In 1926, Austrian physicist Erwin Schrödinger (1887–1961) used the wave-particle duality of the electron to develop and solve a complex mathematical equation that accurately described the behaviour of the electron in a hydrogen atom.

The quantum mechanical model of the atom comes from the solution to Schrödinger’s equation. Quantization of electron energies is a requirement to solve the equation. This is unlike the Bohr model, in which quantization was assumed with no mathematical basis.

Table of Contents

## Key Points of Quantum Mechanical Model

- Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength λ, given by the following equation: λ= mv/h
- Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves.
- Schrödinger’s equation, H^ ψ=Eψ, can be solved to yield a series of wave function ψ, each of which is associated with an electron binding energy, EEE.
- The square of the wave function, ψ2 squared represents the probability of finding an electron in a given region within the atom.
- An atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time.
- The Heisenberg uncertainty principle states that we can’t know both the energy and position of an electron. Therefore, as we learn more about the electron’s position, we know less about its energy and vice versa.
- Electrons have an intrinsic property called spin, and an electron can have one of two possible spin values: spin-up or spin-down.
- Any two electrons occupying the same orbital must have opposite spins.

## Features of Quantum Mechanical Model

There are some features of this model:

1. The energy of an electron is quantized, i.e. an electron can only have certain specific values of energy.

2. The quantized energy of an electron is the allowed solution of the Schrödinger wave equation. And it is the result of the wavelike properties of the electron.

3. As per Heisenberg’s Uncertainty principle, an electron’s exact position and momentum cannot be determined. So the only probability of finding an electron at a position can be determined, and it is |ψ|a that point where ψ represents the wave function of that electron.

4. An atomic orbital is the wave function (ψ) of an electron in an atom. Whenever a wave function describes an electron, it occupies an atomic orbital. As an electron can have many wave functions, there are many atomic orbitals for the electron. Because every wave function or atomic orbital have some shape and energy associated with it. All the information about the electron in an atom is stored in its orbital wave function ψ, and quantum mechanics makes it possible to extract this information out of ψ.

5. The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function. i.e., | ψ |2 at that point. | ψ |2 is known as probability density and is always positive.

## Postulates of Quantum Mechanical Model

The postulates of Schrödinger’s atomic model are the following:

1. The electrons behave like standing waves distributed in space according to the wave function Ψ.

2. The electrons move within the atom in describing orbitals. Because these are areas where the probability of finding an electron is considerably higher. The referred probability is proportional to the square of the wave function Ψ 2.

3. The electronic configuration of the Schrödinger atomic model explains the periodic properties of atoms and the bonds they form.

4. However, the Schrödinger atomic model does not contemplate the spin of electrons, nor does it consider the variations in the behaviour of fast electrons due to relativistic effects.

## The Quantum Mechanical Model of the Atom

The following terms are related to this model.

### 1. Standing Waves

A major problem with Bohr’s model was that it treated electrons as particles that existed in precisely defined orbits. However, based on de Broglie’s idea that particles could exhibit wavelike behaviour, Austrian physicist Erwin Schrödinger theorized that the behaviour of electrons within atoms could be explained by treating them mathematically as matter waves. This model, which is the basis of the modern understanding of the atom, is known as the quantum mechanical or wave mechanical model.

There are only certain allowable states or energies that an electron in an atom can have is similar to a standing wave. So, We will briefly discuss some properties of standing waves to get a better intuition for electron matter waves.

You are probably already familiar with standing waves from stringed musical instruments. **For example**, when a string is plucked on a guitar, the string vibrates in the shape of a standing wave such as the one shown below. Notice that points of zero displacement, or nodes, occur along with the standing wave. The nodes are marked with red dots. Since the string in the animation is fixed at both ends, this leads to the limitation that only certain wavelengths are allowed for any standing wave. As such, the vibrations are quantized.

### 2. Schrodinger’s Equation

On a very simple level, we can think of electrons as standing matter waves that have certain allowed energies. Because Schrödinger formulated a model of the atom that assumed the electrons could be treated at matter waves. While we won’t be going through the math in this article, the basic form of Schrödinger’s wave equation is as follows:

H^ ψ=Eψ

where, ψ is called a wave function;

H, with, hat, on top is known as the Hamiltonian operator;

and E is the electron’s binding energy.

Therefore, Solving Schrödinger’s equation yields multiple wave functions as solutions, each with an allowed value for E

Interpreting exactly what the wave functions tell us is a bit tricky. Due to the Heisenberg uncertainty principle, it is impossible to know for a given electron both its position and its energy. Since knowing the energy of an electron is necessary for predicting the chemical reactivity of an atom, chemists generally accept that we can only approximate the electron’s location.

How do chemists approximate the location of the electron? The wave functions that are derived from Schrödinger’s equation for a specific atom are also called atomic orbitals. Chemists define an atomic orbital as the region within an atom encloses where the electron is likely to be 90% of the time. In the next section, we will discuss how electron probabilities are determined.