波粒二象性：練習題

In a photoelectric effect experiment, monochromatic light of frequency \(f\) and intensity \(I\) shines on a metal surface, producing a photoelectric current. If the intensity is doubled to \(2I\) but the frequency is halved to \(0.5f\), and given that \(0.5f\) is below the threshold frequency of the metal, what will happen to the photoelectric current?

The theoretical resolution of a Transmission Electron Microscope (TEM) is limited by the de Broglie wavelength of the accelerated electrons. To improve the resolution by a factor of 3 (i.e., making the minimum resolvable detail 3 times smaller), the accelerating voltage \(V\) should be increased by a factor of:

In classical wave theory, which property of light was expected to determine the maximum kinetic energy of the emitted photoelectrons, despite experimental evidence showing otherwise?

(a) Frequency
(b) Intensity
(c) Polarization
(d) Speed of light

When monochromatic light of frequency \(f\) is incident on a metal surface, the stopping potential is \(V_1\). When light of frequency \(2f\) is used on the same surface, the stopping potential becomes \(V_2\). Which of the following expressions correctly gives the work function \(\phi\) of the metal?

A photon and an electron have the same de Broglie wavelength \(\lambda\). If \(E_p\) is the energy of the photon and \(E_e\) is the kinetic energy of the electron, which of the following expressions correctly represents the ratio \(\frac{E_p}{E_e}\)?
(Assume \(m\) is the mass of the electron, \(c\) is the speed of light, and \(h\) is Planck's constant.)

A proton is moving at a speed such that its de Broglie wavelength is exactly \(1.0 \times 10^{-11} \text{ m}\). Calculate the momentum of the proton.
(Given: Planck's constant \(h = 6.63 \times 10^{-34} \text{ J s}\))

A photon and an electron are found to have the same de Broglie wavelength \(\lambda\). If the wavelength of both is reduced by half, explain which particle experiences a greater absolute change in its momentum.

The work function of a specific metal surface is \(2.30 \text{ eV}\). Determine the threshold frequency of this metal.
(Given: \(h = 6.63 \times 10^{-34} \text{ J s}\), \(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\))

When a metal surface is illuminated with monochromatic light of wavelength \(400 \text{ nm}\), the stopping potential for the emitted photoelectrons is found to be \(0.80 \text{ V}\).

(a) Calculate the work function of the metal in \(\text{eV}\).
(b) Determine the threshold frequency of this metal.
(Take \(h = 6.63 \times 10^{-34} \text{ J s}\), \(c = 3.00 \times 10^8 \text{ m s}^{-1}\), and \(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\))

In Bohr's model of the hydrogen atom, an electron moves in a circular orbit around the nucleus.

(a) Use the postulate of angular momentum quantization, \(m_e vr = \frac{nh}{2\pi}\), to show that the circumference of the \(n\)-th orbit is equal to \(n\) times the de Broglie wavelength of the electron.
(b) For the first orbit (\(n=1\)), the radius is \(5.29 \times 10^{-11} \text{ m}\). Calculate the speed of the electron in this orbit.
(c) Calculate the de Broglie wavelength of the electron in this first orbit and verify the relationship derived in part (a).