De Broglie equation and Heisenberg's uncertainty principle

De Broglie equation

In 1924, de Broglie pointed out that electron has dual nature i.e. behaves as both matter particle and wave- likethe light. This is also known as "Dual nature of electrons"

de-Broglie equation

The de-Broglie expression for the calculation of wavelength (\(\lambda)\) of the wave associated with an electron.

For a particle showing wave nature E = hν

Here, E = energy

h = plank's constant, h = 6.62× 10^-34

ν = frequency of radiation

we know, c =ν× \(\lambda)\) where, \(\lambda)\) = wavelength

ν = \(\frac{c}{lambda}\)

E = \(\frac{hc}{\lambda}\)

From the mass-energy equivalence relation,

E = mc²

Now mc²= \(\frac{hc}{\lambda}\)

\(\lambda\) = \(\frac{h}{mc}\)

For a particle of velocity v, we replace c by v in above equation

\(\lambda)\) = \(\frac{h}{mv}\)

Above equation shows the wave nature and particle nature of matter.

Now we know, (p) = mass (m)× velocity(v)

we get, \(\lambda = \frac{h}{p}\)

Heisenberg's uncertainty principle

According to this principle, the position and momentum of a particle like an electron, proton etc cannot be simultaneously and precisely. If the position of an electron is calculated to precision then there will be an error in momentum and vice-versa.

Mathematically this principle can be expressed as:

Δx×ΔP≥ \(\frac{h}{4π}\) ...............(i)

Here,Δx = uncertainty involved in position

ΔP = uncertainty involved in momentum and

h = plank's constant

The above equation (i) shows that the product of uncertainty involved in position (Δx) and momentum (ΔP) is equal or greater than \(\frac{h}{4π}\). From above equationΔx andΔP are inversely proportional to each other this means that ifΔx is very small,ΔP would be large.

The realtion (i) is written as:

Δx×ΔP = \(\frac{h}{4π}\) .............(ii)

NowΔP = m×Δv (m = mass of the particle andΔv = uncertainty in velocity) in eqation (ii), get

Δx× (m×Δv) = \(\frac{h}{4π}\)

This principle is applicable to only small particle.

The postulates of wave Mechanical Mode of atom

Wave or quantum Mechanical Model is based on the following postulates:

1. Heisenberg uncertainty principle governs atomic particles. The term probability distribution is used to describe the motion of an electron.
2. A stream of electrons is associated with a wave whose wavelength is given by the de Broglie relationship:

\(\lambda= \(\frac{h}{mv}\)

3. Electron wave is compared to standing wave formed by a vibrating string fixed between two points.

Schrodinger time independent wave equation

When matter passes wave, then this be explained in terms of wave equation. Hence, matter wave can be explained in terms of stationary wave equation. The wave function for such wave is given by

ψ = A sin \(\frac{2\pi x}{\lambda}\) .......(i)

Differtiating the equation (i) with respect tox

\(\frac{dΨ}{dx}\) = A \(\frac{d}{dx}\) sin \(\frac{2\pi x}{\lambda}\)

\(\frac{dΨ}{dx}\) = A \(\frac{2\pi}{\lambda}\) cos \(\frac{2\pi x}{\lambda}\)

Again differentiating the above equation with respect to x

\(\frac{d}{dx}\) (\(\frac{dΨ}{dx}\)) = \(\frac{2\pi x}{\lambda}\) \(\frac{d}{dx}\) cos \(\frac{2\pi x}{\lambda}\)

or, \(\frac{d^2Ψ}{dx^2}\) = \(\frac{2\pi A}{\lambda}\)(-\(\frac{2\pi}{\lambda}\)) sin \(\frac{2\pi x}{\lambda}\)

or, \(\frac{d^2Ψ}{dx^2}\) = - \(\frac{4\pi^2}{\lambda^2}\) A sin \(\frac{2\pi x}{\lambda}\)

or, \(\frac{d^2Ψ}{dx^2}\) =- \(\frac{4\pi^2}{\lambda^2}\)Ψ .........(ii)

Here, equation (ii) is used when an electron moves in only one dimension. An elelctron may move in three dimensions x, y and z so this equation becomes.

∴ \(\frac{∂^2Ψ}{∂x^2}\) + \(\frac{∂^2Ψ}{∂y^2}\) +\(\frac{∂^2Ψ}{∂z^2}\) = - \(\frac{4\pi^2Ψ}{\lambda^2}\)WhereΨ is the wave function of electron and (\(\lambda)\) is the wave length.

