Problem \(\PageIndex{1}\) Consider the operator \(\hat A\) defined in Equation \(11.1.1\) as \(\hat A=\hat x + \dfrac{d}{dx}\). Is it linear or non-linear? Justify. Problem \(\PageIndex{2}\) Which of these functions are eigenfunctions of the operator \(-\frac{d^2}{dx^2}\)? Give the corresponding eigenvalue when appropriate. In each case \(k\) can be regarded as a constant. \[f_1(x)=e^{ikx} \nonumber \] \[f_2(x)=\cos(kx) \nonumber \] \[f_3(x)=e^{-kx^2} \nonumber \] \[f_4(x)=e^{ikx}-cos(kx) \nonumber \] Problem \(\PageIndex{3}\) In quantum mechanics, the \(x\), \(y\) and \(z\) components of the angular momentum are represented by the following operators: \[ \begin{align*} \hat{L}_x &=i\hbar\left(\sin\phi\frac{\partial}{\partial \theta}+\frac{\cos\phi}{\tan \theta}\frac{\partial}{\partial\phi}\right) \\[4pt] \hat{L}_y &=i\hbar\left(-\cos\phi\frac{\partial}{\partial \theta}+\frac{\sin\phi}{\tan \theta}\frac{\partial}{\partial\phi}\right) \\[4pt] \hat{L}_z &=-i\hbar\left(\frac{\partial}{\partial \phi}\right) \end{align*} \] The operator for the square of the magnitude of the orbital angular momentum, \(\hat{L}^2=\hat{L}^2_x +\hat{L}^2_y+\hat{L}^2_z\) is: \[\hat{L}^2=-\hbar^2\left(\frac{\partial^2}{\partial \theta^2}+\frac{1}{\tan \theta}\frac{\partial}{\partial\theta}+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}\right) \nonumber \] a) Show that the three 2p orbitals of the H atom are eigenfunctions of both \(\hat{L}^2\) and \(\hat{L}_z\), and determine the corresponding eigenvalues. \[\psi_{2p0}=\frac{1}{\sqrt{32\pi a_0^3}}r e^{-r/2 a_0}\cos\theta \nonumber \] \[\psi_{2p+1}=\frac{1}{\sqrt{64\pi a_0^3}}r e^{-r/2 a_0}\sin\theta e^{i\phi} \nonumber \] \[\psi_{2p-1}=\frac{1}{\sqrt{64\pi a_0^3}}r e^{-r/2 a_0}\sin\theta e^{-i\phi} \nonumber \] b) Calculate \(\hat{L}_x\psi_{2p0}\). Is \(\psi_{2p0}\) and eigenfunction of \(\hat{L}_x\)? c) Calculate \(\hat{L}_y\psi_{2p0}\). Is \(\psi_{2p0}\) and eigenfunction of \(\hat{L}_y\)? Problem \(\PageIndex{4}\) Prove that \[\left[\hat{L}_z,\hat{L}_x\right]=i\hbar \hat{L}_y \nonumber \] Problem \(\PageIndex{5}\) For a system moving in one dimension, the momentum operator can be written as \[\hat p = i \hbar \frac{d}{dx} \nonumber \] Find the commutator \([\hat x, \hat p]\) Note: \(\hbar\) is defined as \(h/{2 \pi}\), where \(h\) is Plank’s constant. It has been defined because the ratio \(h/{2 \pi}\) appears often in quantum mechanics. Problem \(\PageIndex{6}\) We demonstrated that \(\psi_1s\) is not an eigenfunction of \(\hat T\). Yet, we can calculate the average kinetic energy of a 1s electron, \(\left \langle T \right \rangle\). Use Equation \(11.3.1\) to calculate an expression for \(\left \langle T \right \rangle\). Problem \(\PageIndex{7}\) Use the Hamiltonian of Equation \(11.3.5\) to calculate the energy of the electron in the 1s orbital of the hydrogen atom. The normalized wave function of the 1s orbital is: \[\psi=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0} \nonumber \] Problem \(\PageIndex{8}\) The expression of Equation \(11.3.1\) can be used to obtain the expectation (or average) value of the observable represented by the operator \(\hat{A}\). The state of a particle confined in a one-dimensional box of length a is described by the following wavefunction: \[\psi(x)=\begin{cases} \sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a} \right )& \mbox{ if } 0\leq x\leq a \\ 0 &\mbox{otherwise} \end{cases} \nonumber \] The momentum operator for a one-dimensional system was introduced in Problem \(\PageIndex{5}\). a) Obtain an expression for \(\hat{p}^2\) and determine if \(\psi\) is an eigenfunction of \(\hat{p}\) and \(\hat{p}^2\). If possible, obtain the corresponding eigenvalues. Hint: \(\hat{p}^2\) is the product \(\hat{p}\hat{p}\). b) Determine if \(\psi\) is an eigenfuction of \(\hat{x}\). If possible, obtain the corresponding eigenvalues. c) Calculate the following expectation values: \(\left \langle x \right \rangle\), \(\left \langle p^2 \right \rangle\), and \(\left \langle p \right \rangle\). Compare with the eigenvalues calculated in the previous questions.