波粒二象性：練習題

The de Broglie wavelength of a particle is given by \(\lambda = \frac{h}{p}\). If the momentum, \(p\), of an electron is reduced to one-third of its original value, how does its de Broglie wavelength change?

Which phenomenon provides the most direct evidence that electromagnetic radiation, such as light, possesses a particulate (photon) nature?

A beam of electrons produces a diffraction pattern after passing through a crystalline lattice. If the speed of the electrons is doubled, how does the de Broglie wavelength $(\lambda)$ and the angle of diffraction $(\theta)$ change?

Which row in the table correctly identifies the phenomena that demonstrate the wave nature of matter and the particulate nature of electromagnetic radiation?

A beam of protons has a momentum of $4.42 \times 10^{-23} \text{ kg m s}^{-1}$. Calculate the de Broglie wavelength associated with these protons.

(Use Planck constant $h = 6.63 \times 10^{-34} \text{ J s}$)

A proton is accelerated to a velocity of \(2.0 \times 10^5 \text{ m s}^{-1}\). Given that the mass of a proton is \(1.67 \times 10^{-27} \text{ kg}\), calculate the de Broglie wavelength associated with the proton. (Use \(h = 6.63 \times 10^{-34} \text{ J s}\)).

An electron and a neutron are both travelling with the same non-relativistic kinetic energy. Determine which particle has the longer de Broglie wavelength and justify your answer based on their masses.

A proton and an alpha particle (where the mass of the alpha particle is approximately four times that of the proton) have the same kinetic energy. Calculate the ratio of the de Broglie wavelength of the proton, \(\lambda_p\), to the de Broglie wavelength of the alpha particle, \(\lambda_{\alpha}\).

(a) A baseball of mass \(0.145 \text{ kg}\) is thrown at a speed of \(40.0 \text{ m/s}\). Calculate the de Broglie wavelength associated with the baseball.

(b) Explain why wave properties, such as diffraction, are not observed for macroscopic objects like the baseball, but are observable for electrons accelerated through a few hundred volts.

(Use: Planck constant, \(h = 6.63 \times 10^{-34} \text{ J s}\))

Electrons are accelerated from rest through a potential difference \(V_1 = 150 \text{ V}\). Protons are accelerated from rest through a potential difference \(V_2 = 500 \text{ V}\).

(a) Derive an expression for the de Broglie wavelength \(\lambda\) of a particle of mass \(m\) and charge \(e\) accelerated through a potential difference \(V\), showing that \(\lambda \propto 1/\sqrt{V}\).

(b) Calculate the de Broglie wavelength of the electrons accelerated through \(V_1 = 150 \text{ V}\).

(c) If both the electrons and protons pass through identical diffraction apertures, compare qualitatively the expected diffraction patterns based on their calculated wavelengths. (You may assume the rest masses and charges of the electron and proton are known).

(Use: Planck constant, \(h = 6.63 \times 10^{-34} \text{ J s}\); Mass of electron, \(m_e = 9.11 \times 10^{-31} \text{ kg}\); Mass of proton, \(m_p = 1.67 \times 10^{-27} \text{ kg}\))