why the potential energy becomes more negative.

Never ignore the minus sign. It tells you the direction in which the energy is changing.

I understand each graph individually. But the atom has both types of energy at the same time. So… which one does Nature actually care about?

These graphs describe the two competing tendencies inside the atom. Nature doesn’t minimize kinetic or potential energy individually. It minimizes their sum, the total energy:

This gives us the following graph.

I can see the lowest point on the graph. But how can I find its exact value?

Whenever a smooth curve reaches a maximum or minimum, its slope becomes zero,

Ah… Solving the equation gives us:

And that’s the famous Bohr radius!

Numerically: 0.529 Å.

So the size of the hydrogen atom is not an arbitrary number.

Exactly.

It emerges naturally from the balance between quantum kinetic energy and electrostatic attraction.

This article is Episode 2 of the Satya–Chandra series Why Matter Exists.

I’ve been thinking about our last conversation.

You said the electron doesn’t collapse because the lowest allowed state is

But that doesn’t satisfy me.

Why not?

Because it feels like a mathematical trick.

We simply wrote

And declared

Then we concluded the electron cannot reach the nucleus.

Nature doesn’t work because of our equations.

Our equations are supposed to describe nature.

Why does nature forbid it?

This is where modern quantum mechanics gives us a deeper answer.

The electron is not merely a tiny particle. It also behaves like a quantum wave.

According to Heisenberg’s uncertainty principle:

Here, and represent the uncertainties in the electron’s position and momentum, respectively.

Now imagine if we confine an electron to a region as tiny as a nucleus, about   across. That means, according to Heisenberg’s principle, such a tiny uncertainty in position demands an enormous uncertainty in its momentum and therefore an enormous kinetic energy.

Satya:

Enormous??? Can we actually calculate how impossible it is for the electron to stay inside the nucleus?

Chandra:

Certainly,

Whoo… that’s too high.

Yes,

to squeeze an electron into the nucleus, nature demands an enormous kinetic energy.

Ah! So the electron doesn’t stay outside because the proton stops attracting it…it stays outside because squeezing it into the nucleus would require an enormous amount of energy!

But one mystery still remains…

If the electron cannot fall into the nucleus, why does it settle at one particular distance?

The answer lies in one of the most remarkable numbers in physics—the Bohr radius.

Episode 3: Why is the Bohr Radius About 0.53 Å ?

This article is Episode 1 of the Satya–Chandra series Why Matter Exists.

Electrons repel electrons. Protons repel protons. How is matter stable then?

Oh! I see. It’s because a proton attracts an electron.

But… if so, then why doesn’t electron fall into the nucleus?

Umm… according to classical physics it should collapse.

Yet, here we are.

An orbiting electron is an accelerating charge. Accelerating charges radiate energy. Radiating energy means losing energy. Losing energy means spiralling inward.

Ah…. Bohr…..

Exactly, Bohr proposed something outrageous. He proposed that an electron could occupy only certain allowed orbits. Only those orbits whose angular momentum is an integral multiple of are allowed, i.e.

And remember,

Therefore, there is no allowed orbit below the ground state.

Oh! I see. That means an electron is not allowed to possess just any energy. Only certain discrete energy levels are permitted. It can jump only between these allowed levels.

So there is a lowest allowed energy state n=1, called ground state. There is simply no allowed state corresponding to n=0.

But why is there a lowest one at all?

Why couldn’t nature have allowed one more orbit, even closer to the nucleus?

An excellent question.

Bohr answered where the electron can exist.

He never really explained why those orbits exist in the first place.

That question would eventually lead physicists to one of the deepest principles in quantum mechanics.

We will chase that mystery in our next conversation.