DP IB Physics: SL

E: Nuclear and quantum physics

E.2 Quantum physics

DP IB Physics: SL E: Nuclear and quantum physics E.2 Quantum physics Linking questions:
a)	How can particles diffract?
b)	What are the defining features and behaviours of waves?
c)	What evidence indicates the diffraction of a wave?
d)	How is photon scattering off an electron similar to and how is it different from the collision of two solid balls?
e)	Can the Bohr model help explain the photoelectric effect? (NOS)
f)	How did the explanation of the photoelectric effect lead to the falsification that light was purely a wave? (NOS)
g)	Why is Compton scattering more convincing evidence for the particle nature of light than that from the photoelectric effect? (NOS)

a) How can particles diffract?
Solution:
Wave-particle duality describes the phenomenon of particles diffracting because they are wave-like. Diffraction is the process by which particles spread out or bend around an obstruction or aperture that is similar in size to their wavelength.
When the size of the obstruction or aperture is near or less than the wavelength of the particle, this bending is most obvious.
Diffraction, a behaviour typically associated with waves, can occur in particles like protons, electrons, and even large molecules.
This unexpected outcome is explained by quantum mechanics, which states that all matter exhibits wave–particle duality.
Figure 1 Wave – particle duality
Wave-Particle Duality:
According to quantum mechanics, all particles, including atoms and electrons, are capable of acting both like particles and like waves.
[math]\lambda = \frac{h}{p} = \frac{h}{mv}[/math]
[math]λ[/math] = wavelength of the particle
[math]h[/math] = Planck^’ sconstant which is equals to [math]6.62 × 10^{-34} Js[/math]
– Particles with slower or lower masses have longer wavelengths and are more likely to diffract.
Wavelength and Diffraction:
A particle’s de Broglie wavelength, or wavelength, is inversely proportional to its momentum. Diffraction happens when this wavelength is similar to the size of an obstruction or gap.
b) What are the defining features and behaviours of waves?
Solution:
Every wave exhibits a few distinctive behaviours. They are susceptible to diffraction, interference, reflection, and refraction.
Anything that reflects, refracts, diffracts, and interferes is considered a wave because of these fundamental characteristics.
Figure 2 Wave’s features
⇒ Features of waves:

Wavelength	The distance between two consecutive points in phase (e.g. crest to crest). Measured in meters (m)
Frequency (f)	How many wave cycles pass a point per second. Measured in hertz (Hz).
Amplitude	Maximum displacement from the equilibrium position. Related to wave energy.
Speed (v)	The rate at which the wave propagates through a medium: [math]v = fλ[/math]
Period (T)	Time taken for one full wave cycle: [math]T = \frac{1}{\lambda}[/math]
Phase	Describes the position of a point within the wave cycle (important in interference).
Direction	The direction in which the wave energy is moving.

