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8.E: Central Potentials (Exercises)

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    15775
    1. A particle of mass \(m\) is placed in a finite spherical well: \[V(r) = \left\{ \begin{array}{lcl} -V_0&\mbox{\hspace{1cm}}&\mbox{for} r\leq a\\ 0&&\mbox{for} r>a \end{array} \right. ,\] with \(V_0>0\) and \(a>0\). Find the ground-state by solving the radial equation with \(l=0\). Show that there is no ground-state if \(V_0\,a^{\,2}< \pi^{\,2}\,\hbar^{\,2}/(8\,m)\).

    2. Consider a particle of mass \(m\) in the three-dimensional harmonic oscillator potential \(V(r)=(1/2)\,m\,\omega^{\,2}\,r^{\,2}\). Solve the problem by separation of variables in spherical coordinates, and, hence, determine the energy eigenvalues of the system.

    3. The normalized wavefunction for the ground-state of a hydrogen-like atom (neutral hydrogen, \({\rm He}^+\), \({\rm Li}^{++}\), et cetera.) with nuclear charge \(Z\,e\) has the form \[\psi = A\,\exp(-\beta\,r),\] where \(A\) and \(\beta\) are constants, and \(r\) is the distance between the nucleus and the electron. Show the following:

      1. \(A^2=\beta^{\,3}/\pi\).

      2. \(\beta = Z/a_0\), where \(a_0=(\hbar^{\,2}/m_e)\,(4\pi\,\epsilon_0/e^{\,2})\).

      3. The energy is \(E=-Z^{\,2}\,E_0\) where \(E_0 = (m_e/2\,\hbar^{\,2})\,(e^{\,2}/4\pi\,\epsilon_0)^2\).

      4. The expectation values of the potential and kinetic energies are \(2\,E\) and \(-E\), respectively.

      5. The expectation value of \(r\) is \((3/2)\,(a_0/Z)\).

      6. The most probable value of \(r\) is \(a_0/Z\).

    4. An atom of tritium is in its ground-state. Suddenly the nucleus decays into a helium nucleus, via the emission of a fast electron that leaves the atom without perturbing the extranuclear electron, Find the probability that the resulting \({\rm He}^+\) ion will be left in an \(n=1\), \(l=0\) state. Find the probability that it will be left in a \(n=2\), \(l=0\) state. What is the probability that the ion will be left in an \(l>0\) state?

    5. Calculate the wavelengths of the photons emitted from the \(n=2\), \(l=1\) to \(n=1\), \(l=0\) transition in hydrogen, deuterium, and positronium.

    6. To conserve linear momentum, an atom emitting a photon must recoil, which means that not all of the energy made available in the downward jump goes to the photon. Find a hydrogen atom’s recoil energy when it emits a photon in an \(n=2\) to \(n=1\) transition. What fraction of the transition energy is the recoil energy?

    7. Show that the most probable value of \(r\) in the hydrogen ground state is \(a_0\).

    8. Let \(R_{nl}(r)= v_{nl}(r/a_0)/(r/a_0)\), where \(R_{nl}(r)\) is a properly normalized radial hydrogen wavefunction corresponding to the conventional quantum numbers \(n\) and \(l\), and \(a_0\) is the Bohr radius. [ex4 .fgh]

      1. Demonstrate that \[\frac{d^{\,2}v_{nl}}{dy^{\,2}} = \left[\frac{l\,(l+1)}{y^{\,2}} -\frac{2}{y} + \frac{1}{n^{\,2}}\right]v_{nl}.\]

      2. Show that \(v_{nl}\sim y^{\,1+l}\) in the limit \(y\rightarrow 0\).

      3. Demonstrate that \[\left(\frac{1}{n^{\,2}}-\frac{1}{m^{\,2}}\right)\int_0^\infty v_{nl}(y)\,v_{ml}(y)\,dy = 0.\]

      4. Hence, deduce that \[\int_0^\infty r^{\,2}\,R_{nl}(r)\,R_{ml}(r)\,dr=0\] for \(n\neq m\).

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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