Wave-Particle Duality and de Broglie Hypothesis

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Questions and Answers

Which phenomenon provides evidence that the true nature of light requires considering both wave and particle properties?

Interference
Refraction
Diffraction
Photoelectric effect (correct)

According to de Broglie's hypothesis, the wavelength associated with a particle is directly proportional to its momentum.

False (B)

What experimental evidence led to the acceptance of de Broglie's hypothesis, confirming the wave properties of particles?

Davisson and Germer experiment

According to Bragg's diffraction law, the condition for constructive interference is given by $d sin θ = nλ$, where d is the inter-atomic spacing, θ is the angle of incidence, n is an integer, and λ is the ________.

wavelength Signup and view all the answers

Match each term with its corresponding description:

de Broglie wavelength = The wavelength associated with a moving particle. Photoelectric effect = The emission of electrons when light shines on a material. Bragg's diffraction = The diffraction of X-rays by crystal lattices. Momentum = Product of mass and velocity. Signup and view all the answers

In the Davisson-Germer experiment, what was the primary purpose of using a nickel crystal?

To scatter electrons (D) Signup and view all the answers

According to de Broglie, if the kinetic energy of particle increases, its wavelength will also increase.

False (B) Signup and view all the answers

What is the relationship between the frequency of a matter wave and the total relativistic energy of the particle it is associated with?

$f = E/h$ Signup and view all the answers

What is the significance of the agreement between the calculated wavelength and that derived from Bragg's diffraction law in the context of the de Broglie hypothesis?

It provides evidence supporting the wave nature of electrons, consistent with the de Broglie hypothesis. (A) Signup and view all the answers

A wave packet representing a quantum particle is formed solely through destructive interference of numerous waves in a localized region of space.

False (B) Signup and view all the answers

Explain how the superposition of numerous waves leads to the formation of a wave packet representing a quantum particle. Hint: Consider the roles of constructive and destructive interference.

When numerous waves superpose, they create a region of constructive interference where the amplitudes add up, forming a localized wave packet. Outside this region, destructive interference cancels out the waves, confining the wave packet to a small area, thus representing a quantum particle. Signup and view all the answers

According to classical predictions, what is the effect of oscillating electromagnetic waves on electrons?

Electrons experience oscillations and re-radiate in all directions, and radiation pressure causes acceleration in the direction of wave propagation. Different electrons move at different speeds after the interaction. (C) Signup and view all the answers

In the mathematical representation of two waves, $y_1 = A \cos(k_1x - \omega_1t)$ and $y_2 = A \cos(k_2x - \omega_2t)$, the term $\Delta k$ represents the difference in , while $\Delta \omega$ represents the difference in .

wave numbers, angular frequencies Signup and view all the answers

In Compton scattering, the incident photon's energy ($E_o$) is equal to the scattered photon's energy ($E'$) plus the kinetic energy (K) of the electron, represented as $E_o = E' + K$. This equation reflects the principle of energy creation during the collision.

False (B) Signup and view all the answers

Match the mathematical terms with their descriptions in the context of wave packet analysis.

$\Delta k$ = Difference in wave numbers $\Delta \omega$ = Difference in angular frequencies $\frac{\Delta k}{2}x - \frac{\Delta \omega}{2}t$ = Modulation term influencing the envelope of the wave packet $\frac{k_1 + k_2}{2}x - \frac{\omega_1 + \omega_2}{2}t$ = Carrier wave representing the average wave properties within the wave packet Signup and view all the answers

In the context of Compton scattering, what physical quantities are conserved during the interaction between a photon and an electron?

total energy and total linear momentum Signup and view all the answers

The Compton shift equation, which describes the change in wavelength of a photon after scattering from an electron, is given by $λ' - λ_o = \frac{h}{mc}(1 - cos θ)$. In this equation, $θ$ represents the ______ angle of the scattered photon.

scattering Signup and view all the answers

Which of the following equations correctly represents the relativistic relationship between the total energy (E), momentum (p), and rest mass (m) of an electron?

$E = \sqrt{p^2c^2 + m^2c^4}$ (B) Signup and view all the answers

Match the variable with the correct expression in the context of Compton scattering and relativistic equations:

$λ_o$ = Wavelength of the incident photon $p_o$ = Momentum of the incident photon $E'$ = Energy of the scattered photon $γ$ = Relativistic factor $1/\sqrt{1 - v^2/c^2}$ Signup and view all the answers

Given that $γ = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, how does the kinetic energy (K) of an electron relate to its total relativistic energy (E) and rest energy ($mc^2$) in Compton scattering?

K = E - mc^2 (C) Signup and view all the answers

Light exclusively exhibits diffraction and interference phenomena, which can be entirely explained by its particle nature.

False (B) Signup and view all the answers

Which of the following best describes the significance of Planck's constant ($h$) derived from fitting Planck's law to experimental blackbody radiation data?

It represents the smallest unit of energy that can be emitted or absorbed as electromagnetic radiation, demonstrating energy quantization. (A) Signup and view all the answers

According to the Rayleigh-Jeans law, the intensity of radiation emitted by a blackbody decreases as the wavelength approaches zero, preventing the ultraviolet catastrophe.

False (B) Signup and view all the answers

Explain how Planck's hypothesis of quantized energy levels for oscillators in the cavity walls of a blackbody resolves the ultraviolet catastrophe predicted by classical physics.

