Physics Textbooks (257)

Consider the Hamiltonian of a spinless particle of charge \(e\). In the presence of a static magnetic field, the interaction terms can be generated by $$ \mathbf{p}_{\text {operator }} \rightarrow \mathbf{p}_{\text {operator }}-\frac{e \mathbf{A}}{c} $$ where \(\mathbf{A}\) is the appropriate vector potential. Suppose, for simplicity, that the magnetic field \(\mathbf{B}\) is uniform in the positive \(z\)-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment \((e / 2 m c) \mathbf{L}\) with the magnetic field \(\mathbf{B}\). Show that there is also an extra term proportional to \(B^{2}\left(x^{2}+y^{2}\right)\), and comment briefly on its physical significance.

A $12-cm$long thin rod has the nonuniform charge density $\lambda \left(x\right)=(2.0\mathrm{nC}/\mathrm{cm})e^{-\left|x\right|/(6.0\mathrm{cm})}$, where localid="1648623923714" $x$is measured from the center of the rod. What is the total charge on the rod?

Hint: This exercise requires an integration. Think about how to handle the absolute value sign.

Question: Obtain Eq. 9.20 directly from the wave equation by separation of variables.

In addition to the Coulomb interaction, there exists another, called the hyperfine interaction, between the electron and proton in the hydrogen atom. The Hamiltonian describing this interaction, which is due to the magnetic moments of the two particles is, $$ H_{h f}=A \mathrm{~S}_{1} \cdot \mathrm{S}_{2} \quad(A>0) $$

We consider the emission by a star of light of a narrow frequency band with centre frequency \(\bar{\omega}\) and corresponding wavelength \(\bar{\lambda}=c 2 \pi / \bar{\omega}\) The star is an extended spatially incoherent source. Let \(I(x, y)\) be the intensity on the star's surface orientated towards the earth. The aim of the exercise is to determine \(I(x, y)\) by stellar interferometry. Let \(U_{0}(x, y, t)\) be the field emitted at the surface of the star. Then the mutual coherence function at points \(S_{1}=\left(x_{1}, y_{1}\right), S_{2}=\left(x_{2}, y_{2}\right)\) on the surface of the star is: $$ \begin{aligned} \Gamma\left(S_{1}, S_{2}, \tau\right) &=\left\langle U_{0}\left(S_{1}, t\right)^{*} U_{0}\left(S_{2}, t+\tau\right)\right\rangle \\ &=I\left(x_{1}, y_{1}\right) e^{i \omega \tau} \delta\left(x_{1}-x_{2}\right) \delta\left(y_{1}-y_{2}\right) \quad \text { for all } \tau \end{aligned} $$ a) Let \(z_{e}\) be the distance of the star from earth. Use the quasi- monochromatic approximation to derive the field in a point \(P_{e}=\left(x_{e}, y_{e}\right)\) on earth. b) Show that the mutual coherence function in two points \(P_{e}=\left(x_{e}, y_{e}\right)\) and \(\tilde{P}_{2}=\left(\tilde{x}_{e}, \tilde{y}_{e}\right)\) on earth is for \(\tau=0\) given by $$ \Gamma\left(P_{e}, \tilde{P}_{e}, \tau=0\right)=\iint I\left(x_{1}, y_{1}\right) e^{2 \pi i\left(\frac{x_{e}-\tilde{x}_{e}}{\lambda z_{e}} x_{1}+\frac{y_{e}-\bar{y}_{e}}{\lambda z_{c}} y_{1}\right)} d x_{1} d y_{1} $$ i.e. the mutual coherence function between points on earth for time delay \(\tau=0\) can be expressed in the Fourier transform of the intensity \(I(x, y)\) emitted by the star, evaluated at spatial frequencies $$ \xi=\frac{x_{e}-\tilde{x}_{e}}{\lambda z_{e}} \text { and } \eta=\frac{y_{e}-\tilde{y}_{e}}{\lambda z_{e}} . $$ c) Explain how the mutual coherence for time delay \(\tau=0\) can be measured on earth using interferometry and how this can lead to retrieving the intensity of the star. d) What determines the resolution that can be achieved?