## Heisenberg’s Uncertainty Principle

Werner Heisenberg introduced his uncertainty principle in 1927 which is a direct consequence of the dual nature of electrons.

**Heisenberg’s uncertainly principle:** it is impossible to measure simultaneously both the position and the momentum of a sub-atomic particle like an electron, accurately, at any instant of time.

**Explanation:** If at a particular moment, the uncertainty in position and the uncertainty in the momentum of a sub-atomic particle be Ax and Ap respectively, then it can be

shown that the product of these two uncertainties must be at least equal to or greater than \(\frac{h}{4 \pi}.\)

Mathematically it can be expressed as \(\Delta x \times \Delta p \frac{h}{4 \pi}\) [where h = Planck’s constant]

\(\text { or, } \Delta x \times \Delta(m v) \frac{h}{4 \pi} \quad \text { or, } \Delta x \times m \Delta v \frac{h}{4 \pi}\)**Heisenberg’s uncertainty principle**

Werner Heisenberg introduced his uncertainty principle in 1927 which is a direct consequence of the dual nature of electrons.

**Heisenberg’s uncertainly principle:** it is impossible to measure simultaneously both the position and the momentum of a sub-atomic particle like an electron, accurately, at any instant of time.

**Read and Learn More CBSE Class 11 Chemistry Notes**

**Explanation:** If at a particular moment, the uncertainty in position and the uncertainty in the momentum of a sub-atomic particle are Ax and Ap respectively, then it can be shown that the product ofthese two uncertainties must be at least equal to or greater than \(\frac{h}{4 \pi}.\)

Mathematically it can be expressed as, \(\Delta x \times \Delta p \frac{h}{4 \pi}\) [where h = Plancks constant ….[1]

⇒ \(\text { or, } \Delta x \times \Delta(m v) \frac{h}{4 \pi} \quad \text { or, } \Delta x \times m \Delta v \frac{h}{4 \pi}\)

⇒\(\text { or, } \Delta x \times \Delta v \frac{h}{4 \pi m}\)

[since m is constant]

Suppose we are going to measure simultaneously both the position and momentum of an electron in an atom. If an attempt is made to measure the position of the electron with high accuracy, then the measured value of the momentum will be less accurate and vice versa.

It should be realized that the uncertainty principle is not due to any limitation of the measuring instrument but is the consequence of the dual nature of moving particles and electromagnetic radiation (light).

Heisenberg’s uncertainty principle rules out the concept of a fixed circular path with definite position and momentum electrons in an atom as proposed by Bohr.

It should be remembered that the uncertainty principle is applicable to the position and momentum of a particle along the same axis.

Thus, if Ax represents the uncertainty in position along the x-axis then Ap must be the uncertainty in momentum along the x-axis (let it be represented as Ap ).

Thus \(\Delta x \times \Delta p_x \frac{h}{4 \pi}\)

Similarly, \(\Delta y \times \Delta p_y \frac{h}{4 \pi} \text { and } \Delta z \times \Delta p_z \frac{h}{4 \pi}\)

The uncertainty principle can also be applied to the conjugate pair, energy, and time. If Af represents the uncertainty in measuring the lifetime of a state and AE represents the uncertainty in measuring its energy in that state, then according to uncertainty principle \(\Delta E \times \Delta t \frac{n}{4 \pi}.\)

**Uncertainty Principle For Macrocophic Objects:** Theoretically, the uncertainty principle holds good for objects of all sizes, but in reality, it has no significance for macroscopic objects (big objects).

To realize this, let us consider a particle of mass of 1 mg, and for this, the approximate value ofthe product of Ax and Aw is given by, \(\begin{aligned}

\Delta x \cdot \Delta v \approx \frac{h}{4 \pi m} & =\frac{6.626 \times 10^{-34} \mathrm{~kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-1}}{4 \times 3.14 \times 10^{-6} \mathrm{~kg}} \\

& =0.53 \times 10^{-28} \mathrm{~m}^2 \cdot \mathrm{s}^{-1}

\end{aligned}\)

Thus the product of Ax and Av is extremely small. In other words, for objects of ordinary size, the uncertainties in position and momentum are very small as compared to the size of the object and the momentum of the object respectively.

Hence from the practical point of view, the values of these uncertainties may be taken as zero.

This means that the position and velocity of large objects can be measured almost accurately at any instant in time.

since in everyday life, we come across big objects only, it can be concluded that Heisenberg’s uncertainty principle has no significance in everyday life.

For a subatomic particle such as an electron, we have

\(\begin{aligned}\Delta x \cdot \Delta v \approx \frac{h}{4 \pi m} & =\frac{6.626 \times 10^{-34} \mathrm{~kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-1}}{4 \times 3.14 \times 10^{-6} \mathrm{~kg}} \\

& =0.58 \times 10^{-4} \mathrm{~m}^2 \cdot \mathrm{s}^{-1}

\end{aligned}\)

This value is quite large so neither the uncertainty in position nor the uncertainty in velocity of the sub-atomic particle can be neglected.

For example, if the uncertainty in the position of the electron is 10-4m, uncertainty in its velocity will be ~ 0.58 m.s-1, which is quite significant.

It is for this reason, that Bohr’s concept of a fixed circular path with a definite velocity needs modification.

**Electrons cannot rollers In Ihn nucleus:** The diameter of the Nucleus is of the order 10-15m. For the electron to reside within the nucleus, the maximum uncertainty in

its position should be 10-15m,i.e., Ax = 10-15m.

⇒ \(\begin{aligned}

\Delta x \times \Delta p &\frac{h}{4 \pi} \quad \text { or, } \Delta x \times m \Delta v \frac{h}{4 \pi} \text { or, } \Delta v \frac{h}{4 \pi m \times \Delta x} \\

\text { or, } \Delta v &\frac{6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}}{4 \times 3.14 \times 9.1 \times 10^{-31} \mathrm{~kg} \times 10^{-15} \mathrm{~m}} \\

& =5.7 \times 10^{10} \mathrm{~m} \cdot \mathrm{s}^{-1}

\end{aligned}\)

This value is much higher than the velocity of light (3 x 108m-s_1) which is not possible. Thus, an electron cannot reside within the nucleus.

**Reasons for the failure of Bohr’s atomic model:**

In Bohr’s model of an atom, the electron is regarded as a charged particle moving with definite velocity in a circular orbit of a definite radius.

This is not supported by Heisenberg’s uncertainty principle according to which it is impossible to determine both the velocity and the position of an electron simultaneously with certainty.

Furthermore, Bohr’s model does not take into consideration the concept of the dual nature of an electron (which is a sub-atomic particle). Due to such inherent weakness, Bohr’s atomic model lost its significance