 # Towards Quantum Mechanical Model of Atom

Although Bohr’s model of the atom is the most commonly used model, scientists continued to develop new and improved models for the atom. Two concepts that contributed significantly to the development of improved models were – the Dual behaviour of matter and the Heisenberg uncertainty principle. Let’s learn about these concepts in more detail.

### Suggested Videos         ## Dual Behavior of Matter

In 1924, de Broglie, the French physicist proposed that matter, like radiation, should show dual behaviour i.e. both wave-like and particle-like properties. This means that just like photons, electrons should also have a wavelength as well as momentum. From this, de Broglie gave an equation that relates wavelength and momentum of a material particle –

λ = h/mv = h/p

where h is the Planck’s constant (6.626 x 10-34), m is the mass of the particle, v is the velocity and p is the momentum of the particle.

This prediction of de Broglie’s was experimentally proven when it was found that an electron undergoes diffraction, a property characteristic of waves. This knowledge has been used to construct an electron microscope. Just as an ordinary microscope uses the wave nature of light, an electron microscope uses the wave-like behaviour of electrons.

An electron microscope achieves a magnification of about 15 million times and therefore is a powerful tool in scientific research. According to de Broglie, every object in motion has a wave-like character. The wave properties of ordinary objects are hard to detect because their wavelengths are so short.

However, experimental detection of the wavelengths associated with electrons and other sub-atomic particles is possible. For example, to find out the wavelength of a ball of mass 0.1 kg moving with a velocity of 10ms-1, we will use de Broglie’s equation –

λ = h/mv = (6.626 x 10-34)/ (0.1 x 10) = 6.626 x 10-34 m

## Heisenberg’s Uncertainty Principle

In 1927, the German physicist, Werner Heisenberg came up with the uncertainty principle as a result of concepts like the dual behaviour of matter and radiation. Source: Wikimedia

His principle states that – It is impossible to determine, simultaneously, the exact position and exact momentum (or velocity) of an electron. He gave the following equation –

∆x × ∆px ≥ h/4π   OR

∆x × ∆(mvx) ≥ h/4π   OR

∆x ×∆v≥ h/4πm

where ∆x is the uncertainty in position and ∆por ∆vis the uncertainty in momentum (or velocity) of the particle. If ∆x is small, then ∆vx is large, while if ∆vis small then ∆x will be large. Thus, is we physically measure the electron’s position or velocity, the outcome is always fuzzy or a blur.

We can best explain the uncertainty principle with an example. Suppose you are asked to measure the thickness of a sheet of paper with an unmarked ruler. Obviously, the results will be inaccurate and meaningless. To do this correctly, you will have to use a graduated or marked ruler with units smaller than the thickness of the sheet of paper.

Similarly, in order to measure the position of an electron, we need to measure with a meter stick with units smaller than the dimensions of an electron. We can observe an electron by illuminating it with “light” or electromagnetic radiation. This “light” should have a wavelength smaller than the dimensions of an electron.

The collision of the high momentum photons of light (p=h/λ) with the electrons, change the energy of the electrons. This way, we will be able to calculate the position of the electron but we will still know very little about the velocity of the electron after a collision.

## The Significance of Uncertainty Principle

• It rules out the existence of definite paths or trajectories of electrons and other similar particles. The trajectory of an object involves the location and velocity of the object. If we know the location of a body at a particular instant and we know its velocity and the forces acting on it, we can determine where the body will be sometime later.
Since it is not possible to determine the position and velocity of an electron at any given instant to an arbitrary degree of precision, it is not possible to know the trajectory of an electron.
• The effect of Heisenberg Uncertainty Principle is significant only for the motion of microscopic objects and is negligible for that of macroscopic objects. For example, if we apply the uncertainty principle to an object of mass 1 milligram (10-6 kg), then

∆v.∆x = h/4πm = 6.626 x10-34 Js/(4 x 3.1416 x 10-6kg) ≈ 10-28 m2s-1

• For milligram-sized or heavier objects, the associated uncertainties are of hardly any real consequence. The reason for this is that the value of ∆v∆x obtained is small and insignificant.
• The precise statements of the position and momentum of electrons have to be replaced by the statements of probability, that the electron has at a given position and momentum.

For a microscopic object like an electron, ∆v.∆x is much larger and the uncertainties are of real consequence. For example, for an electron with a mass of 9.11 x 10-31 kg, according to the Heisenberg uncertainty principle –

∆v.∆x = h/4πm = 6.626 x10-34 Js/(4 x 3.1416 x 9.11 x 10-31kg) = 10-4 m2s-1

This means that if you try to find the location of an electron to an uncertainty of about 10-8m, then the uncertainty ∆v in velocity would be –

10-4m2s-1/10-8m ≈ 104 ms-1

This value is so large that the idea of electrons moving in Bohr’s orbits does not hold good. This is why the precise statements of the position and momentum of electrons have to be replaced by the statements of probability, that the electron has at a given position and momentum. This is what happens in the quantum mechanical model of the atom.  ## Reasons for Failure of Bohr Model

Let’s understand the reasons for the failure of Bohr’s model –

• Bohr’s model describes an electron as a charged particle that moves around the nucleus in well-defined circular orbits. The model fails to consider the wave character of the electron.
• An orbit is a clearly defined path which is defined completely only if the position and velocity of the electron are known exactly at the same time. According to Heisenberg’s uncertainty principle, this is not possible. Therefore, Bohr’s model of the hydrogen atom not only ignores the dual behaviour of matter but also contradicts Heisenberg’s uncertainty principle.

Thus, we did not extend Bohr’s model to other atoms because of these limitations. New models were needed that could account for the wave-particle duality of matter and be consistent with Heisenberg’s principle. This led to the advent of quantum mechanics.

## Solved Example For You

Question: An electron is moving with a velocity of 2.05 x 107 ms-1. What is its wavelength?

Solution: According to de Broglie’s equation, we know that –

λ = h/mv

where λ is the wavelength, h is the Planck’s constant, m is the mass of the particle and, v is the velocity.

We also know that v = 2.05 x 10ms-1, h = 6.626 x10-34 Js, m = 9.10939 x 10-31 kg

So substituting the appropriate values, we get –

λ = h/mv = 6.626 x10-34 Js/(9.10939 x 10-31 kg) (2.05 x 10ms-1)

λ = 3.548 x 10-11 m

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