A LevelPhysicsCIETopic Questions22. Quantum Physics22.3 Wave-Particle Duality

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First teaching 2023

First exams 2025

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Wave-Particle Duality (CIE A Level Physics)

Topic Questions

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1a
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After the wave-particle duality of light had been confirmed by experiment, Louis de Broglie hypothesised the existence of matter waves.

Outline the meaning of the 'de Broglie wavelength'.

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1b
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The de Broglie hypothesis was later confirmed experimentally. High-speed electrons were fired at a metal target and were found to scatter in a particular pattern. 

State the phenomenon observed in this experiment and state the evidence this provides about the nature of moving electrons.

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1c
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In a similar experiment, electrons fired at a thin graphite sheet are found to produce a pattern of concentric circles when they strike a fluorescent screen, as shown in Fig 1.1.

2-5-s-q--q3b-medium-aqa-a-level-physics

Fig 1.1

Outline how the radius of the concentric rings would change if the speed of the electrons decreases.

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1d
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Calculate the de Broglie wavelength of an electron travelling at a speed of 3.18 × 107 m s–1. 

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2a
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Once the wave–particle duality of light had been confirmed by experiment, Louis de Broglie proposed that if light could sometimes behave as a wave, and other times a particle, then perhaps all other things have this wave–particle duality. 

He suggested that electrons – which are considered to be particles – could also behave like waves do. This was confirmed experimentally in 1926. 

State the experimental evidence which suggests electrons have wave properties.

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2b
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Louis de Broglie not only proposed a wave–particle duality for all matter, he also wrote an equation to show the precise relationship between the momentum of a particle p and its associated wavelength, λ.  

Describe in words how wavelength λ is related to momentum p.

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2c
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Calculate the de Broglie wavelength of an electron which moves at a velocity of  2.5 × 107 m s–1.

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2d
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A simplified diagram of an electron gun and bulb is shown in Fig 1.1 below:

2-5-s-q--q3d-easy-aqa-a-level-physics

Fig 1.1

The electrons are produced from a very hot filament, and are directed toward a thin graphite film, all of which are set up inside an evacuated glass bulb.  

A very characteristic pattern is formed at the back of the bulb, which demonstrates electrons have been diffracted.

(i)
Describe the pattern observed at the back of the bulb
(ii)
State why the bulb must be evacuated

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3a
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The wave–particle duality of matter has far reaching implications. 

The de Broglie equation, which quantifies wave–particle duality, is given by the equation:

lambda space equals space h over p

   (i)     Identify the quantity represented by each symbol in the de–Broglie equation 

[1]

   (ii)    State the SI unit for each symbol

[1]

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3b
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In an electron diffraction tube, high speed electrons are incident on a thin slice of polycrystalline material. Fig 1.1 shows the pattern of bright concentric rings that is formed on the fluorescent screen at the back of the tube:

2-5-s-q--q4c-easy-aqa-a-level-physics

Fig 1.1

Explain why the electrons diffract.

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3c
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A student wonders if high energy electrons could be used to investigate the nature of protons which have an estimated diameter of 0.84 × 10–15 m. 

Calculate the momentum of the electrons that would be suitable for this investigation.

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3d
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Using your answer from part (c):

(i)
Calculate the theoretical speed required of these electrons.

[2]

(ii)
Therefore, explain why it is not possible for electrons to be used to study protons in detail.

[1]

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1a
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An experiment to investigate how the de Broglie wavelength λ of an electron varies with its velocity v  is carried out. The results from this experiment are shown in Table 1.1. 

v / 104 km s–1

λ / 10–11 m

1.4

4.9

2.5

2.8

3.5

2.1

Table 1.1 

Discuss whether the data in Table 1.1 is consistent with the de Broglie equation: 

lambda space equals space fraction numerator h over denominator m v end fraction

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1b
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Using the data in Table 1.1 and the accepted value for Planck’s constant, comment on the accuracy of the results.

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1c
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The experiment required electrons to travel at very high speeds. To achieve this, they were first accelerated through a very high voltage, V.

Show that the de Broglie wavelength λ  of these electrons is related to the accelerating voltage V  by the expression:

                     λ ∝ fraction numerator 1 over denominator square root of V end fraction

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2a
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The Planck constant h is an important fundamental constant in quantum physics.

Determine the S.I. base units for h.

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2b
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A researcher is investigating the de Broglie wavelength of charged particles. The charged particles are accelerated through a potential difference V.  The de Broglie wavelength λ of these particles is then determined by the researcher. Each particle has mass m and charge q.

Show that the de Broglie wavelength λ is given by the expression:

lambda to the power of 2 space end exponent equals fraction numerator h squared over denominator 2 m q end fraction cross times 1 over V

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2c
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The researcher plots data points on a graph of λ2 against 1 over V, as shown in Fig. 1.1.

q19b-ii-paper-2-nov-2020-ocr-a-level-physics

Fig. 1.1

(i)
Calculate the percentage uncertainty in λ for the data point circled on the grid.

(2)

(ii)
Draw a straight line of best fit through the data points.

(1)

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2d
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The charge q on the particle is 2e, where e  is the elementary charge.
Use your best fit line from part (c) to show that the mass m of the particle is about 10−26 kg.

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1a
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State the formula for the de Broglie wavelength λ of a moving particle. 

State the meaning of any other symbol used. 

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1b
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Electrons accelerate through a potential difference, pass through a thin crystal and are then incident on a fluorescent screen.

The pattern in Fig. 1.1 is observed on the fluorescent screen.

22-2-1b-m-electron-diffraction-sq-cie-a-level
Fig. 1.1 not to scale
(i)
State the name of the phenomenon shown by the electrons at the crystal.
[1]
 
(ii)
State what this phenomenon shows about the nature of electrons.
[1]
(iii)
Suggest why the thin crystal causes the phenomenon in (b)(i).
[1]

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1c
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The potential difference used to accelerate the electron is changed.

The new pattern observed on the screen is shown in Fig. 1.2.

22-2-1c-m-electron-diffraction-change-sq-cie-a-level
Fig 1.2
 

State and explain the change that has been made to the potential difference to create the pattern shown in Fig. 1.2.

 

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2a
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State one piece of experimental evidence for: 

(i)
the wave nature of matter
[1]
(ii)
the particulate nature of electromagnetic radiation.
[1]

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2b
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Calculate the de Broglie wavelength λ of an alpha-particle moving at a speed of 5.9 × 107 m s−1.

 
λ = ............................................ m 

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2c
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The speed v of the alpha-particles in (b)(i) is gradually reduced to zero.  

On Fig. 1.1, sketch the variation with v of λ.

 
22-2-2c-m-lambda-v-graph-axis-sq-cie-a-level
Fig 1.1

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3a
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Explain what is meant by the de Broglie wavelength.

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3b
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A gamma-ray photon of energy 2.6 × 10−18 J is incident on an isolated stationary electron, as illustrated in Fig. 1.1.

 22-2-3b-m-electron-deflection-sq-cie-a-level
Fig. 1.1
 

The photon is deflected elastically by the electron through angle θ. The deflected photon has a wavelength of 590 nm.

(i)
On Fig. 1.1, draw an arrow to indicate a possible initial direction of motion of the electron after the photon has been deflected
[1]
(ii)
Calculate the energy of the deflected photon.
 
photon energy = ........................................ J [2]

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3c
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Calculate the speed of the electron after the photon has been deflected.

 
speed = ........................................... m s−1

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3d
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Explain why the magnitude of the final momentum of the electron is not equal to the change in magnitude of the momentum of the photon.

 

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