Photons and the Quantum Nature of Light

21. Photons

Photons are fundamental particles of light, representing the quantum nature of electromagnetic radiation. Understanding photons is crucial for comprehending how light interacts with matter and forms the basis of quantum mechanics.

21.1 Quantum Nature of Radiation

Before the 20th century, light was primarily understood as a wave, exhibiting phenomena like diffraction and interference. However, certain experiments, particularly those involving the interaction of light with matter, could not be explained by the classical wave theory. Max Planck introduced the concept that energy is quantized, meaning it can only exist in discrete packets or 'quanta'.

Albert Einstein later extended this idea to light, proposing that light itself consists of discrete energy packets called photons. Each photon carries an energy (E) directly proportional to its frequency (ν) and inversely proportional to its wavelength (λ):

E = hν = hc/λ

Where:

• h is Planck's constant (approximately 6.626 x 10^-34 J·s)
• c is the speed of light in a vacuum (approximately 3.00 x 10⁸ m/s)
• ν (nu) is the frequency of the light
• λ (lambda) is the wavelength of the light

This revolutionary idea established that light exhibits both wave-like and particle-like properties, a concept known as wave-particle duality.

Key Properties of Photons:

• Zero Rest Mass: Photons have no rest mass, meaning they can only exist in motion.
• Speed of Light: In a vacuum, photons always travel at the speed of light, c.
• Energy and Momentum: They carry energy (E = hν) and momentum (p = E/c = h/λ).
• Electrically Neutral: Photons have no electric charge, so they are not affected by electric or magnetic fields.
• Quantized Energy: The energy of a photon is discrete and depends only on its frequency.

21.2 Einstein’s Photoelectric Equation; Stopping Potential

The photoelectric effect is the emission of electrons from a material when light shines on it. Classical wave theory failed to explain several observations of this effect:

• Existence of a Threshold Frequency: No electrons are emitted if the light's frequency is below a certain minimum (threshold frequency), regardless of its intensity.
• Instantaneous Emission: Electron emission is almost instantaneous if the frequency is above the threshold, even at very low light intensities.
• Kinetic Energy Dependence: The maximum kinetic energy of the emitted electrons depends only on the frequency of the light, not its intensity.
• Intensity Dependence: The number of emitted electrons (photocurrent) is proportional to the intensity of the light.

Einstein successfully explained the photoelectric effect in 1905 by applying Planck's quantum hypothesis. He proposed that when a photon strikes the surface of a metal, it transfers all its energy to a single electron. If this energy is sufficient to overcome the binding forces holding the electron in the metal, the electron is ejected.

Einstein's Photoelectric Equation:

The energy of an incident photon (hν) is used in two ways:

1. To overcome the work function (Φ or W0) of the metal, which is the minimum energy required to eject an electron from its surface.
2. To provide the emitted electron with kinetic energy (Kmax).

hν = Φ + Kmax

Or, the maximum kinetic energy of the photoelectron is:

Kmax = hν - Φ

Where:

• Φ = hν0, where ν0 is the threshold frequency. Below this frequency, no photoemission occurs.

If the energy of the incident photon (hν) is less than the work function (Φ), no electrons will be emitted, regardless of the intensity of the light.

Stopping Potential (V_s):

To measure the maximum kinetic energy of the photoelectrons, a retarding potential is applied to stop them. The minimum negative potential applied to the anode with respect to the cathode that is just sufficient to stop the most energetic photoelectrons from reaching the anode is called the stopping potential (or cut-off potential), Vs.

At the stopping potential, the work done by the electric field (eVs) is equal to the maximum kinetic energy of the photoelectrons:

Kmax = eVs

Combining with Einstein's equation:

eVs = hν - Φ

This equation shows that a plot of stopping potential (Vs) versus frequency (ν) should be a straight line with a slope of h/e and a y-intercept of -Φ/e.

Example:

Light of wavelength 400 nm is incident on a metal plate with a work function of 2.5 eV. Calculate the maximum kinetic energy of the emitted photoelectrons and the stopping potential.

Given:

• λ = 400 nm = 400 x 10^-9 m
• Φ = 2.5 eV
• h = 6.626 x 10^-34 J·s
• c = 3.00 x 10⁸ m/s
• e = 1.602 x 10^-19 C (for converting J to eV or vice-versa)

Step 1: Calculate the energy of the incident photon (hν).

E = hc/λ = (6.626 x 10-34 J·s)(3.00 x 108 m/s) / (400 x 10-9 m)

E = 4.97 x 10-19 J

Step 2: Convert photon energy to eV.

E (in eV) = (4.97 x 10-19 J) / (1.602 x 10-19 J/eV) ≈ 3.10 eV

Step 3: Calculate the maximum kinetic energy (K_max).

Kmax = E - Φ = 3.10 eV - 2.5 eV = 0.60 eV

Step 4: Calculate the stopping potential (V_s).

Vs = Kmax / e = 0.60 eV / e = 0.60 V

So, the maximum kinetic energy is 0.60 eV, and the stopping potential is 0.60 V.

21.3 Measurement of Planck’s Constant

The photoelectric effect provides a direct experimental method to determine Planck's constant (h). By rearranging Einstein's photoelectric equation in terms of stopping potential:

eVs = hν - Φ

Vs = (h/e)ν - (Φ/e)

This equation is in the form of a straight line y = mx + c, where:

• y corresponds to Vs
• x corresponds to ν
• The slope m corresponds to h/e
• The y-intercept c corresponds to -Φ/e

Experimental Procedure:

1. Setup: A photocell (a vacuum tube containing a photosensitive cathode and an anode) is used. Monochromatic light (light of a single frequency) from a source is directed onto the cathode.
2. Vary Frequency: The frequency (or wavelength) of the incident light is varied using different color filters or a monochromatic light source.
3. Measure Stopping Potential: For each frequency, the stopping potential (Vs) is measured. This is done by applying a variable retarding voltage to the anode and determining the voltage at which the photocurrent just drops to zero.
4. Plot Graph: A graph is plotted with the stopping potential (Vs) on the y-axis and the frequency (ν) of the incident light on the x-axis.
5. Determine Slope: The resulting graph is a straight line. The slope of this line is calculated (ΔVs / Δν).
6. Calculate Planck's Constant: Since the slope of the Vs vs. ν graph is h/e, Planck's constant h can be determined by multiplying the measured slope by the elementary charge e (which is a known constant, 1.602 x 10^-19 C).

h = (Slope) × e

Additionally, the work function Φ of the material can be found from the y-intercept (-Φ/e) or by extrapolating the line to the x-axis to find the threshold frequency ν0 (where Vs = 0 and hν0 = Φ).

This experiment, performed by Millikan and others, provided strong evidence for the particle nature of light and the validity of quantum theory, allowing for a precise determination of Planck's constant.