Heisenberg uncertainty principle-definition, formula & significance

 Heisenberg principle-definition, equation, importance & limitation

Imagine you were walking along a road casually, and suddenly a double-decker bus crossed you. You can guess its location and speed without much hassle. Meanwhile, you saw a tiny ant is passing nearby. And I am damn sure you stare seriously to find the ant on the floor. Why so? If the same puzzle blinks in a school student's mind, he can convince himself as the bus is large and the ant is small. So, there is a considerable difference in our vision based on their size to locate them properly. Surprisingly, you won't realize the scientific proposal behind it until you understand Heisenberg’s uncertainty principle.

What does Heisenberg's principle clarify for us?

It is impractical to measure the position and momentum of a nano-sized particle simultaneously without error. It is an uncertainty theory by the German physicist Werner Heisenberg who proposed it in 1927. It is appropriate for only microscopic particles such as electrons, protons, and neutrons at sub-atomic (or) atomic and molecular levels. It is due to their wave-particle duality at the quantum level. Likewise, it is an exception to macroscopic objects whose wave character is negligible in motion.

Heisenberg’s principle states that the product of uncertainties in the measurement of conjugate quantum variable pairs whose dimensional unit is joule sec is always greater than or equal to a fixed value. And the value of the constant quantity is h/4π. As a result, the product of an error in the simultaneous measurement of the position and momentum of a quantum particle is h/4π.

Formula:

(Δx) x (Δp) ≥ ^h⁄_4π

(Or) (Δx) x (mΔv) ≥ ^h⁄_4π

(Δx) x (Δv) ≥ ^h⁄_4πm

Where,

Δx = Uncertainty in the position

Δv = Uncertainty in the velocity

h = Planck’s constant

The above equation shows an inversely proportional relationship between the errors. Consequently, finding the exact position of a nanoparticle leads to uncertainty in its momentum when measured together. Alternatively, if a particle's velocity is known accurately, its location is fixed with error. Though we can measure the position or momentum of matter waves accurately without error, their concurrent measurement gives errors in both.

(Δx) x (Δv) = ^{(6.626 x 10-34 Joule second)}⁄_{(4 x 3.142)m Kg}

(Δx) x (Δv) = ^{(0.527 x 10-34)} ⁄ _m

Where,

m = mass of the particle

From the above, the product of uncertainties in position and velocity relates inversely to the body's mass. As a result, the measurement error is prominent in microscopic particles rather than macroscopic particles.

It also applies to the concurrent measurement of energy and time for a microscopic particle in a particular energy level. If the system is in a fixed state for so long, it is impractical to determine its energy with much less error. Conversely, when the system stays in a specific energy state for a finite time, more precise is its energy calculation.

(ΔE) x (Δt) ≥ ^h⁄_4π

Overview of Heisenberg's uncertainty principle:

Every moving object possesses a wave character following the de-Broglie hypothesis. It proved both the wave-particle nature of materials. As a result, it generated uncertainties in the simultaneous measurement of conjugate physical pairs. Heisenberg's principle discussed the possibilities of finding the position and momentum of any moving object without error. It is attainable for massive materials but not for tiny invisible particles.

The position of a particle is most accurate when the undulation of the wave is high. Due to increased waveforms with different wavelengths, their velocity is unpredictable. It poses uncertainty in the determination of the momentum of the object. Following the de-Broglie principle, it is clear that the finite wavelength value of any matter wave displays indefinite momentum.

Contrarily, when the wavelength of an object is indistinct, its momentum is defined accurately. But the enhanced disturbance in the particle's motion leads to higher impreciseness in its location evaluation.

Finally, when the particle is a wave packet, its position and momentum concurrent computation shows an error for both.

Theory of Heisenberg’s principle:

Heisenberg’s principle reveals practical restrictions in determining the position and momentum of an invisible nano-sized particle concomitantly with absolute accuracy. The visibility of an object is the primary criterion for positioning it. It is the magic of light, which hides or highlights an item from human eyes. Light radiations of appropriate wavelengths, when falling on an item, they are visible to the naked human eye only when the reflected light ray wavelength lies between 380 nm to 750 nm. Hence, adjusting the wavelength of incident light radiation is a prevalent technique for fixing an object’s position.