Let us denote (\(\frac{∂^2Ψ}{∂x^2}\) + \(\frac{∂^2Ψ}{∂y^2}\) +\(\frac{∂^2Ψ}{∂z^2}\)) by Laplacian operator∇²

∴∇²Ψ = - \(\frac{4\pi^2Ψ}{\lambda^2}\) .................(iii)

The de - Broglie equation states that

(\(\lambda)\) = \(\frac{h}{mv}\) (where 'h' is Plank's constant, 'm' is the mass of an electron and 'v' its velocity)

Substituting the value of (\(\lambda)\) in equation (iii), we get

∇²Ψ = - \(\frac{4\pi^2 m^2v^2Ψ}{h^2}\) = 0 ...............(iv)

However, the total energy of the system 'E' is made up of the kinetic energy (K) plus potential energy (V)

E = K + V

or, K = E - V

But, K.E. = \(\frac{1}{2}\) mv²

So,\(\frac{1}{2}\) mv²= (E -V)

v²= \(\frac{2}{m}\) (E - V)

Substituting the value of v²in equation (iv), we get

∇²Ψ+ \(\frac{8\pi^2 m(E - V)Ψ}{h^2}\) = 0

This equation is Schrodinger's time independent wave equation and finds it's application in problems concerning the energy of electron.

Physical significance of wave function (Ψ)

The wave functionΨ is the measure of anamplitude of thewave. Solution which is physically possible for the wave equation must be properties like

1. Ψ must be continuous
2. must be finite
3. must be single valued
4. must be real
5. The probability of finding the electron overall space from +∞ to -∞ must be equal to one. The probability of finding an electron at a given point, x, y, z isΨ²

The probability at any point must be real quantity. Now, the value ofΨ can be positive, negative or imaginary burΨΨ^∗has a real value whereΨ^∗is a complex congugate ofΨ. IfΨ = a+ib thenΨ^∗= a-ib, where a and b are real numbers and i = \(\sqrt (-1)\).

Eigen values and Eigen fuctions

The particular values of wave fuction Ψ which yields stisfactory solution of Shrodinger's wave equation are called Eigen fuction and the values of the energy which correspond to the Eigen fuction are called Eigen values.

Quantum numbers

The integral numbers which describe the energy of an elctron in an orbit, the position of the electron from the nucleus, shape and number of orientations of an electron cloud (orbital) round its own axis and the direction of the spinning of the electron round its own axis. Thus quantum numbers are the identification numbers, for an individual electron in an atom.

They are four in numbers and a set of these four quantum number is needed to describe the electron completelyl

1. Principal quantum numbers : This quantum number denotes the size of an orbital and is denoted by 'n'. The principle shell or major energy level to which an electron belongs is denoted by this quantum number. It can have only positive value when n = 1,2,3,4. The letter used to denote are K, L, M, N respectively. If for an electron n =4, it is in the 'N' shell.
2. Azimuthal quantum number : It determines the angular momentum of electron. So also called as angular momentum quantum number. It is denoted by 'L'.

It represents the subshell to which an electron belongs. Different values of l = 0, 1, 2, 3 ,,,, are symbolised by the letter s, p, d, f respectively. The values of 'l' depends on 'n' and ranges from 0 to (n-1). The total number of different value of 'l' is equal to n. Thus, if n = 1, l = 0, if n = 2, l = 0,1 and if n = 3, then l = 0, 1, 2.

3. Magneticquantum number : It determines the different directions or orientations of subshell which are permitted.These results from the still finer split of subshell. It is denoted by the letter 'm'.The value of 'm' is equal to -1 tp+1 including 0.

4. Spin quantum number : The electron in its motion in orbit mayrotates about its own axis either in a clockwise or in an anticlockwise direction. It is denoted by letter 's' or m_sand can have only two values for an electron i.e. +\(\frac{1}{2}\) and -\(\frac{1}{2}\). For clockwise and anticlockwise spin respectively. The clockwise spin is represented by an arrow (\(\uparrow\)) and anticlockwise spin is represented by an arrow (\(\downarrow\)). An orbital can accommodate maximum two electrons with opposite spin.

Radial Distribution function

The measure of the probability of finding the electron in a spherical shell between the distances r and r+ dr from the nucleus is called radial distribution function. The probability of electron being at a distance r from the nucleus is 4πr²R².

References

Lee J.D., Concise Inorganic Chemistry,5^thedition, Chapman and Hall, London, 1996

Cotton F.A., Wilkinson G., Murillo C.A., Bochman M., Advanced Inorganic Chemistry, 3^rdedition, John WIley and Sons, Pvt., Ltd., 2007

https://en.wikipedia.org/wiki/Quantum_number

http://www.chemteam.info/Electrons/deBroglie-Equation.html

https://www.theguardian.com/science/2013/nov/10/what-is-heisenbergs-uncertainty-principle