Welcome to Quantum Physics!
Up to this point in Physics, light has behaved like a continuous wave, rippling across space, reflecting, refracting, and diffracting. But at the start of the 20th century, physicists discovered experiments that wave theory simply could not explain. This led to the birth of quantum physics—the science of the very small.
Don't worry if this topic sounds mind-bending at first! We will break down every concept step-by-step, starting from light behaving as tiny packets of energy, moving to how light kicks electrons out of metals, and finishing with how matter itself can behave like a wave.
1. The Photon Model and the Electronvolt
What is a Photon?
In the quantum model, electromagnetic (EM) radiation is not emitted as a continuous wave. Instead, it is emitted and absorbed in discrete packets (or "quanta") of energy called photons.
Think of a photon as a tiny, indivisible bundle of pure energy traveling at the speed of light, \(c = 3.00 \times 10^8 \text{ m s}^{-1}\).
Photon Energy Equations
The energy \(E\) of a single photon is directly proportional to its frequency \(f\):
\(E = hf\)
Where:
• \(E\) = energy of the photon in joules (\(\text{J}\))
• \(h\) = Planck constant \(= 6.63 \times 10^{-34} \text{ J s}\)
• \(f\) = frequency of the radiation in hertz (\(\text{Hz}\))
Since the wave speed equation tells us that \(c = f\lambda\) (so \(f = \frac{c}{\lambda}\)), we can also write the photon energy in terms of wavelength \(\lambda\):
\(E = \frac{hc}{\lambda}\)
Where:
• \(c\) = speed of light in a vacuum (\(3.00 \times 10^8 \text{ m s}^{-1}\))
• \(\lambda\) = wavelength of the radiation in metres (\(\text{m}\))
Important Insight: Higher frequency (or shorter wavelength) means each individual photon carries more energy. For example, an ultraviolet photon carries significantly more energy than a visible red light photon.
The Electronvolt (\(\text{eV}\))
Because the joule (\(\text{J}\)) is far too large a unit when dealing with individual photons and electrons, physicists use a much smaller unit: the electronvolt (\(\text{eV}\)).
Definition: One electronvolt (\(1 \text{ eV}\)) is the kinetic energy gained by an electron when it is accelerated through a potential difference of \(1 \text{ volt}\).
Using \(W = QV\), where \(Q = e = 1.60 \times 10^{-19} \text{ C}\) and \(V = 1 \text{ V}\):
\(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\)
Converting Between Joules and Electronvolts:
• 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 energies back into joules (\(\text{J}\)) before substituting them into standard SI equations involving the Planck constant \(h\)!
Key Takeaway: Section 1
Light consists of packets of energy called photons where \(E = hf = \frac{hc}{\lambda}\). For atomic-scale interactions, energy is often measured in electronvolts (\(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\)).
2. The Photoelectric Effect
What is the Photoelectric Effect?
The photoelectric effect is the emission of electrons (called photoelectrons) from the surface of a metal when electromagnetic radiation of high enough frequency is shone onto it.
Demonstrating the Effect: The Gold-Leaf Electroscope
A classic experiment uses a negatively charged zinc plate attached to a gold-leaf electroscope:
1. The electroscope is given a negative charge, causing the gold leaf to be repelled and rise.
2. When visible light is shone on the zinc plate, nothing happens—the leaf stays up, regardless of how bright the light is.
3. When ultraviolet (UV) light is shone on the zinc plate, the gold leaf falls immediately.
4. This shows that UV light causes electrons to escape from the zinc plate, discharging the electroscope.
Why Classical Wave Theory Failed
Classical wave theory predicted that energy from a wave should accumulate over time, meaning any frequency of light should eventually eject an electron if left long enough or made bright enough. But experimental observations showed:
• Instantaneous emission: Photoelectrons are released with no detectable time delay as soon as radiation of sufficient frequency hits the metal.
• Existence of a Threshold Frequency (\(f_0\)): No electrons are emitted if the incident frequency is below a minimum frequency \(f_0\), no matter how intense the light is.
• Kinetic energy depends on frequency, not intensity: Increasing the brightness (intensity) does not increase the maximum kinetic energy of the emitted electrons; only increasing the frequency does.
• Intensity determines the rate of emission: Above the threshold frequency, increasing intensity increases the number of photoelectrons emitted per second (the photocurrent).
