Central Forces: Spring-2026
HW 08: Due W4 D5

  1. Quantum Numbers on the Sphere Consider an arbitrary state for a quantum particle confined to the surface of a sphere written as a superposition of spherical harmonics: \begin{equation} \left|{\Psi}\right\rangle =\sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell c_{\ell m}\left|{\ell, m}\right\rangle \end{equation}
    1. Suppose you wanted to calculate the probability that an energy measurment of this state would yield \(E_{17}\). What coefficients \( c_{\ell m} \) would you need to find to calculate this? Write an expression for this probability.
    2. Suppose you wanted to calculate the probability that a measurement of the \(z\)-component of angular momentum would yield \(5\hbar\). What coefficients \( c_{\ell m} \) would you need to find to calculate this? Write an expresson for this probability.
  2. Sphere Questions Consider the following normalized state for the rigid rotor given by: \begin{equation} \left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left\vert 1, -1\right\rangle + \frac{1}{\sqrt{3}}\left\vert 1, 0\right\rangle + \frac{i}{\sqrt{6}}\left\vert 0, 0\right\rangle \end{equation}
    1. Write the state as a superposition of spherical harmonics \(Y_l^m\)
    2. What is the probability that a measurement of \(L_z\) will yield \(2\hbar\)? \(-\hbar\)? \(0\hbar\)?
    3. If you measured the z-component of angular momentum to be \(-\hbar\), what would the state of the particle be immediately after the measurement is made? What about if it yields \( 0\hbar \)?
    4. What is the expectation value of \(L_z\) in the original state \(\left|{\psi}\right\rangle \)?
    5. What is the expectation value of \(L^2\) in the original state \(\left|{\psi}\right\rangle \)?

    6. What is the expectation value of the energy in the original state \(\left|{\psi}\right\rangle \)?

  3. Sphere Consider the normalized function: \begin{equation} f(\theta,\phi)= \begin{cases} N\left(\frac{\pi^2}{4}-\theta^2\right) & 0<\theta<\frac{\pi}{2} \\ 0 & \frac{\pi}{2}<\theta<\pi \end{cases} \end{equation} where \begin{equation} N=\frac{1}{\sqrt{\frac{\pi^5}{8} +2\pi^3-24\pi^2+48\pi}} \end{equation}
    1. Find the coefficients of the \(\left|\ell,m\right\rangle=\left|0,0\right\rangle\), \(\left|1,-1\right\rangle\), \(\left|1,0\right\rangle\), and \(\left|1,1\right\rangle\) terms in the spherical harmonic expansion of \(f(\theta,\phi)\). It is helpful to remember that Mathematica has a built-in function for spherical harmonics: SphericalHarmonicY[l, m, \[Theta], \[Phi]].
    2. If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that a measurement of the square of the total angular momentum will yield \(2\hbar^2\)? \(4\hbar^2\)?
    3. If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that the particle can be found in the region \(0<\theta<\frac{\pi}{6}\) and \(0<\phi<\frac{\pi}{6}\)?