Planck's hypothesis posits that energy is emitted or absorbed in discrete packets (quanta), limiting the number of high-frequency (short-wavelength) modes that can be excited, thereby preventing the intensity from diverging to infinity as predicted by the Rayleigh-Jeans Law. Signup and view all the answers

According to Stefan's Law, the power radiated by a blackbody is directly proportional to the fourth power of its absolute ______.

temperature Signup and view all the answers

Match the concepts with their descriptions:

Stefan's Law = The total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's thermodynamic temperature. Wien's Displacement Law = The black body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature. Rayleigh-Jeans Law = An early attempt to describe the spectral radiance of electromagnetic radiation emitted by a black body; fails at short wavelengths. Planck's Law = Describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature; accurately predicts the black body spectrum. Signup and view all the answers

Which of the following statements accurately describes the relationship between the wavelength at which maximum blackbody radiation intensity occurs ($\lambda_m$) and the temperature ($T$) of the blackbody, according to Wien's Displacement Law?

$\lambda_m$ is inversely proportional to $T$. (B) Signup and view all the answers

How does Planck's Law modify the classical understanding of energy emission and absorption in blackbody radiation?

It proposes that energy is emitted or absorbed in discrete packets (quanta) proportional to the frequency of the radiation. (A) Signup and view all the answers

Consider a blackbody at temperature $T_1$ emitting radiation with peak wavelength $\lambda_1$. If the temperature is doubled to $2T_1$, what happens to the new peak wavelength $\lambda_2$?

$\lambda_2 = \lambda_1 / 2$ (it halves) (B) Signup and view all the answers

What is the relationship between group velocity ($v_g$) and phase velocity ($v_p$) described?

$v_g = v_p - \lambda \frac{dv_p}{d\lambda}$ (B) Signup and view all the answers

What is the formula for phase velocity ($v_p$) in terms of angular frequency ($\omega$) and wave number ($k$)?

$v_p = \frac{\omega}{k}$ (B) Signup and view all the answers

The group speed ($v_g$) of a wave packet is always greater than the phase speed ($v_p$) of the individual waves within the packet.

False (B) Signup and view all the answers

In the double-slit experiment, what does the variable 'd' represent in the equation $d \sin \theta = m \lambda$?

slit separation Signup and view all the answers

The relationship between group speed ($v_g$) and particle speed ($u$) indicates that $v_g$ is equal to ____.

u Signup and view all the answers

Match the following terms with their corresponding descriptions:

Phase Speed ($v_p$) = Speed at which the crest of an individual wave moves. Group Speed ($v_g$) = Speed of the wave packet. $\omega$ = Angular frequency. $k$ = Wave number. Signup and view all the answers

Given $\omega = 2 \pi f$ and $k = \frac{2 \pi}{\lambda}$, and knowing $v_p = \frac{\omega}{k}$, which of the following is an alternative expression for $v_p$?

$v_p = f \lambda$ (B) Signup and view all the answers

In the context of the relationship between group speed ($v_g$) and particle speed ($u$), which equation correctly represents the group speed in terms of the derivative of energy ($E$) with respect to momentum ($p$)?

$v_g = \frac{dE}{dp}$ (C) Signup and view all the answers

In the double-slit experiment, increasing the wavelength ($\lambda$) of the electrons used will decrease the separation between the interference fringes observed.

False (B) Signup and view all the answers

According to the material, what variables determine group speed?

$\Delta\omega$ and $\Delta k$ Signup and view all the answers

What is the fundamental implication of the Heisenberg uncertainty principle regarding simultaneous measurements of a subatomic particle's position and momentum?

The act of measuring a particle's position inevitably disturbs its momentum, and vice versa, leading to inherent uncertainties. (D) Signup and view all the answers

According to Wien's displacement law, as the temperature of a black body increases, the wavelength at which its emission spectrum peaks shifts towards longer wavelengths.

False (B) Signup and view all the answers

A blackbody emits radiation. If the temperature of the blackbody doubles, by what factor does the total power radiated by the blackbody increase, according to the Stefan-Boltzmann law? Give your answer as a number.

16 Signup and view all the answers

In the context of the photoelectric effect, the minimum energy required to remove an electron from a metal surface is known as the ______.

work function Signup and view all the answers

What is the effect on a photon's wavelength when it undergoes Compton scattering and transfers some of its energy to an electron?

The wavelength increases because the photon loses energy. (B) Signup and view all the answers

Which of the following scenarios best illustrates wave-particle duality?

Electrons behaving as particles when detected at a specific location but exhibiting interference patterns that are characteristic of waves. (B) Signup and view all the answers

Match the scientist with their contribution to quantum physics:

Heisenberg = Uncertainty Principle Wien = Displacement Law Stefan = Stefan Constant Signup and view all the answers

What is the relationship between the lifetime of an excited atomic state and the line width ($\Delta f$) of the emitted radiation, as described by the uncertainty principle?

The line width is inversely proportional to the lifetime of the excited state; shorter lifetimes result in broader line widths. (C) Signup and view all the answers

Flashcards

Black-body Radiation

Electromagnetic radiation emitted by a black body.

Stefan's Law

Hotter objects emit more energy per unit area than colder objects.