# 11.4: Problems

• • Contributed by Marcia Levitus

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.