Welcome to Quantum Physics!
Welcome to one of the most exciting and mind-bending chapters in your AS Physics course. Up until now, you have looked at the world through the lens of classical physics, where waves are waves and particles are particles. In this chapter, we enter the quantum realm, where energy comes in tiny discrete packets, light can behave like a stream of particles, and moving particles (like electrons) can behave like waves!
Don't worry if these ideas seem strange or counter-intuitive at first. Even Albert Einstein found them puzzling! We will break down every concept step-by-step, use real-world analogies, and highlight the key equations you need for your exam.
1. The Photon Model of Electromagnetic Radiation
What is a Photon?
In classical physics, light is described as a continuous wave. However, to explain how light interacts with matter on an atomic scale, physicists discovered that electromagnetic radiation behaves as if it is made up of tiny, discrete packets of energy. These "packets" or "quanta" of electromagnetic energy are called photons.
Key Concept: A photon is a quantum (discrete packet) of electromagnetic energy.
Energy of a Photon
The energy \(E\) carried by a single photon depends entirely on the frequency \(f\) of the electromagnetic radiation.
\(E = hf\)
Since the wave equation tells us that \(c = f\lambda\) (where \(c\) is the speed of light and \(\lambda\) is the wavelength), we can substitute \(f = \frac{c}{\lambda}\) to get:
\(E = \frac{hc}{\lambda}\)
Where:
• \(E\) = photon energy in Joules (\(\text{J}\))
• \(h\) = Planck's constant \(= 6.63 \times 10^{-34}\text{ J s}\)
• \(f\) = frequency of the radiation in Hertz (\(\text{Hz}\))
• \(c\) = speed of light in a vacuum \(= 3.00 \times 10^8\text{ m s}^{-1}\)
• \(\lambda\) = wavelength of the radiation in metres (\(\text{m}\))
Analogy: Imagine a vending machine that only accepts exact coins. Electromagnetic waves do not transfer energy like a smooth stream of water; instead, energy is delivered in individual "energy coins" (photons). A higher frequency wave means each individual coin has a higher value.
The Electron-Volt (\(\text{eV}\))
Because the Joule is a very large unit compared to the tiny energy of a single photon, physicists use a much smaller unit: the electron-volt (\(\text{eV}\)).
Definition: One electron-volt (\(1\text{ eV}\)) is the energy gained by an electron when it accelerates through a potential difference of \(1\text{ volt}\).
\(W = QV \implies 1\text{ eV} = (1.60 \times 10^{-19}\text{ C}) \times (1\text{ V}) = 1.60 \times 10^{-19}\text{ J}\)
Conversion Rules:
• To convert from \(\text{eV}\) to \(\text{J}\): multiply by \(1.60 \times 10^{-19}\)
• To convert from \(\text{J}\) to \(\text{eV}\): divide by \(1.60 \times 10^{-19}\)
Common Mistake to Avoid: Always convert photon energy back into Joules (\(\text{J}\)) before using formulas that involve Planck's constant \(h\) or the speed of light \(c\)!
Quick Summary: Photon Model
• Photons are discrete packets of electromagnetic energy.
• Energy is directly proportional to frequency: \(E = hf = \frac{hc}{\lambda}\).
• \(1\text{ eV} = 1.60 \times 10^{-19}\text{ J}\).
2. The Photoelectric Effect
What is the Photoelectric Effect?
The photoelectric effect is the emission of electrons from the surface of a metal when electromagnetic radiation of a sufficiently high frequency is directed at the metal. The emitted electrons are called photoelectrons.
The Gold-Leaf Electroscope Experiment
A classic demonstration uses a clean zinc plate placed on top of a negatively charged gold-leaf electroscope:
1. When the electroscope is negatively charged, the gold leaf rises because like charges repel.
2. If visible light (even very bright visible light) is shone onto the zinc plate, the leaf stays raised (no charge is lost).
3. When ultraviolet (UV) radiation is shone onto the plate, the leaf falls immediately.
4. This happens because UV radiation causes electrons to be knocked out of the zinc plate, neutralizing the excess negative charge.
Key Experimental Observations & Why Wave Theory Failed
Classical wave theory predicted that energy would accumulate gradually over time as the wave hit the surface. However, the experimental findings completely contradicted wave theory:
1. Threshold Frequency (\(f_0\)):
• Observation: No photoelectrons are emitted if the incident radiation frequency is below a specific value called the threshold frequency (\(f_0\)), no matter how intense (bright) the light is.
• Failure of Wave Theory: Wave theory predicted that even low-frequency light should eventually transfer enough energy to eject electrons if left shining long enough.
2. Instantaneous Emission:
• Observation: If \(f \ge f_0\), photoelectrons are emitted instantaneously (with no detectable time delay).