Einstein's Explanation (The Photon Theory)
Albert Einstein explained these observations by proposing that energy transfer occurs on a one-to-one interaction basis: one single photon transfers all its energy to one single electron.
To escape the surface of the metal, an electron must overcome the attractive electrostatic forces holding it there. The minimum energy required to escape is called the work function (\(\Phi\)).
Everyday Analogy: Imagine a vending machine where a drink costs \(\Phi = £2.00\). The machine only takes one coin at a time. If you put in a \(£1.00\) coin (a low-frequency photon), nothing happens and no drink comes out. If you insert a \(£3.00\) coin (a high-frequency photon), you get your drink and \(£1.00\) change (kinetic energy of the electron)!
Einstein's Photoelectric Equation
Applying the principle of conservation of energy to a one-to-one photon-electron interaction gives:
\(hf = \Phi + E_{k(\text{max})}\)
Where:
• \(hf\) = total 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 (\(\text{J}\))
Since \(E_{k(\text{max})} = \frac{1}{2}mv_{\text{max}}^2\), we can also write:
\(hf = \Phi + \frac{1}{2}mv_{\text{max}}^2\)
Key Terms Defined:
• Work Function (\(\Phi\)): The minimum energy required to liberate an electron from the surface of a metal.
• Threshold Frequency (\(f_0\)): The minimum frequency of incident radiation required to cause photoelectric emission. At threshold frequency, electrons escape with zero kinetic energy (\(E_{k(\text{max})} = 0\)):
\(\Phi = hf_0\)
• Threshold Wavelength (\(\lambda_0\)): The maximum wavelength that will cause photoelectric emission:
\(\lambda_0 = \frac{hc}{\Phi}\)
Stopping Potential (\(V_s\))
The maximum kinetic energy of emitted photoelectrons can be measured experimentally by applying a negative (retarding) potential to a collector plate. The minimum potential required to reduce the photocurrent to zero is called the stopping potential (\(V_s\)).
At this point, the work done by the electric field equals the maximum kinetic energy of the electrons:
\(E_{k(\text{max})} = eV_s\)
Where \(e = 1.60 \times 10^{-19} \text{ C}\) and \(V_s\) is the stopping potential in volts (\(\text{V}\)).
The Graph of \(E_{k(\text{max})}\) versus Frequency (\(f\))
Rearranging Einstein's equation into the straight-line form \(y = mx + c\):
\(E_{k(\text{max})} = hf - \Phi\)
• Gradient: Planck's constant (\(h\)) — identical for all metals.
• x-intercept: Threshold frequency (\(f_0\)).
• y-intercept: Negative of the work function (\(-\Phi\)).
Key Takeaway: Section 2
The photoelectric effect provides definitive proof of the particle nature of light. One photon interacts with one electron: \(hf = \Phi + E_{k(\text{max})}\). Light must be at or above threshold frequency \(f_0\) to liberate an electron.
3. Atomic Energy Levels and Spectra
Discrete Energy Levels
Electrons in an isolated atom (such as a gas) cannot have just any amount of energy. They can only occupy fixed, discrete energy levels. We say atomic energy levels are quantised.
• The lowest energy level an electron can occupy is called the ground state (\(n = 1\)).
• When an electron absorbs energy and moves to a higher level, the atom is in an excited state.
• Energy levels are given negative values (e.g., \(-13.6 \text{ eV}\)) because zero energy represents the state where an electron is completely free from the nucleus (known as ionisation).
Photon Emission (De-excitation)
An electron in an excited state is unstable and will eventually transition (fall) down to a lower energy level. In doing so, it emits a single photon with energy exactly equal to the difference between the two energy levels:
\(\Delta E = E_1 - E_2 = hf = \frac{hc}{\lambda}\)
Where \(E_1\) is the initial higher energy level and \(E_2\) is the final lower energy level.
Photon Absorption (Excitation)
An electron can move from a lower energy level to a higher energy level by absorbing a photon. However, the photon's energy \(hf\) must exactly match the difference \(\Delta E\) between the two levels. If the photon has slightly more or slightly less energy, it cannot be absorbed.
Line Spectra: Emission vs. Absorption
• Emission Line Spectrum: Produced by a hot, low-pressure gas. When electrons drop to lower energy states, they emit photons of specific frequencies. When viewed through a diffraction grating or prism, this appears as a series of bright coloured lines on a dark background.