The reflected light of massive objects falls in the visible region for almost all incident electromagnetic radiations. Consequently, the evaluation of the position of a macroscopic object is independent of incident radiation wavelength. A moving macro and semi-macro object hold a high momentum due to its heavy mass. As a result, when high-energy light is stuck with a moving material of appreciable size, the photon’s energy cannot alter the object’s momentum. Therefore, knowing the position and velocity combinedly for heavy objects is possible with much less error. Keeping it in mind, we can conclude that Heisenberg’s principle fails for routine life materials.

But, it is controversial in the case of invisible particles. The sub-atomic and atomic particles are visible only if shorter wavelength electromagnetic radiations are employed. If we utilize energetic gamma rays to locate a nano-sized particle, the photon energy alters its momentum due to the negligible mass of the microparticle. In this situation, Heisenberg’s uncertainty will come into play where the product of errors in the concurrent computation of both location and linear momentum of the nanoparticle is greater than or equal to a constant number, and it is 0.527 x 10^-34 joule second.

Experimental verification of Heisenberg's uncertainty principle:                            

Calculating the position and momentum of a moving electron both at the same instant of time outlines the impreciseness in their measurement. And it is a piece of practical evidence that elucidates Heisenberg's uncertainty.

Error in the determination of the exact position of moving electron:

Irradiate the revolving valence shell electrons of a cesium atom with gamma rays. The fast-moving gamma photons eject the outer shell cesium electrons. The emitted electrons then bounce into the microscope with a light flash. It helps us to locate the exact position of the moving electron. But the laws of optics restrict determining the electron location precisely. As a result, Δx is the error in the measurement of electron position that is equal to the microscope resolving power.

(Δx) = ^λ ⁄ _2sinθ

Error in the determination of the exact momentum of moving electron:

The energetic gamma photon passes on some part of its kinetic energy to the emitted electron during their collision. As a result, the moving electron recoils into the microscope with increased velocity. It enhances the colliding electron linear momentum.

The wavelength of gamma photon = λ₁

Momentum of gamma photon (m₁v₁) = h/ λ₁   (According to the de-Broglie equation)

The new wavelength of motioned electron = λ

The moving electron enters the microscope either through the x-direction or y-direction. As momentum is a vector quantity, its value is direction specific.

Momentum imparted to the mobile electron along X-axis is

mv = ^h⁄_λ1 + (-^{h sinθ} ⁄ _λ)

mv = ^h⁄_λ1 - ^{h sinθ} ⁄ _λ

Momentum imparted to the mobile electron along y-axis is

mv = ^h⁄_λ1 + ^{h sinθ} ⁄ _λ

Error for change in momentum of the electron along the x-direction is

Δp_x = (^h⁄_λ1 + ^{h sinθ} ⁄ _λ) - (^h⁄_λ1 - ^{h sinθ} ⁄ _λ)

Δp_x = ^{2h sinθ}⁄_λ

Heisenberg's principle applies to the enumeration of the position and momentum of moving objects along the same axis. Consequently, Δx is the error in the measurement of mobile electron position in the x-direction, and Δp_x is uncertainty in the computation of the momentum of electrons in the same direction.

(Δx) x (Δp_x) = (^λ⁄_{2 sinθ} ) x  (^{2h sinθ}⁄_λ   )

From the above equation, we can say that by increasing the wavelength of the moving electron or decreasing the semi-vertical angle of cone light of the electron, the accuracy in measurement of electron position rises, but at the cost of uncertainty in the electron momentum. On solving the above equation, we get;

(Δx) x (Δp_x) = h

Hesse Kennard and Hermann Wey, in 1928, derived the standard deviation of position and momentum of microscopic objects and concluded as below;

(σ_x) (σ_y) ≥ ^ħ⁄₂

(σ_x) (σ_y) ≥ ^h⁄_4π

Where, ħ is reduced Planck’s constant and its value is h/2π.

All these experimental facts suggest the impossibility of concurrent measurements of the position and momentum of microscopic particles with absolute accuracy.

The physical significance of Heisenberg's uncertainty principle:

(1) Expressed the errors in the measurement of physical conjugate pairs for microscopic materials:

Heisenberg developed a mathematical relation to knowing the inaccuracies in the enumeration of position and momentum or energy and time of micro sized matter. The product of these errors is equal to half of the reduced Planck's constant value. Hence, it played a prominent role in the development of quantum mechanics.

According to the Heisenberg uncertainty rule, a nano sized particle is visible only when using photons of shorter wavelengths rather than longer wavelengths. It brings a variation in their momentum to define their location precisely. Conversely, by utilizing less energetic light radiations, we can calculate the microscopic particles' velocity with much less error, but their position is uncertain as they are invisible. Thus, it is impractical to define the trajectory of a moving electron.

