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Heisenberg's Uncertainty Principle - Formula, Mathematical Forms, Practise Problems and Frequently Asked Questions (FAQs)

Heisenberg's Uncertainty Principle - Formula, Mathematical Forms, Practise Problems and Frequently Asked Questions (FAQs)

Imagine a blind man walking into a room which consists of just a chair and a sack of marbles kept far away from each other. What are the chances that he would collide with both the chair and the sack of balls? 

Knowing that two objects are kept far away from each other, the chances of the blind man colliding with both of them are quite minuscule. 

However, if I was to pose a similar question on the odds of the blind man hitting either of the two objects, the answer would be somewhat higher in magnitude.

Classical physics presupposes that exact simultaneous values can be assigned to all physical quantities, but it was something Heisenberg somehow could not make peace with it. 

But, according to quantum mechanics, the more precisely the position of a particle is given, the less precisely one can say what its momentum is, and vice versa.

If we want to measure any object, the x-component of its momentum with uncertainty (p), at the same time we can’t measure its x-position (x) more accurately than pxh4. Heisenberg’s uncertainty principle is a basic theory that explains why it is impossible to measure more than one quantum variable simultaneously. 

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Table of contents

  • Heisenberg uncertainty principle for an electron 
  • Heisenberg uncertainty principle formula
  • Another mathematical form of Heisenberg uncertainty principle
  • Practice problems
  • Frequently Asked Questions-FAQs

Heisenberg uncertainty principle for an electron 

For position measurement, light projected on electron and observe the reflected light using a microscope carefully, the uncertainty in position is given by the wavelength of the light used for conducting the experiment. So to calculate the position with minimum deviation, it is mandatory to use light having a shorter wavelength. But for momentum measurement light having a longer wavelength is used because it imparts less momentum (kick) to electrons (the shorter the wavelength, the higher the energy and the larger the change in momentum).

Thus, at the moment when the position of the particle is accurately known, Heisenberg argued, its momentum can’t be accurately known. At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum.

Heisenberg replaced the concept of definite orbits with the concept of probability.

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Heisenberg uncertainty principle formula

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m = mass of particle

h= Planck's constant

If the position of a particle is measured precisely, i.e. x 0 then p. If the momentum of the particle is measured precisely. i.e. p0 then x . 

Another mathematical form of Heisenberg uncertainty principle

We know,

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(from Newton's second law, rate of change of momentum is equal to F (Force Displacement = = Energy )

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Where, F = force

 E = uncertainty in energy

t = uncertainty in time

Practice problems

Q 1. Uncertainty in position is four times the uncertainty in momentum. So, uncertainty in velocity is

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Answer: (D)

we know, 

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Q 2. Calculate the uncertainty in the position of an electron if the uncertainty in its velocity is 1

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Answer: (A)

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Q 3. The uncertainty in measuring the speed of a particle is 1 m/sec. The least uncertainty in measuring its position will be

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Answer: (A)

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Q 4. Which mathematical expression correctly represents Heisenberg uncertainty principle 

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Answer: (D) 

all expressions represent Heisenberg uncertainty principle
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Frequently Asked Questions-FAQs

Q 1. Where could a real-life analogy of Heisenberg's Uncertainty principle be validated?
Answer: The rollercoaster ride found in amusement parks serves as an analogy for how the uncertainty principle works at scales much smaller than this. When the rollercoaster car reaches the peak of the hill, we could take a snapshot and know its location. But the snapshot alone would not give us enough information about its speed. As the rollercoaster car descends the hill, we can measure its speed over time but would be less certain about its position. The uncertainty principle is a trade-off between two complementary variables, such as position and speed.

Q 2. What violates the Heisenberg uncertainty principle?
Answer:
If an object travelling through spacetime can loop back in time in a certain way, then its trajectory can allow a pair of its components to be measured with perfect accuracy, violating Heisenberg’s uncertainty principle.

Q 3. What is the significance of Heisenberg's uncertainty principle?
Answer: Heisenberg’s uncertainty principle is not an instrumental error, rather it is a fundamental error. Heisenberg’s uncertainty principle rules out the existence of a definite path of electrons. Heisenberg’s uncertainty principle introduced the concept of probability of finding the electrons.

Q 4. How did the uncertainty principle affect Bohr’s atomic model?
Answer: according to Bohr path of the electron was defined but after Heisenberg's uncertainty principle it was proved that the position (path) of the electron can’t be predicted accurately.

Related Topic Links:

Millikan’s oil drop experiment

Discovery of neutrons

Thomson Atomic Model

Blackbody radiation

Quantum Mechanical Model

De Broglie's Hypothesis

 

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