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Quantum Mechanical Atomic Model

Last Updated : 21 Dec, 2021
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Schrödinger used the electron’s wave-particle duality to design and solve a difficult mathematical equation that precisely represented the behaviour of the electron in a hydrogen atom in 1926. The solution to Schrödinger’s equation yielded the quantum mechanical model of the atom. The quantization of electron energy is required to solve the equation. In contrast to the Bohr model, quantization was merely assumed with no mathematical basis.

Remember that under the Bohr model, the electron’s exact path was limited to very well-defined circular orbits around the nucleus. The quantum mechanical paradigm, on the other hand, is a significant departure from that. Wave functions, which are solutions to the Schrödinger wave equation, only indicate the chance of detecting an electron at a certain position surrounding the nucleus. Electrons do not travel in simple circular orbits around the nucleus.

The positioning of electrons in the quantum mechanical model of the atom is referred to as an electron cloud. This is how the electron cloud looks:

Consider laying a square piece of paper on the floor with a dot in the centre to symbolize the nucleus. Now, grab a marker and continuously drop it onto the paper, making little marks at each spot the marker lands. The overall arrangement of dots will be about round if you drop the marker enough times. There will be more dots near the nucleus and fewer dots as you travel away from it if you aim pretty efficiently toward the centre. Each dot represents a potential electron location at any given time.

There is no way to tell where the electron is due to the uncertainty principle. An electron cloud has varying densities: high densities where the electron is most likely to be and low densities where the electron is least likely to be.

To more precisely specify the geometry of the cloud, it is common to refer to the region of space where the electron has a 90% chance of being found. This is known as an orbital, a three-dimensional region of space that indicates where an electron is likely to be found.

Bohr’s model of hydrogen

The quantized emission spectra suggested to Bohr that electrons may perhaps only exist within the atom at specific atomic radii and energies. Remember that quantized refers to the fact that energy can only be absorbed and released within a certain range of permitted values rather than with any value at all. 

Bohr deduced an equation from this model that accurately predicted the various energy levels in the hydrogen atom, which corresponded directly to the emission lines in the hydrogen spectrum. Bohr’s model also predicted the energy levels in other one-electron systems, such as He+. It did not, however, explain the electrical structure of atoms with more than one electron. While some physicists attempted to modify Bohr’s model to make it more applicable to more complex systems, they eventually determined that a whole different model was required.

Quantum mechanical model of the atom

Standing waves

One key flaw in Bohr’s model was that it considered electrons as entities with well-defined orbits. Schrödinger postulated that the behaviour of electrons within atoms could be explained mathematically by considering them as matter waves, based on de Broglie’s concept that particles may display wavelike behaviour. This paradigm, which is the foundation of modern atomic understanding, is known as the quantum mechanical or wave mechanical model

The fact that an electron in an atom can only have certain permitted states or energies is analogous to a standing wave. Along with the standing wave, there are points of zero displacements, known as nodes. The nodes are denoted by red dots. Because the string in the animation is fixed at both ends, only specific wavelengths are permitted for any standing wave. As a result, the vibrations have been quantized.

Shapes of atomic orbitals

So far, we’ve been looking at spherical s orbitals. As a result, the key factor influencing an electron’s probability distribution is its distance from the nucleus, r. Other types of orbitals, such as p, d, and f orbitals, however, include the electron’s angular position relative to the nucleus in the probability density. The p orbitals have the shape of dumbbells and are directed along with one of the axes—x, y, or z. With the exception of the d orbital, which resembles a p orbital with a donut going around the centre, the d orbitals have a clover form with four potential orientations.