(2) Described the location of an electron in an atom:

Even Bohr's atomic model and Rutherford's atomic theory insisted on the electron's position in the extra nuclear part. But it is not mathematically proven until the Heisenberg uncertainty principle.

We know, the radius of nucleus = 10^-15 m

Therefore, the error in the calculation of electron position = 10^-15 m

Mass of electron = 9.1 x 10^-31 Kg

By applying Heisenberg’s uncertainty equation, we get;

Δx (mΔv) = ^h⁄_4π

By substituting the values of h, π, m and Δx in the above equation, we have;

Δv = ^{(0.527 x 10-34Joule second)}⁄_{(10^-15 m) (9.1 x 10^-31 kg)}

Δv = 0.0579 x 10¹² m/sec

Δv = 579 x 10⁸ m/sec

We know the speed of light is 3 x 10⁸ m/sec. But our calculated electron’s velocity is much higher than the speed of light which is impossible. So, the electron cannot reside inside the nucleus of an atom.

Now let us check for the outermost shell of an atom:

The size of an atom = 10^-10 m

Hence, the uncertainty in the computation of electron’s position = 10^-10 m

By applying Heisenberg’s uncertainty equation, we get;

Δx (mΔv) = ^h⁄_4π

By substituting the values of h, π, m and Δx in the above equation, we have;

Δv = ^{(0.527 x 10-34Joule second)}⁄_{(10^-10 m) (9.1 x 10^-31 kg)}

Δv = 0.0579 x 10⁷ m/sec

Δv = 579 x 10³ m/sec

It implies the electrons revolve continuously in the valence shell of an atom with less velocity than light speed.

Limitation of Heisenberg's uncertainty principle:

It is appropriate for particles at the atomic and sub-atomic levels of micro size. Since these objects are invisible to the naked human eye, we don't find any use for Heisenberg's principle in our daily life. Visible substances are massive whose velocities do not vary by hitting with a photon of either shorter or longer wavelengths. Besides, their minimal wave character helps to measure their position and momentum together accurately. As a result, we cannot employ Heisenberg's principle on large-sized objects that we see in our routine life.

For example: Calculate the error in the velocity of an iron ball weighing 100 kg with a size of 2 m.

Mass of the iron ball = 100 kg

Uncertainty in the position of the iron ball = 2 m

By applying Heisenberg's uncertainty equation, we get;

Δx (mΔv) = ^h⁄_4π

By substituting the values of h, π, m and Δx in the above equation, we have;

Δv = ^{(0.527 x 10-34Joule second)} ⁄ _{(2 m) (100 kg)}

Δv = 0.002635 x 10^-34 m/sec

Δv = 2635 x 10^-40 m/sec

It proves the velocity of the iron ball is negligible. Hence, we do not observe the ball moving like a wave rather than straight.

Multiple choice questions and answers on the Heisenberg uncertainty principle:

1. The Heisenberg uncertainty principle is successful for________________

1. Hefty bodies
2. Nano-sized particles
3. Celestial objects
4. Electromagnetic radiations

Answer:
Nano-sized particles
Explanation:
The Heisenberg uncertainty principle is a prominent resolution in quantum mechanics that deals with microparticles at atomic and sub-atomic levels.

2. Which of the following pairs are conjugate physical variables whose dimension is joule second in the SI system?

1. Position and momentum
2. Energy and time
3. Both a and b
4. Power and time

Answer:
Both a and b
Explanation:
The SI units of position and momentum are meters and kilogram meters per second. Alternatively, we can adjust it to joule second. Similarly, the SI units of energy and time are joule and second. Consequently, option 3 is the correct answer.

3. The product of an error in the simultaneous measurement of the position and velocity of a microscopic particle is ____________________ quantity

1. Fixed
2. Variable
3. Indefinite
4. Alpha-numerical

Answer:
Variable
Explanation:
The multiplicative product of the concurrent computation of the position and momentum of a microscopic particle is equal to the ratio of Planck's constant and 4π. Yet the error in the simultaneous measurement of position and velocity is the same as the ratio of Planck's constant and 4πm. It changes with the mass of the body. Hence, a variable quantity.

4.Uncertainty in position and momentum of an electron varies___________

1. Directly
2. Inversely
3. Infinitely
4. Equally

Answer:
Inversely
Explanation:
The product of impreciseness in the combined measurement of location and linear momentum of an electron varies inversely. Therefore, a more precise measurement of the position of the electron results in increased uncertainty in the enumeration of its velocity.

