Question

A photon scatters off of a free electron. (a) What is the maximum possible change in wavelength? (b) Suppose a photon scarters off of a free proton. What is the maximum possible change in wavelength now? (c) Which more clearly demonstrates the particle nature of electromagnetic radiation- - collision with an electron or collision with a proton?

Short answer

The maximum change in the wavelength of a photon scattering off an electron is \( \frac{2h}{m_ec} \). When the photon scatters off a proton, the maximum change is \( \frac{2h}{m_pc} \). The scattering off an electron more clearly exemplifies the particle nature of electromagnetic radiation because the change in wavelength is more significant.

Step-by-step solution

Calculate maximum change in wavelength for photon-electron scattering

Firstly, it's needed to use the Compton formula with the electron mass, and looking for the maximum possible change in wavelength, which means setting the scattering angle \( \theta = 180° \) or \( \cos(\theta) = -1 \). So, \( \Delta \lambda = \frac{h}{m_ec}(1 - (-1)) = \frac{2h}{m_ec} \). Here \( h \) is the Planck's constant, \( m_e \) is the electron's rest mass, and \( c \) is the speed of light.

Calculate maximum change in wavelength for photon-proton scattering

Similarly, for a photon scattering off a free proton, the mass of the proton should be inserted into the Compton formula. Since the proton's mass is much greater than the electron's, the change in wavelength will be less: \( \Delta \lambda = \frac{h}{m_pc}(1 - (-1)) = \frac{2h}{m_pc} \). Here \( m_p \) is the rest mass of a proton.

Analyze which scattering more clearly demonstrates the particle nature of electromagnetic light

As the greater change in wavelength would exemplify more clearly the particle nature of electromagnetic radiation, the scattering involving the electron would more clearly demonstrate this. The reason is that the wavelength change is inversely proportional to the mass of the scattered particle, meaning that the lighter the particle, the more significant the change.

Key concepts

Photon-Electron Scattering

Photon-electron scattering, also known as Compton scattering, describes the interaction between a photon and an electron. When a photon collides with an electron, it transfers part of its energy to the electron. This interaction causes the photon's path to change direction.
The Compton formula, \( \Delta \lambda = \frac{h}{m_ec}(1 - \cos\theta) \), is used to calculate the change in wavelength of the photon after scattering. Here, \( h \) is Planck's constant, \( m_e \) is the electron's rest mass, and \( c \) is the speed of light.

• The maximum change in wavelength occurs when the scattering angle, \( \theta \), is \( 180^\circ \).
• The equation simplifies to \( \Delta \lambda = \frac{2h}{m_ec} \).

This significant change in wavelength highlights the process and underpins the quantum nature of light interactions.

Wavelength Change

Wavelength change in photon scattering is a crucial concept in understanding how light behaves like a particle. When a photon collides with a particle like an electron or a proton, its wavelength changes. This change is pivotal in revealing the particle-like behavior of photons.
The Compton effect demonstrates that the change in wavelength depends on the mass of the particle it collides with. The equation \( \Delta \lambda = \frac{h}{mc}(1 - \cos\theta) \) shows how wavelength change is inversely related to the mass of the particle encountered.

• Lighter particles, like electrons, cause a larger wavelength shift.
• Heavier particles, like protons, result in a smaller change.

Assessing this change is critical for experiments and applications that probe the fundamental properties of light and matter.

Particle Nature of Light

The particle nature of light is demonstrated through experiments like Compton scattering, where light behaves not just as a wave but also as particles known as photons. These interactions provide evidence supporting the theory of quantum mechanics.
This wave-particle duality suggests that light can exhibit characteristics of both waves and particles. In photon-electron scattering, because electrons are relatively light, the significant change in wavelength supports the view that light acts as particles.

• The greater the wavelength shift, the more apparent the particle nature.
• Photon scattering on light particles like electrons highlights this duality.

Such phenomena are essential for a deeper understanding of how light interacts with matter at a fundamental level.

Photon-Proton Scattering

Photon-proton scattering is similar in concept to photon-electron scattering, but involves larger particles. When a photon encounters a proton, the mass difference leads to less significant changes in wavelength.
In this scenario, the Compton formula is adjusted with the proton's mass, \( m_p \). The equation \( \Delta \lambda = \frac{2h}{m_pc} \) is used to calculate the wavelength change. Due to the proton's large mass compared to the electron, the maximum possible wavelength change here is much smaller.

• Proton's larger mass results in a smaller wavelength shift.
• This affects how we observe the light's particle nature in the experiment.

Although the change is smaller, studying such interactions is vital for particle physics and aids in understanding forces at play at the subatomic level.