• Absorption Line Spectrum: Produced when continuous white light passes through a cool gas. The atoms absorb specific frequencies matching their energy level transitions. When viewed, this appears as dark lines on a continuous rainbow background.
• Continuous Spectrum: Produced by hot, dense objects (like the filament of a lamp or a star's core), containing all wavelengths across a continuous band.
Did you know? Because every element has a unique set of energy levels, every element produces a unique spectral "fingerprint". This allows astronomers to determine the chemical composition of distant stars millions of light-years away!
Key Takeaway: Section 3
Atoms have discrete, quantised energy levels. A transition between levels involves the emission or absorption of a photon with energy \(\Delta E = hf = \frac{hc}{\lambda}\), creating characteristic line spectra.
4. Wave-Particle Duality and the De Broglie Wavelength
Wave-Particle Duality
By the 1920s, light was proven to have a dual nature:
• It behaves like a wave during propagation (showing interference and diffraction).
• It behaves like a particle (photon) during interaction with matter (such as the photoelectric effect).
In 1924, Louis de Broglie proposed a revolutionary hypothesis: If light waves can behave like particles, can matter particles (like electrons) also behave like waves?
The De Broglie Equation
De Broglie showed that any particle moving with momentum \(p\) has an associated wave nature with a wavelength given by:
\(\lambda = \frac{h}{p} = \frac{h}{mv}\)
Where:
• \(\lambda\) = de Broglie wavelength in metres (\(\text{m}\))
• \(h\) = Planck constant (\(6.63 \times 10^{-34} \text{ J s}\))
• \(p = mv\) = momentum of the particle in \(\text{kg m s}^{-1}\)
• \(m\) = mass of the particle in kilograms (\(\text{kg}\))
• \(v\) = velocity of the particle in \(\text{m s}^{-1}\)
Why don't we notice everyday objects behaving like waves?
Because everyday objects (like a thrown tennis ball) have a large mass \(m\), their momentum \(p\) is large, making their de Broglie wavelength \(\lambda\) unimaginably tiny (e.g., \(10^{-34} \text{ m}\))—far too small to observe diffraction. However, for tiny particles like electrons, \(\lambda\) is on the order of \(10^{-10} \text{ m}\), which is roughly the spacing between atoms in a crystal!
Experimental Evidence: Electron Diffraction
The wave nature of electrons was confirmed experimentally by firing a beam of accelerated electrons through a thin layer of polycrystalline graphite:
1. The spacing between carbon atoms in the graphite acts like a microscopic diffraction grating.
2. The electrons pass through and diffract, producing a pattern of concentric bright and dark diffraction rings on a fluorescent screen.
3. Diffraction is uniquely a wave phenomenon, providing undeniable evidence that electrons exhibit wave properties.
Effect of Changing Electron Speed / Voltage:
• Increasing accelerating voltage \(\implies\) higher speed \(v\) \(\implies\) larger momentum \(p\).
• Since \(\lambda = \frac{h}{p}\), increasing momentum causes the de Broglie wavelength \(\lambda\) to decrease.
• A shorter wavelength causes less diffraction, meaning the diffraction rings contract (move closer to the centre).
Summary of Duality
• Evidence for Light as a Wave: Young's double-slit interference, diffraction.
• Evidence for Light as a Particle: Photoelectric effect.
• Evidence for Matter as a Particle: Deflection in magnetic/electric fields, collision impacts.
• Evidence for Matter as a Wave: Electron diffraction through graphite.
Key Takeaway: Section 4
All matter has both particle and wave properties. A particle of mass \(m\) moving at speed \(v\) has a de Broglie wavelength \(\lambda = \frac{h}{mv}\). Electron diffraction confirms the wave-like behavior of matter.
Quick Summary Checklist
• Photon energy: \(E = hf = \frac{hc}{\lambda}\)
• Energy conversion: \(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\)
• Einstein's photoelectric equation: \(hf = \Phi + E_{k(\text{max})}\)
• Threshold condition: \(\Phi = hf_0\)
• Stopping potential: \(E_{k(\text{max})} = eV_s\)
• Energy level transitions: \(\Delta E = E_1 - E_2 = hf\)
• De Broglie wavelength: \(\lambda = \frac{h}{p} = \frac{h}{mv}\)