Quantum Physics: The Physics of the Very Small
Welcome to Quantum Physics! This is one of the most fascinating (and sometimes mind-bending) topics in A Level Physics. Up until now, we’ve primarily studied Classical Physics, which describes the world of large, everyday objects.
Quantum physics takes us into the realm of the extremely small – atoms, electrons, and photons – where energy is quantised (comes in discrete packets) and particles can act like waves.
Don't worry if these ideas seem strange; they challenge common sense! By the end of this chapter, you will understand the concept of the photon, how light knocks electrons out of metals, and why everything, even you, has a wavelength.
22.1 Energy and Momentum of a Photon
The Particulate Nature of Light
For centuries, light was understood purely as a wave (evidenced by diffraction and interference). However, certain experiments, especially the photoelectric effect, could only be explained if light also behaved like a stream of particles.
Key Concept: Electromagnetic radiation (light) has a particulate nature.
The Photon: A Quantum of Energy
A light particle is called a photon. A photon is defined as a quantum of electromagnetic energy. The word 'quantum' means a fixed, discrete amount.
Think of energy like stairs, not a ramp. In quantum mechanics, energy can only exist at specific levels or in specific packets (quanta). A photon is one of these packets.
Energy of a Photon (\(E = hf\))
The energy \(E\) of a single photon is directly proportional to the frequency \(f\) of the electromagnetic radiation.
Photon Energy Equation:
\[E = hf\]
Where:
\(E\) is the energy of the photon (J)
\(h\) is the Planck constant (\(6.63 \times 10^{-34}\) J s)
\(f\) is the frequency of the radiation (Hz)
Since we know \(c = f\lambda\) (where \(c\) is the speed of light), we can also write the energy equation in terms of wavelength \(\lambda\):
\[E = \frac{hc}{\lambda}\]
The Electronvolt (eV)
Since the energy of a single photon is extremely small (in the order of \(10^{-19}\) J), the joule is often an inconvenient unit. We use the electronvolt (eV).
Definition: The electronvolt (eV) is the energy transferred when an electron moves through a potential difference of 1 Volt.
Conversion: To convert from eV to Joules, we use the elementary charge \(e\):
\[1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\]
Momentum of a Photon
Photons have no mass, yet they carry momentum. This supports the particle model.
Photon Momentum Equation:
\[p = \frac{E}{c}\]
Where:
\(p\) is the momentum of the photon (kg m s\(^{-1}\))
\(E\) is the energy of the photon (J)
\(c\) is the speed of light (\(3.00 \times 10^8\) m s\(^{-1}\))
Quick Tip: You can substitute \(E = hc/\lambda\) into the momentum equation to get \(p = h/\lambda\). This formula links the particle property (momentum \(p\)) to the wave property (wavelength \(\lambda\)) – a key idea of quantum physics!
Key Takeaway (22.1): Photons are discrete energy packets, \(E=hf\). Higher frequency means higher energy. They carry momentum, \(p=E/c\).
22.2 The Photoelectric Effect
The Phenomenon
The photoelectric effect is the emission of electrons (called photoelectrons) from a metal surface when it is illuminated by electromagnetic radiation (light).
Did you know? The photoelectric effect is what makes solar panels work! Light hits the metal (semiconductor), knocking out electrons to create a current.
Failure of Classical Wave Theory
Classical physics predicted that:
1. Brighter light (higher intensity) should cause electrons to be emitted with higher kinetic energy (faster speed).
2. Electrons should be emitted regardless of the frequency, given enough time to absorb the wave energy.
Experimental observations proved classical theory wrong:
- Electrons were emitted immediately, if at all.
- Emission only occurred if the frequency of light was above a specific minimum value.
- The maximum kinetic energy (\(K_{max}\)) of the emitted electrons depended only on the frequency, not the intensity.
Einstein's Photon Explanation (1905)
Einstein explained these results by using Planck's idea that light is quantised (photons).
- One Photon, One Electron: Each photon interacts with only one electron, transferring all its energy (\(hf\)) instantly. This explains the immediate emission.
- Threshold Frequency (\(f_0\)): An electron needs a minimum amount of energy to escape the metal surface. If the incoming photon energy (\(hf\)) is too low, the electron never escapes, regardless of how many low-energy photons arrive (intensity).
Key Definitions
1. Work Function Energy (\(\Phi\)):
The minimum energy required by an electron to escape from a specific metal surface. This energy is constant for a given metal.
2. Threshold Frequency (\(f_0\)):
The minimum frequency of electromagnetic radiation required to cause photoelectric emission. It is related to the work function by \(\Phi = hf_0\).
3. Threshold Wavelength (\(\lambda_0\)):
The maximum wavelength that can cause photoelectric emission. Since \(f_0 = c/\lambda_0\), the threshold wavelength is given by \(\lambda_0 = hc/\Phi\).
Memory Aid: To escape the metal, the electron needs energy \(E_{escape}\). The minimum possible \(E_{escape}\) is the Work Function (\(\Phi\)).
The Photoelectric Equation
This equation applies the conservation of energy to the photon-electron interaction:
Photon Energy = Energy to Escape + Remaining Kinetic Energy
\[hf = \Phi + K_{max}\]
or
\[hf = \Phi + \frac{1}{2}mv_{max}^2\]
Here, \(K_{max}\) (or \(\frac{1}{2}mv_{max}^2\)) is the maximum kinetic energy of the emitted photoelectrons. The maximum energy occurs because some electrons lose energy through collisions before they escape, so they are emitted with less than \(K_{max}\).
