Unit 15: Modern Physics
Weightage : 15–18%
Light with a wavelength of \(5.4\times10^{-7}\,\mathrm{m}\) falls on two slits that are separated by a distance of \(1.0\times10^{-3}\,\mathrm{m}\). An interference pattern forms on a screen \(2\,\mathrm{m}\) away.
What is the energy of each photon of this light?
Correct Answer: \( \boxed{\mathrm{D}} \)
The energy of a photon is given by
\( E=hf=\frac{hc}{\lambda} \)
where
\( h=6.63\times10^{-34}\,\mathrm{J\,s} \), \( c=3.00\times10^{8}\,\mathrm{m/s} \), and \( \lambda=5.4\times10^{-7}\,\mathrm{m} \).
Substituting,
\( E=\frac{(6.63\times10^{-34})(3.00\times10^{8})}{5.4\times10^{-7}} \approx3.68\times10^{-19}\,\mathrm{J} \)
This value is approximately
\( \boxed{4\times10^{-19}\,\mathrm{J}} \).
Therefore, the correct answer is (D).
The graph above shows the wave function \( \psi(x) \) of a particle moving in the region \(0<x<4\,\mathrm{m}\). At which of the following positions is the probability of finding the particle greatest?
Correct Answer: \( \boxed{\mathrm{B}} \)
In quantum mechanics, the probability density of finding a particle is proportional to the square of the magnitude of its wave function:
\( P(x)\propto |\psi(x)|^2 \)
Therefore, the particle is most likely to be found where the magnitude of the wave function, \( |\psi| \), is greatest.
From the graph, the largest magnitude of the wave function occurs at \(x=2.0\,\mathrm{m}\), where
\( \psi=-3 \)
giving
\( |\psi|^2=(-3)^2=9 \),
which is greater than the probability density at any of the other listed positions.
Therefore, the correct answer is (B).
Which of the following phenomena can be better understood by considering the wave-particle duality of electrons?
Correct Answer: \( \boxed{\mathrm{C}} \)
The de Broglie hypothesis states that electrons exhibit wave-like behavior with wavelength
\( \lambda=\dfrac{h}{p} \)
When a beam of electrons is reflected from the regularly spaced atoms in a crystal, the electron waves undergo interference, producing alternating bright and dark intensity patterns. This diffraction pattern is direct evidence of the wave nature of electrons.
Choice (A) is also related to the wave nature of electrons, but this question has been converted to a single-correct multiple-choice question. Choices (B) and (D) describe phenomena explained primarily by the particle nature of electromagnetic radiation (photoelectric effect and Compton scattering, respectively).
Therefore, the correct answer is (C).