Features of Quantum mechanical model of the atom

  1. An electron’s energy is quantized, which means that an electron can only have certain particular energy values.
  2. The quantized energy of an electron is the allowable solution of the Schrödinger wave equation and is the outcome of the electron’s wave-like features.
  3. The precise position and momentum of an electron, according to Heisenberg’s Uncertainty principle, cannot be calculated. So the only chance of finding an electron at a given place is |ψ|2 at that point, where ψ denotes the wave-function of that electron.
  4. The wave-function (ψ) of an electron in an atom is referred to as an atomic orbital. An electron occupies an atomic orbital whenever it is described by a wave function. There are multiple atomic orbitals for the electron since it can have various wave functions. Every wave function or atomic orbital has a form and associated energy. All of the information about an electron in an atom is contained in its orbital wave function ψ, which may be extracted using quantum mechanics.
  5. The likelihood of finding an electron at a given place within an atom is proportional to the square of the orbital wave function, i.e., |ψ|2 at that point. |ψ|2 represents the probability density and is always positive.

Schrodinger Wave equation

The Schrodinger wave equation describes the behaviour of a particle in a force field or the change in a physical parameter over time. The equation’s creator, Erwin Schrödinger, was even awarded the Nobel Prize in 1933. The Schrodinger wave equation is a mathematical expression that describes the energy and position of an electron in space and time while accounting for the electron’s matter wave nature inside an atom. It is based on three factors. They are the classical plane wave equation, Broglie’s matter-wave hypothesis, and energy conservation.

The Schrodinger equation describes in detail the shape of the wave functions or probability waves that influence the motion of some smaller particles. The equation also illustrates how external influences affect these waves. Furthermore, the equation employs the energy conservation idea, which provides information about the behaviour of an electron linked to the nucleus.

Wave Function

Quantum physics, often known as quantum mechanics, is a branch of science concerned with the study and behaviour of matter and light. In quantum physics, the wave function can be used to depict a particle’s wave characteristics. As a result, the quantum state of a particle can be characterised using its wave function.

Properties of Wave Function

  • There can only be one value for ψ and it must be continuous.
  • The energy can be easily calculated using the Schrodinger equation.
  • To establish a probability distribution in 3D space, the wave function equation is applied.
  • If there is a particle, the probability of discovering it is one.
  • A particle’s properties that can be measured should be known.

Physical Significance of Wave Function

The wave function has no physical meaning because it is not a quantity that can be observed. It is, on the contrary, complicated. The wave function is written as ψ(x, y, z, t) = a + ib, while the complex conjugate is written as ψ*(x, y, z, t) = a – ib. The sum of these two is the probability density of detecting a particle in space at any given time. However, ψ2 is the physical interpretation of the wave function because it gives the chance of locating a particle at allocation at a certain time. 

Sample Questions

Question 1: When is the energy of an electron regarded as zero?


When an electron is at an infinite distance from the nucleus, its energy is considered to be zero. The force of attraction between the electron and the nucleus is essentially non-existent at that point. As a result, its energy is regarded as zero.

Question 2: What were the weaknesses or limitations of Bohr’s model of atoms?


  1. It was unable to describe the spectra of multi-electron atoms.
  2. It couldn’t account for the Zeeman and Stark effects.
  3. It was unable to describe the shape of molecules.
  4. It did not follow Heisenberg’s uncertainty principle.

Question 3: Briefly describe the quantum mechanical model of the atom.


  1. It was based on Heisenberg’s uncertainty principle and matter’s dual behaviour.
  2. The energy of electrons in an atom is quantized, which means that it can only have specific values.
  3. The existence of quantized electronic energy levels is a direct effect of electrons’ wave-like features.
  4. The precise position and velocity of an electron in an atom cannot be known at the same time.
  5. The orbitals are filled in increasing energy order.

Question 4: Explain the Aufbau principle.


The orbitals in the ground state of the atoms are filled in the sequence of their rising energy. In other words, electrons occupy the lowest-energy orbital available to them initially and then go on to higher-energy orbitals after the lower-energy orbitals are occupied.

Question 5: What is Hund’s rule of maximum multiplicity?


Electron pairing in p and d orbitals is not possible unless each orbital in a particular subshell contains one electron or is single occupied.

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