5. If an error in the measurement of Δp for a sodium proton is zero, the unpredictability in its position is equal to_________________

1. Zero
2. ^h⁄_4π
3. Infinite
4. 0.5

Answer:
Infinite
Explanation:
The formula for calculating the errors in the position and momentum of a sodium proton by the Heisenberg uncertainty principle is (Δx)(Δp) = ^h⁄_4π
(Δx) = ^h⁄_4π(Δp)
(Δx) = ^h⁄_4π0
(Δx) = ∞

6. Which of the following property is the genesis of the Heisenberg uncertainty principle?

1. Fine structure splitting
2. Photoelectric effect
3. Wave-particle duality
4. Interference of matter waves

Answer:
Wave-particle duality
Explanation:
The wave nature of microscopic matter generates uncertainty in calculating the conjugate physical pairs at once with absolute accuracy.

7. Heisenberg uncertainty principle relates to__________________

1. Transverse waves
2. Longitudinal waves
3. Electromagnetic waves
4. Matter waves

Answer:
Matter waves
Explanation:
Heisenberg's principle deals with moving nanoscopic bodies. And waves that associate with these motioned particles are matter waves.

8. Which of the following object in motion exhibits trajectory by the Heisenberg uncertainty principle?

1. A hydrogen electron
2. Dust particle
3. Car
4. Radioactive radiations

Answer:
A hydrogen electron
Explanation:
According to the Heisenberg principle, a mobile macroscopic substance exhibits a definite motion path. It is due to its preciseness in the concurrent measurement of its location and velocity.

9. The product of impreciseness in knowing the position and momentum of Mercury neutrons is equal to______________________

1. 0.273 x 10^-15m/sec
2. 4.567 x 10^-27joule second
3. 0.527 x 10^-34joule second
4. 1.234 x 10^-34joule second

Answer:
0.527 x 10^-34joule second
Explanation:
The product of an error in the combined computation of the location and linear momentum of Mercury neutrons is equal to h/4π. By substituting the values of Planck's constant and π, it will be 0.527 x 10^-34joule second.

10. Which of the following is the correct Heisenberg uncertainty formula?

1. (Δx) (Δv) = ^ħ⁄_2m
2. (Δx) (Δp) = ^ħ⁄_2m
3. (ΔE) (Δv) = ^ħ⁄₂
4. (Δx) (Δt) = ^ħ⁄_2mv

Answer:
(Δx) (Δv) = ^ħ⁄_2m
Explanation:
The product of simultaneous measuring of uncertainty in position and velocity is equal to the ratio of reduced Planck's constant and 2m.

11. The uncertainty in the position of an electron and Lithium atom is similar. If the error in the computation of electron momentum is 9 x 10⁵, then calculate the Δp_x value of the Lithium atom.

1. 3 x 10⁵
2. 27 x 10⁵
3. 9 x 10⁵
4. 0.333 x 10⁵

Answer:
9 x 10⁵
Explanation:
By applying the Heisenberg uncertainty principle to the revolving electron, we have; (Δx) (Δp_x) = 0.527 x 10^-34
Error in computation of electron momentum =9 x 10⁵
(Δx) (9 x 10⁵) = 0.527 x 10^-34
(Δx) = ^{0.527 x 10-34}⁄_{9 x 10⁵} = 0.0585 x 10^-39
According to the question, the error in the measurement of position of Lithium atom = 0.0585 x 10^-39
According to the Heisenberg uncertainty principle, we have; (Δx) (Δp_x) = 0.527 x 10^-34
(Δp_x) = (^{0.527 x 10-34} ⁄ _{0.0585 x 10^-39}) = 9 x 10⁵

12. When did Heisenberg put forward his uncertainty principle?

1. 1914
2. 1938
3. 1927
4. 1909

Answer:
1927
Explanation:
Werner Heisenberg proposed the uncertainty principle in 1927.

13. Which of the following law explains the wave phenomenon of moving micro sized bodies?

1. Schrodinger wave equation
2. de-Broglie dual nature of matter
3. Heisenberg uncertainty principle
4. Einstein's photoelectric effect

Answer:
de-Broglie dual nature of matter
Explanation:
According to de-Broglie, every moving matter possesses the dual character of both particle and wave.