⇒ Behaviors of waves:
Reflection:
– Waves reverberate off of objects.
– Angle of incidence equals angle of reflection.
– seen in echoes and mirrors.
Refraction:
– When a wave enters a new medium, its direction and speed change.
– Causes light to bend in water or glass.
[math]\frac{\sin i}{\sin r} = \frac{v_1}{v_2}[/math]
Diffraction:
– After travelling through a small opening or around an obstruction, the wave spreads out.
– When gap ≈ wavelength, it is most apparent.
Interference:
– When two or more waves are superimposed, the result is:
– Amplitudes add in constructive interference.
– Destructive interference, where amplitudes cancel or subtract.
c) What evidence indicates the diffraction of a wave?
Solution:
Diffraction is the phenomenon of waves spreading out after passing through an opening or bending around obstructions.
Waves that bend around corners, travel through small openings, or produce interference patterns are examples of diffraction.
Typical instances include light forming patterns after passing through slits, water waves spreading after passing through a gap, and sound diffracting around corners.
Figure 3 The diffraction of waves
The spreading of a wave as it travels through a hole or around an obstruction is known as diffraction.
The emergence of interference or fringe patterns and the shift in wave intensity in areas that would otherwise be in shadow are the most obvious signs of diffraction.
⇒ Evidence of diffraction:
Fringe or interference patterns:
Waves overlap and produce a pattern of alternating bright and dark (light) or loud and quiet (sound) areas when they diffract through a small opening or around an obstruction.
When wavelength ≈ gap width, it is most visible.
Wave Spreading beyond geometric shadow:
Sharp shadows would be cast by obstacles if there was no diffraction and only rays were involved.
Wave behaviour is indicated when waves bend into the shadow region due to diffraction.
d) How is photon scattering off an electron similar to and how is it different from the collision of two solid balls?
Solution:
Momentum and energy are conserved when two solid balls collide and when photons scatter off an electron.
But in photon scattering, a massless particle (the photon) transfers momentum and energy to a mass-containing particle (the electron), changing the photon’s direction and wavelength.
On the other hand, when two massive particles collide with solid balls, their velocities usually change as a result of the transfer of kinetic energy and momentum.
Figure 4 Photon scattering
This analogy highlights the similarities between classical mechanics and quantum particle interactions.
The Compton effect, in which a high-energy photon (such as an X-ray or gamma ray) collides with a stationary electron and changes direction and energy, is a well-known illustration of photon scattering.
⇒ Similarities:
Conservation of Momentum and Energy:
– During the interaction, the system’s overall momentum and energy are preserved in both cases.
Deflection:
– The interacting particles scatter or deflect as a result of both kinds of collisions.
⇒ Difference:
Photon Identity:
The scattered photon and the incoming photon are different in photon-electron scattering. The energy transfer to the electron gives it a different wavelength (energy). The balls in a billiard ball collision stay the same after the impact.
Wavelength Change:
While collisions between billiard balls do not have the same effect, photon-electron scattering—more especially, Compton scattering—causes a change in the photon’s wavelength.
Character of the Interaction:
A quantum mechanical process known as photon-electron scattering occurs when light (photons) interacts with matter (electrons) to exchange energy and momentum.
Collisions between billiard balls are an example of a classical mechanical interaction between macroscopic objects.
e) Can the Bohr model help explain the photoelectric effect? (NOS)
Solution:
Similar to Einstein’s theory of the photoelectric effect, Bohr’s formula is predicated on the idea that a specific quantity of energy is released during a quantum jump.
The discreteness of atomic energy levels was used to explain the quantisation of the electromagnetic field.
No, the photoelectric effect cannot be explained by the Bohr model because it was not intended to deal with the interaction of light with free (or loosely bound) electrons on a metal surface; rather, it offered crucial proof of the particle nature of light, which aided in the development of the quantum theory of light, which is distinct from the Bohr atom.
Figure 5 Photoelectric effect
Electrons are released from a metal surface when light strikes it.
Findings revealed:
– Regardless of intensity, no electrons are released if the frequency of the light falls below a specific threshold.
– Frequency, not intensity, determines the kinetic energy of released electrons.
– High-frequency light emits instantly.
– According to Albert Einstein’s 1905 theory, light is made up of photons, each of which carries energy.
[math]E = hf[/math]
– One electron is struck by one photon:
[math]hf = \phi + KE_{\text{max}}[/math]
Scientists were forced to reconsider light as having both wave and particle properties after the photoelectric effect revealed that the wave theory of light was not comprehensive.
The Bohr model was successful for hydrogen spectra but not for electron emission from metals, demonstrating the limitations of models.
The significance of empirical data in promoting theoretical change is further demonstrated by this case.
In addition to atomic models, quantum electrodynamics was the result of the development of the photoelectric theory.
f) How did the explanation of the photoelectric effect lead to the falsification that light was purely a wave? (NOS)
Solution:
Electron emission from a material brought on by electromagnetic radiation, such as ultraviolet light, is known as the photoelectric effect.
Photoelectrons are electrons released in this way. In order to make conclusions about the characteristics of atoms, molecules, and solids, the phenomenon is investigated in condensed matter physics, solid state physics, and quantum chemistry.
Electronic devices designed for light detection and precisely timed electron emission have made use of the effect.
Figure 6 Photoelectric effect
⇒ Classical wave theory predictions:
Light was thought to be a wave prior to Einstein’s explanation. Classical wave theory states:
The energy of ejected electrons should rise with increasing light intensity.
If a light frequency is strong enough, electrons should eventually be ejected.
Before electrons are released, there ought to be a lag (time for energy to accumulate).
⇒ Experimental observations:
– Even in extremely low-intensity light, electrons are released instantly (no time delay).
– No matter how strong the light is, if the frequency of the light falls below a threshold, no electrons are released.
– The frequency of light, not its intensity, determines the electrons’ kinetic energy.
– These findings were in direct opposition to the wave theory.
The act of the notion that scientific theories must be testable and disproved when evidence contradicts them, is how science advances, according to philosopher Karl Popper.
The notion that light is merely a wave was refuted by the photoelectric effect.
The wave model was insufficient, but it wasn’t completely incorrect.
As a result, wave–particle duality was accepted: depending on the circumstance, light can act as both a wave and a particle.
g) Why is Compton scattering more convincing evidence for the particle nature of light than that from the photoelectric effect? (NOS)
Solution:
Since it offers concrete, quantitative proof that light behaves like a particle with momentum rather than just energy, Compton scattering is more compelling.
The results are consistent with predictions based on conservation of momentum and energy, which are characteristics of particle collisions, and unlike the photoelectric effect, it cannot even be roughly explained by wave theory.
Since photons have momentum and energy, two essential characteristics of particles, Compton scattering offers stronger evidence for the particle nature of light than the photoelectric effect.
Compton scattering demonstrates that photons have momentum as well, as demonstrated by the shift in wavelength and direction of scattered photons during collisions with electrons, whereas the photoelectric effect only demonstrates that light energy is quantised into photons.
Figure 7 Compton scattering
⇒ Photoelectric effect:
Electrons can be ejected from a material by light (photons) with enough energy, according to the photoelectric effect.
The idea that light is made up of energy packets (photons) is supported by the fact that the energy of the ejected electrons depends on the frequency of the light rather than its intensity.
However, because the photoelectric effect mainly concentrates on the energy exchange during the interaction, it does not explicitly show momentum transfer.
⇒ Compton Scattering:
When a photon and an electron collide, the photon’s wavelength (and thus energy) changes, and the electron is scattered. This phenomenon is known as Compton scattering.
The photon is giving the electron some of its energy and momentum, as evidenced by this wavelength shift, which cannot be explained by classical wave theory.
Compton scattering provides more concrete proof of the particle nature of light since it explicitly illustrates this transfer of momentum, which is a defining characteristic of particle behaviour.