Quantum Mechanical Model of an Atom: Class 11 Chemistry Notes

The quantum mechanical model of the atom states that electrons do not move in fixed paths. Instead, they exist in regions, known as orbitals. In these orbitals, there is a greater probability of finding electrons.  

There is a lot to unpack here, and from the earlier Chemistry textbook section on Towards the Quantum Mechanical Model of the Atom, you may recall the basics. 

In the atomic model of Bohr, his idea of electrons moving like planets in fixed circles around the nucleus (much like the sun) turned out to be wrong. This was especially true for atoms with more than one electron. Experiments showed that electrons don’t follow neat tracks at all. Instead, they hang out in regions of space called orbitals, more like fuzzy clouds where they’re most likely to be found.

De Broglie added to this by suggesting electrons act like waves, and inside the atom, they fit as standing waves around the nucleus. 
Heisenberg then showed that it's impossible to find an electron’s position and momentum at the same time. That means exact orbits around the nucleus cannot exist. 
Then, Schrödinger created equations that described these orbitals properly. That gave us the modern picture of electrons as probability clouds instead of tidy circles.

In short, this electron behaviour accounts for what we know as quantum mechanics. In the current syllabus of the Chemistry textbook, there is also a brief mention on the difference between classical and quantum mechanics in section 2.6. In case you would want to learn more, see below. 

Quantum Mechanics vs. Classical Mechanics

In your exams, you may be asked about the features of the atom's quantum mechanical model. Here are the most important ones. 

Quantised Energy Levels in Atoms

One of the biggest ideas in the quantum mechanical model is that electrons can only occupy specific energy levels. They can’t just take on any random value. Bohr introduced this concept where he mentions that the electrons' energy is quantised. Like steps on a staircase. This comes from the wave-like behaviour of electrons, which Schrödinger’s wave equation helped explain. Each energy level is tied to the standing wave patterns that electrons can form around the nucleus.

Atomic Orbitals and Electron Probability

Instead of following neat circular orbits like Bohr once imagined, electrons live in regions of space called orbitals. You can think of an orbital as the 'address' of an electron. Each orbital has a set energy and can hold a maximum of two electrons. The shape of these orbitals is based entirely on solutions to Schrödinger’s equation. They could be spherical, dumbbell-shaped, or more complex. We can never know the exact location of an electron at any moment. That's why probability becomes a mathematical way to understand it. And, the square of the wave function, written as |ψ|², tells us the likelihood of finding the electron in a certain spot.

Role of the Uncertainty Principle in Quantum Mechanics

Heisenberg’s Uncertainty Principle mentions we know that we cannot know an electron's exact position and momentum at the same time. Unlike the Bohr model, electrons don’t travel on fixed paths. Instead, they exist as fuzzy clouds of probability. This principle is basic and too important to the quantum mechanical model.  It explains why we can only talk about where an electron is likely to be, never exactly where it is.

The dual nature of matter posits that particles, such as electrons, exhibit both particle-like and wave-like properties, with their wavelengths being dependent on their momentum. 

(You could further refer to the Law of Conservation of Momentum in Class 11 Physics.)  

So, in your Chemistry textbook, you will find out that Louis de Broglie (1924) proposed that electrons, like light, have wave-particle duality, with wavelength given by: $\lambda =\frac{h}{mv}$ where $h$ is Planck's constant, $m$ is mass, and $v$ is velocity. This was verified by Davisson and Germer's electron diffraction experiments.

Mathematical Significance of the Dual Nature of Matter

• For an electron accelerated by a potential $V$ : $\lambda =\frac{h}{\sqrt[]{2meV}}$
• In Bohr's orbits, de Broglie's wavelength satisfies: $2\pi r=n\lambda ,\mathrm{}mvr=\frac{nh}{2\pi }$
• Significant for microscopic particles, negligible for macroscopic objects.

Heisenberg's Uncertainty Principle states that the position and momentum of a particle cannot be simultaneously measured with exact precision. That introduces an inherent uncertainty in quantum systems.