• Failure of Wave Theory: Wave theory predicted a significant time delay while electrons absorbed enough continuous energy to escape.
3. Maximum Kinetic Energy (\(E_{k(\text{max})}\)):
• Observation: The maximum kinetic energy of the emitted electrons depends only on the frequency of the light, not on the intensity.
• Failure of Wave Theory: Wave theory predicted that brighter (more intense) light would deliver more energy per second, causing electrons to emerge with higher kinetic energy.
4. Effect of Intensity:
• Observation: Increasing the intensity of the radiation increases the number of photoelectrons emitted per second (the photocurrent), provided \(f \ge f_0\), but does not change their maximum speed.
Einstein's Explanation & The Photoelectric Equation
Einstein explained the effect using the photon model based on a one-to-one interaction: one single photon is absorbed by one single electron.
An electron requires a minimum amount of energy to escape the metal surface. This minimum energy is called the work function (\(\Phi\)).
• If the incoming photon energy \(hf\) is less than \(\Phi\), no electron can escape.
• The minimum frequency needed to just liberate an electron is the threshold frequency (\(f_0\)), where: \(\Phi = hf_0\).
• If \(hf > \Phi\), the remaining energy becomes the kinetic energy of the escaping electron.
Einstein's Photoelectric Equation:
\(hf = \Phi + E_{k(\text{max})}\)
or
\(hf = \Phi + \frac{1}{2}mv_{\text{max}}^2\)
Where:
• \(hf\) = energy of the incident photon (\(\text{J}\))
• \(\Phi\) = work function of the metal (\(\text{J}\))
• \(E_{k(\text{max})}\) = maximum kinetic energy of the emitted photoelectron (\(\text{J}\))
• \(m\) = mass of an electron \(= 9.11 \times 10^{-31}\text{ kg}\)
• \(v_{\text{max}}\) = maximum velocity of the emitted electron (\(\text{m s}^{-1}\))
Why is it "maximum" kinetic energy?
Electrons at the very surface of the metal require the least energy (\(\Phi\)) to escape, so they leave with the maximum possible kinetic energy. Electrons deeper inside the metal lose additional energy in collisions on their way to the surface, emerging with less kinetic energy.
Stopping Potential (\(V_s\))
When photoelectrons are emitted in a vacuum tube towards an anode, applying a negative potential to the anode repels the oncoming electrons. The minimum negative potential required to stop even the fastest-moving electrons from reaching the anode is called the stopping potential (\(V_s\)).
At the stopping potential, the work done against the electric field equals the maximum kinetic energy of the photoelectrons:
\(E_{k(\text{max})} = eV_s\)
where \(e = 1.60 \times 10^{-19}\text{ C}\).
Graphical Representation: \(E_{k(\text{max})}\) against Frequency \(f\)
Rearranging Einstein's equation into the standard straight-line form \(y = mx + c\):
\(E_{k(\text{max})} = hf - \Phi\)
When plotted as a graph of \(E_{k(\text{max})}\) (y-axis) against \(f\) (x-axis):
• Gradient: represents Planck's constant (\(h\)) — this is identical for all metals!
• x-intercept: represents the threshold frequency (\(f_0\)).
• y-intercept: represents the negative of the work function (\(-\Phi\)).
Quick Summary: Photoelectric Effect
• One photon interacts with one electron.
• Threshold frequency \(f_0\) is the minimum frequency for emission: \(\Phi = hf_0\).
• Energy equation: \(hf = \Phi + E_{k(\text{max})}\).
• Intensity determines the rate of photon arrival (and thus the rate of electron emission), while frequency determines the energy of each individual photon.
3. Atomic Energy Levels and Line Spectra
Quantized Energy Levels in Atoms
Electrons orbiting an isolated atomic nucleus cannot have just any arbitrary amount of energy. Instead, they can only exist in specific, discrete energy states called energy levels.
• Ground State: The lowest, most stable energy level an electron can occupy.
• Excited State: Any energy level higher than the ground state.
• Ionisation: When an electron absorbs enough energy to completely escape the atom (reaching energy \(= 0\text{ eV}\)).
Convention: Energy level values are given negative signs (e.g., \(-13.6\text{ eV}\)). A negative value represents a bound state, meaning energy must be added to free the electron from the nucleus.
Photon Emission and Absorption
Electrons can transition between these fixed levels by absorbing or emitting photons:
1. De-excitation (Emission):
When an electron drops from a higher energy level (\(E_{\text{high}}\)) to a lower energy level (\(E_{\text{low}}\)), it emits a single photon carrying energy exactly equal to the difference between the two levels:
\(\Delta E = E_{\text{high}} - E_{\text{low}} = hf = \frac{hc}{\lambda}\)
2. Excitation (Absorption):
An electron in a lower energy level can absorb an incoming photon and jump to a higher level, but only if the photon's energy exactly matches the energy difference \(\Delta E\) between the two levels.