14. Which of the following statement best describes the Heisenberg uncertainty principle?

1. It is impossible to know the position and momentum of a hydrogen electron simultaneously without error.
2. It is impossible to measure either the position or momentum of the hydrogen electron at the same instant of time.
3. It is possible to measure the position and momentum of the hydrogen electron precisely concurrently.
4. It is possible to measure either the position or momentum of the hydrogen electron with error.

Answer:
It is impossible to know the position and momentum of a hydrogen electron simultaneously without error.
Explanation:
The product of the error of the two conjugate physical variables with CGS unit erg seconds is equal to a constant quantity.

15. Calculate the error in velocity of the moving Barium electron whose matter wave has a wavelength of 0.01 nm.

1. 5.29 x 10⁶ m/s
2. 5.79 x 10⁶ m/s
3. 5.27 x 10^-22 m/s
4. 5.62 x 10^-23 m/s

Answer:
5.79 x 10⁶ m/s
Explanation:
The uncertainty in the position of Barium electron = 0.01 nm = 10^-11 m
By Heisenberg uncertainty principle, we have; (Δx) (mΔv) = 0.527 x 10^-34 joule second
(Δv) = ^{0.527 x 10-34 joule second}⁄_{(10^-11 m) (9.1 x 10^-31 kg)}
(Δv) = 5.79 x 10⁶ m/s
Because the mass of an electron = 9.1 x 10^-31 Kg

16. If the error in position and momentum of a moving Helium proton are equal, find the uncertainty in its velocity.

1. ^ħ⁄₂
2. ^h⁄₂
3. ^ħ⁄_2m
4. ¹⁄_m^(ħ⁄₂)^1/2

Answer:
¹⁄_m^(ħ⁄₂)^1/2
Explanation:
Let the error in position of Helium electron = x
The error in the momentum of the Helium electron = x (according to the question)
By Heisenberg uncertainty principle, we have; (Δx) (Δp) = ^ħ⁄₂
x² = ^ħ⁄₂
x = (^ħ⁄₂)^1/2
We can write it as; (Δp) = (^ħ⁄₂)^1/2
m(Δv) = (^ħ⁄₂)^1/2
(Δv) = ¹⁄_m(^ħ⁄₂)^1/2

17. Which of the following hypothesis mathematically proved the existence of electrons in orbitals?

1. Bohr's atomic model
2. Schrodinger wave equation
3. de-Broglie principle
4. Heisenberg uncertainty principle

Answer:
Heisenberg uncertainty principle
Explanation:
Measurement of error in the electron's location helped to confirm its occupancy in the orbitals of an atom.

18. A proton travels with the velocity of light. Suppose its velocity measurement is 0.1% accurate. Calculate Δx.

1. 1.05 x 10^-13 m
2. 2.36 x 10^-23 m
3. 6.35 x 10^-15 m
4. 4.23 x 10^-34 m

Answer:
1.05 x 10^-13 m
Explanation:
Uncertainty in the velocity of the proton = 0.1 % of 3 x 10⁸ m/sec
=^0.1⁄₁₀₀(3 x 10⁸)
= 3 x 10⁵ m/sec
By Heisenberg uncertainty principle, we have;
(Δx)(mΔv) = 0.527 x 10^-34 joule second
Mass of the proton = 1.67 x 10^-27 Kg
(Δx) = ^{0.527 x 10-34 joule second}⁄_{(1.67 x 10^-27) (3 x 10⁵)}
(Δx) = 0.105 x 10^-12 m
(Δx) = 1.05 x 10^-13 m

19. Calculate the product of impreciseness of displacement and velocity of moving neutron.

1. 0.567 x 10¹⁷
2. 3.45 x 10⁵
3. 0.3148 x 10^-7
4. 6.54 x 10³

Answer:
0.3148 x 10^-7
Explanation:
Mass of the neutron = 1.674 x 10^-27 Kg
By Heisenberg principle, we have;
(Δx)(mΔv) = 0.527 x 10^-34 joule second
(Δx)(Δv) = ^{0.527 x 10-34 joule second}⁄_{1.674 x 10^-27 Kg}
(Δx)(Δv) = 0.3148 x 10^-7

20. Which of the following is a physical significance of the Heisenberg uncertainty principle?

1. It is difficult to determine the trajectory of microscopic objects
2. It is difficult to determine the trajectory of macroscopic objects
3. It defines the position and momentum of substances
4. Both (a) and (b)

Answer:
It is difficult to determine the trajectory of microscopic objects
Explanation:
As micro-sized particles' location and velocity are not known with certainty by the Heisenberg uncertainty principle.