Explaining Intensity vs. Energy
The quantum model clearly explains why maximum kinetic energy is independent of intensity, but photoelectric current is proportional to intensity:
1. Maximum Kinetic Energy (\(K_{max}\)):
\(\bullet\) \(K_{max} = hf - \Phi\). Since \(\Phi\) is fixed and \(h\) is fixed, the kinetic energy only depends on the frequency (\(f\)) of the individual photons.
\(\bullet\) Increasing the intensity (brighter light) means sending more photons, but each photon still has the same individual energy \(hf\).
2. Photoelectric Current:
\(\bullet\) Current is the rate of flow of charge (electrons). Each successful photon releases one photoelectron.
\(\bullet\) Therefore, the brighter the light (higher intensity, meaning more photons per second), the more electrons are emitted per second, and the greater the photoelectric current.
Quick Review (22.2): The minimum energy to escape is the Work Function, \(\Phi\). Only light above the Threshold Frequency, \(f_0\), works. \(K_{max}\) depends on frequency, current depends on intensity.
22.3 Wave-Particle Duality
The Nature of Light
The previous sections showed a confusing picture:
- Light acts as a wave: Phenomena like interference, diffraction, and polarisation can only be explained by a wave model.
- Light acts as a particle: The photoelectric effect can only be explained by a particle (photon) model.
This dual nature is called wave-particle duality. Light exhibits both properties, but we observe them separately depending on the experiment being performed.
The Wave Nature of Matter (De Broglie Hypothesis)
If light (waves) can behave like particles (photons), then perhaps matter (particles like electrons) can also behave like waves. This idea was proposed by Louis de Broglie in 1924.
Evidence for Matter Waves: Electron Diffraction
The key evidence for the wave nature of particles is electron diffraction.
- When a beam of electrons is passed through a crystalline structure (like graphite), they produce a diffraction pattern (concentric rings).
- Diffraction is a uniquely wave phenomenon. Since electrons produce a diffraction pattern, they must possess wave properties.
De Broglie Wavelength
The wavelength associated with a moving particle is called the de Broglie wavelength (\(\lambda\)). It links the momentum of the particle (a particle property) to its wavelength (a wave property).
De Broglie Wavelength Equation:
\[\lambda = \frac{h}{p}\]
Where:
\(\lambda\) is the de Broglie wavelength (m)
\(h\) is the Planck constant
\(p\) is the momentum of the particle (\(p = mv\))
Analogy: Why don't we notice the wave nature of everyday objects, like a moving tennis ball? Because macroscopic objects have very large masses, resulting in a large momentum (\(p\)). Since \(h\) is tiny (\(10^{-34}\)), the resulting de Broglie wavelength is so infinitesimally small that wave effects are impossible to detect. We only see significant wave behaviour for particles with momentum comparable to Planck's constant (like electrons).
Key Takeaway (22.3): Duality means everything (light and matter) acts as both a wave and a particle. The wave nature of matter is described by the de Broglie wavelength, \(\lambda = h/p\).
22.4 Energy Levels in Atoms and Line Spectra
Discrete Energy Levels
We must move from the classical idea of electrons orbiting freely (like planets) to the quantum model, where electrons can only exist in specific, defined orbits, each corresponding to a specific energy value.
In isolated atoms (like atomic hydrogen), the electrons exist in discrete energy levels. These levels are often shown on an energy level diagram.
- The electron cannot have energy values between these levels.
- The lowest level is the ground state (most stable).
- Higher levels are excited states.
Formation of Line Spectra
Electrons can only move between these energy levels by absorbing or emitting a photon whose energy exactly matches the difference between the levels.
1. Emission Line Spectra
When an atom is heated or excited (e.g., in a gas discharge tube), electrons jump to higher energy levels. When they fall back down to a lower level, they emit a photon.
- Since the energy difference (\(E_1 - E_2\)) is discrete, the energy of the emitted photon (\(hf\)) is also discrete.
- The result is a spectrum showing bright coloured lines on a dark background. Each line corresponds to a specific energy transition (a specific frequency/wavelength).
- Every element has a unique emission spectrum, acting as its 'fingerprint'.
2. Absorption Line Spectra
When white light (containing all frequencies) passes through a cool gas:
- Electrons in the gas atoms absorb photons only if the photon energy (\(hf\)) exactly matches the energy required to jump from a lower level to a higher level.
- The absorbed frequencies are removed from the continuous spectrum, resulting in a spectrum with dark lines on a coloured background.
- The pattern of dark lines in an absorption spectrum is exactly the same as the pattern of bright lines in the emission spectrum for that element.
The Energy Transition Equation
The energy of the emitted or absorbed photon is equal to the difference in the energy levels between which the electron moves.
\[hf = E_1 - E_2\]
Where:
\(E_1\) is the energy of the higher level (J or eV)
\(E_2\) is the energy of the lower level (J or eV)
\(hf\) is the energy of the photon involved in the transition (J or eV)
Important Note: Ensure \(E_1\) and \(E_2\) are in the same units (either Joules or eV) as \(hf\). If energy levels are given in eV, and you need frequency \(f\), first calculate \((E_1 - E_2)\) in eV, convert it to Joules, and then use \(f = E/h\).
Quick Review (22.4): Electrons occupy fixed, discrete energy levels. Jumps between levels require the absorption or emission of a photon with energy equal to the energy difference, \(hf = E_1 - E_2\). This results in characteristic line spectra.