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Wize University Chemistry Textbook > Early Atomic Theory to Quantum Theory

Heisenberg Uncertainty Principal

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The Heisenberg Uncertainty Principle

As electrons were discovered to have both wave and particle properties, Heisenberg found that we cannot determine the exact position and momentum (direction and speed) simultaneously. This is known as the Heisenberg uncertainty principle.

ΔxΔp≥h4π\boxed{\Delta x\Delta p\ge\frac{h}{4\pi}}
∆x is the uncertainty in position
∆p is the uncertainty in momentum
∆p=m∆v where m is the mass (kg) and ∆v is the uncertainty in velocity (m/s)
h is Planck's constant (6.626x10-34 Js or 6.626x10-34 kg m2/s)

The larger the value for ∆x is, the (larger/smaller)
smaller
the value for ∆p is
  • In other words, a larger value for ∆x means we are more uncertain about the position of the particle and less uncertain about the momentum of the particle!

Max Born added to this and said that we can interpret this equation to mean that we cannot say exactly where electrons are but we can say where they are likely to be.

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Example: Wave Particle Duality and the Heisenberg Uncertainty Principle

A beam of electrons is accelerated towards a thin metal foil and diffracts to give the exact same pattern as a blue laser with a wavelength of 480nm. The mass of an electron is 9.11x10-31 kg.

a) What speed were the electrons accelerated to?

λ=hmv\lambda=\frac{h}{mv}

v=hm λ=6.626 ×10−34kg m2 s−1(9.11 ×10−31kg)(480 ×10−9m)=1515 m s−1v=\frac{h}{m\ \lambda}=\frac{6.626\ \times10^{-34}kg\ m^2\ s^{-1}}{\left(9.11\ \times10^{-31}kg\right)\left(480\ \times10^{-9}m\right)}=1515\ m\ s^{-1}





b) If we measure the speed to a precision of ± 1% what is the minimum uncertainty in the position of the electron?

ΔxΔp≥h4 π\Delta x\Delta p\ge\frac{h}{4\ \pi}

Δx(mΔv)≥h4 π\Delta x\left(m\Delta v\right)\ge\frac{h}{4\ \pi}

Δx ≥h4 π m Δv=6.626 ×10−34kg m2 s−14 π (9.11 ×10−31kg)(15.15 m s−1)=3.82 ×10−6m\Delta x\ \ge\frac{h}{4\ \pi\ m\ \Delta v}=\frac{6.626\ \times10^{-34}kg\ m^2\ s^{-1}}{4\ \pi\ \left(9.11\ \times10^{-31}kg\right)\left(15.15\ m\ s^{-1}\right)}=3.82\ \times10^{-6}m

Practice: Uncertainty of Position

A circus performer is shot out of a cannon and leaves the cannon with a velocity of 6.2±\pm0.1 m/s. If the performer weighs 82 kg, what is the minimum uncertainty in their position?


Extra Practice
Overall concept testing
Choose the incorrect statement:
Nature of electron
Choose the correct statement.
Uncertainty Principle
The uncertainty in the momentum Δp of a football traveling at 40 m/s is 1 ⋅\cdot 10−6 of its momentum. What is its uncertainty in position Δx? Mass=0.40 kg
The Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle
As electrons were discovered to have both
wave
and
particle
properties, Heisenberg found that we cannot determine the
exact position
and
momentum
(direction and speed) simultaneously.
ΔxΔp≥h4π\boxed{\Delta x\Delta p\ge\frac{h}{4\pi}}
Wave Particle Duality and the Heisenberg Uncertainty Principle
A beam of electrons is accelerated towards a thin metal foil and diffracts to give the exact same pattern as a blue laser with wavelength of 480nm. (me = 9.11 x 10-31 kg)
a) What speed were the electrons accelerated to?'
b) If we measure the speed to a precision of ± 1% what is the minimum uncertainty in the position of the electron?
Quantum Model: Heisenberg Uncertainty Principle
The hardest slapshot ever recorded in North America was 175 km/h ± 2 %. If an average hockey puck weighs 170 g, what is the uncertainty in the position of the puck after it has been shot?
Heisenberg principle: Uncertainty of Position
A softball is thrown and the velocity is measured at 112 km/h. The error in the measurement is 0.5%. A softball weighs 212 g, what is the limit of the accuracy in the measurement of the softballs position?
Heisenberg principle: Uncertainty of Position
A circus performer is shot out of a cannon and leaves the cannon with a velocity of 6.2±\pm0.1 m/s. If the performer weighs 82 kg, what is the minimum uncertainty in their position?
Elements: Binding Energy
An element is bombarded with a beam of X-rays of wavelength 0.75664nm. Upon irradiation, electrons are ejected with a velocity of 2.35 x 107 m/s.
What is the binding energy for the element?
Electromagnetic Radiation: Energy of Blue Laser Pointers
An argon ion laser produces a blue laser beam with a wavelength of 458 nm. What is the energy of this light expressed in kJ/mol?