12.1.9 The Uncertainty Principle

• The quantisation of energy levels and their associated wave functions leads to some surprising outcomes
  • One of these outcomes is known as the uncertainty principle

• The uncertainty principle states that:

• This leads to two consequences related to:
  • Position and momentum
  • Energy and time

Position & Momentum

• One example is that it is not possible to know the position and the momentum of a quantum particle precisely

• This can be represented in equation form as:

• Where:
  • Δx = change in displacement
  • Δp_x = change in momentum
  • h = Planck's constant

• This equation shows that:
  • The better the position of the particle is known, the less precise is the knowledge of its momentum

• The value of Planck’s constant is small so this limitation only becomes important for small particles within the quantum regime - it does not apply in the Newtonian mechanics that you have used to calculate momentum for larger objects

Energy & Time

• There is a similar relationship between energy and time where both cannot be known precisely

Wavefunctions

• This principle can be demonstrated by looking at the wavefunctions associated with quantum particles that are nearly 'free'
  • For example, those associated with the energy levels in a 'swimming pool' potential energy well

• The graph of wavefunction vs position shows that while the momentum is known precisely the position of the particle is unknown

The wave function of a free particle with a fixed energy. The wave function shows it has a single value of momentum while the probability distributions shows that the particle can be anywhere in space.

• For a particle in a 'swimming pool' potential well, its energy levels are close together
  • Hence, the wave function can be a mixture of the wavefunctions associated with a number of closely spaced energy levels

• For a range of five neighbouring energy levels, the wavefunction can be obtained by adding the individual wavefunctions
  • This is a result of the principle of superposition

• This gives rise to a wave packet, meaning the particle is more likely to be in a particular region of space
• This can be shown by a graph of the probability distribution vs position
  • This helps to visualise how the approximate uncertainty in the position, Δx and momentum Δp_x can be determined

A wave packet made from the wavefunctions associated with five closely spaced energy levels. The probability distribution is shown along with the uncertainty in position and momentum.

• Consider the wavefunctions associated with five and ten energy levels with the probability distribution and the estimates of Δx and Δp_x
• Comparing the two sets of graphs, it can be seen that:
  • As the momentum becomes less well known the position becomes more well known

A wave packet made from the wavefunctions associated with five closely spaced energy levels. The probability distribution is shown along with the uncertainty in position and momentum.

Worked Example

Step 1: Write down the uncertainty equation for position and momentum

Step 2: Identify the known uncertainty from the question

• • The slit is 10⁻⁹ m wide, so the uncertainty in the vertical component of the position of the electrons at the slit, Δx = 10⁻⁹ 

Step 3: Rearrange the equation to find Δp and calculate

Step 4: Calculate the momentum, p

• • Mass of electron, m = 9.1 × 10⁻³¹ kg

• • Momentum, p = mv = 9.1 × 10⁻³¹ × 10⁶ = 9.1 × 10⁻²⁵ N s

Step 5: Substitute change in momentum (step 3) and momentum (step 4) into the equation for change in direction

Step 6: Apply the deviation of angle to find the length on the screen

• • Distance from slits to screen, D = 2.0 m
  • Angle of deviation from line of beam, θ = 0.06 rad

• • Therefore:

length on screen = 2θ × D = (2 × 0.06) × 2.0 = 0.24 m

Step 7: Write the final answer out in full

• • The length of the screen where appreciable numbers of electrons are expected to be observed = 24 cm