# 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

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

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

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.