IB DP Physics: HL复习笔记12.1.9 The Uncertainty Principle

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:
There are certain pairs of physical quantities that are cannot be known precisely at the same time
  • 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
    • Δpx = 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
  • Where:
    • ΔE = change in energy
    • Δt = change in time
    • h = Planck's constant

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

12-1-9-uncertainty-1-ib-hl

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 Δpx can be determined

12-1-9-uncertainty-2-ib-hl

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 Δpx
  • Comparing the two sets of graphs, it can be seen that:
    • As the momentum becomes less well known the position becomes more well known

12-1-9-uncertainty-3-ib-hl

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

A narrow beam of electrons with velocity 106 m s−1 are directed towards a slit which is 10-9 m wide.

The electrons are observed on a screen placed 2.0 m from the slit.

Estimate the length of the area of the screen where electrons will be seen in appreciable numbers.

 

 

 

 

 

 

 

 

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