In this chapter, you will learn about Heisenberg uncertainty principle. Neil Bohr’s theory assumes an electron as a material particle of small mass revolving round the nucleus in circular orbits situated at a fixed distance from the nucleus with a certain velocity or momentum. Since the electron is considered as a material particle, its position and momentum can be determined with a great accuracy.
But, when an electron is considered in the form of a wave as suggested by Louis de Broglie, it is not possible to know simultaneously its exact position in the space at any instant and its velocity more precisely since the wave is extending throughout a region of space.
This means that if you attempt to make the exact measurement of the position of an electron, the exact measurement of its velocity becomes uncertain and vice versa. In 1927, a German physicist, Werner Heisenberg had suggested in his principle that the measurement of the position and momentum of an electron (or any particle) cannot be precisely measured simultaneously. This concept is known after his name as Heisenberg’s uncertainty principle. So, let’s understand his principle what he said.
Heisenberg’s Uncertainty Principle
According to Heisenberg uncertainty principle, it is impossible to measure simultaneously the exact position and exact velocity (or momentum) of a sub-atomic particle like electron and neutron.
If the velocity (or momentum) be measured very accurately, the measurement of the position of the sub-atomic particle correspondingly becomes less precise. On the other hand if position is determined with a great accuracy, the momentum becomes less accurately known or uncertain.
Thus, the certainty of determination of one property introduces uncertainty of determination of the other. The uncertainty in measurement of position, Δx, and the uncertainty of determination of momentum, Δp (or Δmv), are related by Heisenberg’s relationship as:
Δx * Δp ≥ h/4π
or, Δx * Δmv ≥ h/4π
Where,
- Δx represents the error (or uncertainty) in the measurement of position.
- Δp represents the error (or uncertainty) in the measurement of momentum.
- ℏ is the Planck’s constant.
- The sign ≥ indicates equal to or greater than.
The above relation is called uncertainty relation. From this relation, it is clear that if Δx is small (i.e. if the position of a atomic or subatomic particle is measured with accuracy), Δp will be large (i.e. the momentum will be measured less accurately or with less accuracy) and vice-versa. Thus, if one quantity is measured with accuracy, the other quantity will be measured less accurately.
Uncertainties for Large and Small Particles
1. For large particle: Let us consider a moving ball of mass 100 g. Then, the product of uncertainties of the measurement of its position and velocity is given by :
Δx * Δv = h/4π*m = 6.626 * 10-34 kg m2 s-1 / 4 * 3.14 * 10-1 kg = 10-33 m2 s-1.
Obviously, this uncertainty product is very small and thus negligible. Therefore, the uncertainty of measurements for large objects is practically nil. In other words, the uncertainty product is negligible in case of large objects.
2. For small atomic and subatomic particles: Let us consider an electron which is a subatomic particle and has mass equal to 9.1 * 10-31 kilogram. In this case, the product of the uncertainties of measurement of its position and velocity is as follows:
Δx * Δv = h/4π*m = 6.626 * 10-34 kg m2 s-1 / 4 * 3.14 * 9.1 * 10-31 kg = 10-4 m2 s-1.
Obviously, this value is large enough and can be thus in no way negligible. Thus, it is very clear that the Heisenberg uncertainty principle is only applicable for small particles like electron and neutron.
Uncertainty in the Measurement of Energy of Particle
According to Heisenberg uncertainty principle,
Δx * Δp ≥ h/4π
Multiplying and dividing the left hand side of above uncertainty relation, we will get
Δx/v * Δp * v ≥ h/4π
or, Δt * ΔE ≥ h/4π (for energy and time)
Where ΔE = Δp * v = uncertainty in the measurement of energy of the particle. While Δt = Δx / v = uncertainty in the measurement of time at which the measurement is made.
Similarly, ΔΦ * Δϕ ≥ h/4π (for angular motion)
Based on the Heisenberg uncertainty principle, the Bohr model of the atom, which gives a fixed position and definite velocity to an electron in a fixed orbit, is no longer valid. The best we can describe is the probability of locating an electron with a probable velocity in a given region of space at a specific time.
The space or three-dimensional region around the nucleus where there is the highest probability of finding an electron with a specific energy is called an atomic orbital.
Solved Examples on Heisenberg Uncertainty Principle
Example 1: Calculate the uncertainty in the position of an electron if the uncertainty in the velocity is 5.7 × 105 msec–1.
Solution: According to Heisenberg’s uncertainty principle,
Δx * Δp = h/4π
or Δx * mΔv = h/4π
or Δx = h/(4πm * Δv)
Here, Δv = 5.7 × 105 msec–1 (given), h = 6.6 × 10–34 kg m2 sec–1 and m = 9.1 × 10–31 kg
On substitution we get
Δx = 6.6 × 10–34 / (4 * 3.14 * 9.1 × 10–31 * 5.7 × 105) = 6.6 × 10–34 / 4 * 3.14 * 9.1 * 5.7
Δx = 1 × 10–10 m (Ans.)
Example 2: Calculate the mass of the particle of the uncertainty in the position and velocity of a particle are 10–10 m and 5.27 × 10–24 msec–1, respectively.
Solution: According to Heisenberg uncertainty principle,
Δx * Δp = h/4π
m = h/(4π * Δx * Δp)
Here, h = 6.6 × 10–34 kg m2 sec–1
Δx = 1 × 10–10 m
Δν = 5.27 × 10–24 msec–1
Substituting the values, we get
m = 6.6 × 10–34 kg m2 sec–1 / (4 * 3.14 * 1 × 10–10 * 5.27 × 10–24) = 0.10 kg = 100 grams (Ans.)