Line Spectra as Evidence for Discrete Levels
When light passes through a diffraction grating or prism, it splits into its component wavelengths. There are two main types of line spectra:
• Emission Line Spectrum: A series of sharp, bright coloured lines on a dark background. Produced when excited gas atoms de-excite, emitting photons of specific discrete wavelengths.
• Absorption Line Spectrum: A continuous rainbow spectrum crossed by dark lines. Produced when white light passes through a cooler gas; the gas atoms absorb photons of exact frequencies corresponding to transitions between their energy levels.
Why is this important? Every chemical element has a unique set of energy levels, resulting in a unique spectral "fingerprint". This allows astronomers to identify the chemical composition of distant stars!
Quick Summary: Energy Levels & Spectra
• Energy levels in atoms are discrete (quantized).
• Transition formula: \(\Delta E = hf = \frac{hc}{\lambda}\).
• Line spectra provide direct evidence that electron energy levels are discrete.
4. Wave-Particle Duality & Matter Waves
What is Wave-Particle Duality?
Wave-particle duality is the principle that both light and matter exhibit properties of both waves and particles depending on the experiment being conducted.
• Light as a Wave: Exhibited in phenomena like diffraction and interference (e.g., Young's double-slit experiment).
• Light as a Particle: Exhibited in the photoelectric effect (photons transferring discrete packets of energy).
The de Broglie Hypothesis
In 1924, Louis de Broglie proposed that if light waves can behave like particles, then particles of matter (such as electrons) should also behave like waves!
The wavelength associated with a moving particle is called its de Broglie wavelength (\(\lambda\)):
\(\lambda = \frac{h}{p} = \frac{h}{mv}\)
Where:
• \(\lambda\) = de Broglie wavelength (\(\text{m}\))
• \(h\) = Planck's constant \(= 6.63 \times 10^{-34}\text{ J s}\)
• \(p\) = momentum of the particle (\(\text{kg m s}^{-1}\))
• \(m\) = mass of the particle (\(\text{kg}\))
• \(v\) = velocity of the particle (\(\text{m s}^{-1}\))
Experimental Evidence: Electron Diffraction
Wave properties only become noticeable when a wave interacts with an obstacle or aperture roughly equal in size to its wavelength.
Because electrons have very small mass, their de Broglie wavelength when accelerated through a few thousand volts is around \(10^{-10}\text{ m}\) — which is approximately the spacing between carbon atoms in a graphite crystal!
The Experiment:
1. A beam of electrons is accelerated through a potential difference in an electron gun.
2. The beam passes through a thin sheet of polycrystalline graphite.
3. The regular atomic spacing acts as a diffraction grating.
4. The electrons strike a fluorescent screen, producing a pattern of concentric circular diffraction rings.
Conclusion: Since diffraction is strictly a wave phenomenon, this experiment provides conclusive proof that moving electrons possess wave properties.
What happens if electron speed increases?
If the accelerating voltage is increased, the electron velocity \(v\) increases \(\implies\) momentum \(p\) increases \(\implies\) de Broglie wavelength \(\lambda = \frac{h}{mv}\) decreases \(\implies\) the diffraction rings become smaller (tighter together).
Did you know? Everyday objects (like a moving tennis ball) also have a de Broglie wavelength! However, because their mass \(m\) is so large, their wavelength is astronomically tiny (around \(10^{-34}\text{ m}\)), far too small to undergo observable diffraction.
Quick Summary: Wave-Particle Duality
• Light acts as a wave (interference, diffraction) and as a particle (photoelectric effect).
• Matter also has wave properties: \(\lambda = \frac{h}{mv}\).
• Electron diffraction through polycrystalline graphite proves that matter behaves as a wave.
• Higher electron speed \(\implies\) shorter wavelength \(\implies\) less diffraction.
Chapter Review & Formula Checklist
Key Formulas to Master:
• Photon energy: \(E = hf = \frac{hc}{\lambda}\)
• Energy conversion: \(1\text{ eV} = 1.60 \times 10^{-19}\text{ J}\)
• Threshold frequency & Work function: \(\Phi = hf_0\)
• Einstein's equation: \(hf = \Phi + E_{k(\text{max})}\)
• Stopping potential: \(E_{k(\text{max})} = eV_s\)
• Energy level transitions: \(\Delta E = E_1 - E_2 = hf = \frac{hc}{\lambda}\)
• de Broglie wavelength: \(\lambda = \frac{h}{p} = \frac{h}{mv}\)