Applications of uncertainty principle

• According to Heisenberg’s uncertainty principle it is impossible to determine precisely and simultaneously two canonically conjugate variables.
• If we measure one physical quantity precisely then the other will have a great uncertainty.
• This principle is observed only in microscopic particles, not in macroscopic bodies.
• In this article we will discuss about the following applications of uncertainty principle.

1. Particle in a box
2. Bohr’s orbit and principle of uncertainty
3. Ground state energy of hydrogen atom
4. Non-existence of electron in the nucleus
5. Binding energy of an electron in an atom
6. Photon emission from an excited atom

           Δx × Δp ≥ ħ / 2

Particle in a box

• Let a particle of mass m be lie in a box having side a.
• Since the particle can be anywhere in the box, and therefore the maximum uncertainty of finding the particle in this box is Δx = a.
• From Heisenberg’s uncertainty principle

           Δx × Δp ≈ ℏ

• Therefore the minimum uncertainty of momentum will be

Bohr’s orbit and principle of uncertainty

• If ΔJ and Δφ are the uncertainty in angular momentum and the uncertainty in angular displacement of the particle then from uncertainty principle, we have

           ΔJ × Δφ ≈ ℏ

• If electron revolve in first Bohr’s orbit then the uncertainty in its angular momentum will be 5% of ℏ

• Since the value of angle perpendicular to angular momentum can not be greater than 2π = 6.28 rad.
• So the position of electron in Bohr’s orbit is completely uncertain.

First Bohr radius

∵      Δx × Δp ≈ ℏ

∴      Δp ≈ ℏ / Δx

But  Δx = r

∴      Δp ≈ ℏ / r

• Now, uncertainty in kinetic energy is

• And uncertainty in potential energy is

• Therefore the uncertainty in total energy is

           ΔE = ΔK + ΔU

• Since in ground state energy is always minimum, therefore E_min =ΔE

• Thus the radius of first Bohr orbit of hydrogen atom is 0.53 Å, which is same as given by Bohr.

Ground state energy of hydrogen atom

• The minimum energy of electron is always in ground state

           E_min = ΔE

• Since the radius of first Bohr orbit

• This energy is same as that obtained from Bohr model of hydrogen atom.

Non- existence of electron in nucleus

• The electron revolves around the nucleus in their fixed stable orbit, they never lie in nucleus.
• Let an electron be exist in atomic nucleus and radius of nucleus is of the order of 2 × 10^-14 m

∴      (Δx)_max ≈ 2 × 10^-14

∵      Δx × Δp ≈ ℏ

• If minimum uncertainty in momentum of electron is 5.27 × 10^-21 kg.m/s, then the momentum of electron should be

• From relativistic theory

• If an electron exists in the nucleus, it must have minimum energy of about 9.88 MeV, but the maximum K.E. with which a β-particle (electron) emitted from radioactive nuclei is about 2-3 MeV.
• So a free electron can’t be reside inside the nucleus.

Binding energy of an electron in an atom

• In hydrogen electron revolves around the nucleus in an orbit of radius about 0.5 Å.

∴      (Δx)_max ≈ 5 × 10^-11 m

• Thus binding energy of an electron in an atom is about 15 eV.

Photon emission from an excited atom

• The electron excited to higher energy level by absorbing energy and it comes back to ground by emitting the photon within 10 ns.

∴      Δt ≈ 10 × 10^-9= 10^-8s

∵      ΔE × Δt ≈ ℏ

• Therefore the uncertainty in frequency is