Heisenberg Uncertainty Principle: Applications, Limitations

Heisenberg Uncertainty Principle states that it is impossible to determine the exact position and the momentum of a sub-atomic particle simultaneously with great accuracy.

Bohr’s theory considers an electron as a particle. If so, it would be quite feasible to determine its exact position and momentum. However, quantum theory suggests that an electron with a fixed position and momentum is no longer possible. Therefore, we are unable to determine the exact position of the particle or its rate of movement. The uncertainty of the position and the uncertainty of the momentum are complementary. This is known as the uncertainty principle and was proposed by German scientist Werner Heisenberg in 1927.

Heisenberg’s uncertainty principle (or simply the uncertainty principle) is one of several mathematical inequalities that assert a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables such as position x and momentum p, can be known.

Heisenberg’s uncertainty principle limits the accuracy of simultaneous measurement of position and momentum. The more precise our location measurement is, the less precise our momentum measurement will be, and vice versa. According to Heisenberg’s uncertainty principle, it is impossible to estimate both the position and velocity accurately for a particle that displays both particle and wave characteristics.

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Mathematical expression for Heisenberg uncertainty principle

Let ∆x be the uncertainty (error) in the measurement of position and ∆p be the uncertainty in the

measurement of the momentum of a sub-atomic particle like an electron. Then according to this

principle these two values are related as,

∆x × ∆p ≥ h/4∏

Or, ∆x × m ∆v ≥ h/4∏

since p = m × v

∆p = m ∆v

Or, ∆x × ∆v ≥ h/4∏m

This is an uncertainty relation which implies that the product of these two uncertain values may

greater or equal to h/4∏ but never less than that.

This uncertainty relationship says that, if ∆x is small I.e. if the position of a particle is measured more

accurately then ∆p would be large I.e. momentum is measured less accurately. On the other hand

if ∆p is small i.e. if momentum is measured more accurately, then, ∆x would be large I.e. position

of a particle is measured less accurately. In other words, if one quantity is measured more accurately, another quantity is measured less accurately, i.e. certainty in measuring one quantity introduces uncertainty in measuring another. This fact can be shown as:

If the position of a sub-atomic particle like an electron is measured accurately, then ∆x = 0

Now, from the uncertainty relation,

∆x × ∆p ≥ h/4∏

∆p ≥ h/4∏∆x

∆p ≥ h/4∏ x 0

= ∞

∆p = ∞ shows that the uncertainty in measuring momentum becomes infinite, i.e., the momentum of that particle cannot be measured when the particle’s position is certain x = 0, i.e., absolutely known.

Application of the Heisenberg uncertainty principle

I. Nonexistence of electrons in the nucleus

For example, let’s suppose the uncertainty in the position of an electron is ∆x= 10^-15m.

According to the Heisenberg uncertainty principle,

∆x × ∆p = h/4∏

Or ∆v = h/4∏∆x m

M = mass of electron (i.e., 9.1 x 10^-31 kg)

∆x= 1 x10^-15m

h = 6.6 x 10-34 JS

∆v = h/4∏∆x m

= 6.6 x 10-34/ 4 x 3.14 x 1 x10^-15x 9.1 x 10^-31

= 5.77 x 10¹⁰ ms^-1

The value of uncertainty in ∆v is much higher than the velocity of light (3.0 x 10⁸ ms-1) and hence an electron cannot be found within the atomic nucleus.

II. Stability

Consider atoms, which have negatively charged electrons orbiting a positively charged nucleus. According to classical logic, the two opposite charges should attract each other, causing everything to collapse into a ball of particles. The Heisenberg uncertainty principle explains why this does not occur. If an electron gets too close to the nucleus, its position in space is accurately known, and hence the error in measuring its position is negligible. This implies that the error in calculating its momentum would be significant. The electron would travel quickly enough to escape the atom entirely in that situation.

III. Radioactive decay

Heisenberg’s uncertainty principle can also explain alpha decay, a kind of nuclear radiation. Alpha particles are made up of two protons and two neutrons that are emitted by heavy nuclei such as uranium-238. Typically, these are bonded inside the heavy nucleus and require a lot of energy to break the bonds that hold them in place. However, an alpha particle inside a core has a highly well-defined velocity but a less well-defined position. That means there is a small but non-zero chance that the particle will find itself outside the nucleus at some point, even though it does not have enough energy.

Limitation of the Heisenberg uncertainty principle

The uncertainty principle is significant for a small particle but not for a large particle.

Verification

Validity of the uncertainty principle that whether it is applied to a small particle or large,

particle can be tested by calculating either ∆x or ∆v from the uncertainty principle. So, let

calculate ∆v for both small particles and large particles assuming ∆x = 10^-10m for both particles.

A. ∆v for large particle

Let us suppose the ball has mas 1 kg and uncertainty in the measurement of position is 10^-10 m.

Now from the Heisenberg uncertainty principle,

∆x × ∆v ≥ h/4∏m

Or, ∆v ≥ h/4∏m∆x

= 0.527 X 10^-24 m/s

This value is insignificant and can be ignored. It means that when we measure the Uncertainty in the velocity of a large particle (assuming that its position is certain), the value is very small, i.e. certain. As a result, we can accurately determine the position and velocity of a large particle. As a result, the Heisenberg uncertainty principle does not hold for large particles.

B. ∆v for small particle

Let us suppose the electron has mas 9.109 x 10^-31 Kg and uncertainty in the measurement of position is 10^-10 m.