Werner Heisenberg (1927) formulated: $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{h}{4\pi },\mathrm{}\mathrm{\Delta }x\cdot \mathrm{\Delta }v\ge \frac{h}{4\pi m}$ where $\mathrm{\Delta }x$ is position uncertainty, $\mathrm{\Delta }p=m\mathrm{\Delta }v$ is momentum uncertainty. Similarly, energy and time uncertainty: $\mathrm{\Delta }E\cdot \mathrm{\Delta }t\ge \frac{h}{4\pi }$

This principle limits classical precision in electron measurements. The NCERT Solutions for Chapter 2 Chemistry help you practice numerical problems based on the formula for Heisenberg uncertainty.  

Mathematical Significance of the Heisenberg Uncertainty Principle

• Challenges Bohr's fixed orbits, favouring probabilistic electron clouds.
• Impacts spectroscopic and quantum calculations.

JEE problems may test uncertainty calculations or conceptual implications. Have a look at the JEE Main analysis. 

The Schrödinger wave equation describes electrons as three-dimensional waves, with solutions providing probability distributions of electron positions.

Your NCERT textbook mentions that Erwin Schrödinger (1926) proposed:

$\frac{{\partial }^2\mathrm{\Psi }}{\partial x^2}+\frac{{\partial }^2\mathrm{\Psi }}{\partial y^2}+\frac{{\partial }^2\mathrm{\Psi }}{\partial z^2}+\frac{8{\pi }^{2}m}{h^2}(E-V)\mathrm{\Psi }=0$

or

${\nabla }^2\mathrm{\Psi }+\frac{8{\pi }^{2}m}{h^2}(E-V)\mathrm{\Psi }=0$

where $\mathrm{\Psi }$ is the wave function, $E$ is total energy, $V$  is potential energy, and ${\mathrm{\Psi }}^2$ represents electron probability density.

Key Features of the Schrödinger Equation

• ${\mathrm{\Psi }}^2$ indicates the likelihood of finding an electron, defining atomic orbitals.
• Solutions yield quantum numbers describing electron states.

JEE questions may focus on orbital probability or quantum numbers. Check for such in previous JEE Mains papers. 

In general, quantum numbers are integers or half integers that specify an electron's energy, orbital shape, orientation, and spin in an atom.

In Class 11 Chemistry, you learn that there are four quantum numbers which define electron states in the following way. 

• Principal Quantum Number ( $n$ ): Determines energy and size ( $n=\mathrm{1,2},\dots$ ). Max electrons: $2n^2$ .
• Azimuthal Quantum Number ( $l$ ): Defines orbital shape ( $l=0$ to $n-1$ ). Subshells: $l=0$ (s), 1 (p), 2 (d), 3 (f). Max electrons: $2(2l+1)$ .
• Magnetic Quantum Number $\left(m_l\right)$ : Specifies orbital orientation $\left(m_l=-l\right.$ to $\left.+l\right)$ . Total values: $2l+1$ .
• Electron Spin Number ( $m_s$ ): Indicates electron spin ( $+1/2$  or $-1/2$ ).

Key Features of Quantum Numbers

• Orbitals are designated as $nl$ (e.g., $2\mathrm{p}:n=2,l=1$ ).
• Spin angular momentum: $\sqrt[]{s(s+1)}\frac{h}{2\pi }$ .

JEE problems often involve assigning quantum numbers or calculating electron capacities.

Orbitals are regions of high electron probability. And, quantum numbers determine their shapes, and that influences chemical bonding.

There are four types of orbital shapes defined in the Structure of Atoms chapter. These are

• s-orbital $(l=0)$ : Spherical, no nodes for 1 s , radial nodes $=n-l-1$ .
• p-orbital $(l=1)$ : Dumbbell-shaped, three orientations $\left(p_x,p_y,p_z\right)$ , one nodal plane.
• d-orbital $(l=2)$ : Double dumbbell, five orientations (e.g., $d_{xy},d_{z^2}$ ), two nodal planes.
• f-orbital $(l=3)$ : Complex, seven orientations, three nodal planes.

Key Features of Orbitals

• Radial probability: $R=4\pi r^{2}dr{\mathrm{\Psi }}^2$ .
• Size increases with $n:1s<2s<3s$ .