diff --git a/includes/angmomentum-ladder-operators.tex b/includes/angmomentum-ladder-operators.tex index fa2c409..4f212bb 100644 --- a/includes/angmomentum-ladder-operators.tex +++ b/includes/angmomentum-ladder-operators.tex @@ -25,7 +25,7 @@ \section*{Angular Momentum Raising and Lowering Operators\sectionmark{Exercise: \part What are $[\hat M_z,\hat M_+]$ and $[\hat M_z,\hat M_-]$? - \begin{solution}[2in] + \begin{solution}[3in] \end{solution} \end{parts} diff --git a/includes/exchange-symmetry.tex b/includes/exchange-symmetry.tex index 18824aa..1d02e54 100644 --- a/includes/exchange-symmetry.tex +++ b/includes/exchange-symmetry.tex @@ -12,7 +12,7 @@ \section*{Symmetry with Respect to Exchange\sectionmark{Exercise: Exchange Symme \begin{questions} - \question The permutation operator, $\hat P_{12}$, swaps particles one and 2, e.g. $\hat P_{12} \psi(q_1,q_2) = \psi(q_2,q_1)$. + \question The permutation operator, $\hat P_{12}$, swaps particles 1 and 2, e.g. $\hat P_{12} \psi(q_1,q_2) = \psi(q_2,q_1)$. \begin{parts} \part If we apply the permutation operator twice, what should we get? In other words, what is $\hat P_{12}^2 \psi(q_1,q_2)$? diff --git a/includes/matrixmech-eigenstates.tex b/includes/matrixmech-eigenstates.tex index cc6b610..a964c6a 100644 --- a/includes/matrixmech-eigenstates.tex +++ b/includes/matrixmech-eigenstates.tex @@ -47,7 +47,7 @@ \section*{Eigenvalues, Eigenstates, and Eigenvectors\sectionmark{Exercise: Eigen \begin{solution}[1.25in] \end{solution} - \part Now, assume we only know the matrix elements of $\hat H$ in some other basis, $\{\ket i\}$. What are the matrix elements in this basis (i.e. what is $H_{ij}$? + \part Now, assume we only know the matrix elements of $\hat H$ in some other basis, $\{\ket i\}$. What are the matrix elements in this basis (i.e. what is $H_{ij}$)? \begin{solution}[1.25in] \end{solution} diff --git a/includes/matrixmech-intro.tex b/includes/matrixmech-intro.tex index 60c26ae..53b0e80 100644 --- a/includes/matrixmech-intro.tex +++ b/includes/matrixmech-intro.tex @@ -41,7 +41,7 @@ \section*{Describing States and Operators with Matrices\sectionmark{Exercise: In \question \begin{parts} - \part If we calculate matrix elements of $\hat A$ using one basis set (e.g. the particle-in-a-box eigenstates), will it be the same as if we use a different basis set (e.g. the harmonic oscillator eigenstates)? Why or why not? + \part If we calculate matrix elements of $\hat A$ using one basis set (e.g. the particle-in-a-box eigenstates), will they be the same as if we use a different basis set (e.g. the harmonic oscillator eigenstates)? Why or why not? \begin{solution}[2.5in] \end{solution} diff --git a/includes/perturbation-first-order.tex b/includes/perturbation-first-order.tex index a03187d..2b025ed 100644 --- a/includes/perturbation-first-order.tex +++ b/includes/perturbation-first-order.tex @@ -33,7 +33,7 @@ \section*{First-Order Perturbation Theory\sectionmark{Exercise: First-Order Pert \begin{solution}[1.25in] \end{solution} - \part Because the equation from part (a) must hold for \emph{any} value of $\lambda$, the terms that scale as $\lambda$ on the left side must be equal to the ones that scale as $\lambda$ on the right side $\lambda \neq 0$ (the same would be true for the terms that scale as $\lambda^2$, but we haven't included all of those, so we'll ignore them for now). + \part Because the equation from part (a) must hold for \emph{any} value of $\lambda$, the terms that scale as $\lambda$ on the left side must be equal to the ones that scale as $\lambda$ on the right side (the same would be true for the terms that scale as $\lambda^2$, but we haven't included all of those, so we'll ignore them for now). If you include only the terms that have exactly one factor of $\lambda$, what equation results? diff --git a/includes/the-H-atom.tex b/includes/the-H-atom.tex index e2a6d3c..f57a40f 100644 --- a/includes/the-H-atom.tex +++ b/includes/the-H-atom.tex @@ -16,7 +16,7 @@ \section*{The Schr\"odinger Equation for the Hydrogen Atom\sectionmark{Exercise: \question A ``hydrogen-like'' atom consists of a nucleus of charge $+Ze$ and mass $m_n$ and an electron of charge $-e$ and mass $m_e$. \begin{parts} - \part If the nucleus is at coordinates $\vec{R_n} = (x_n,y_n,z_n)$ and the electron is at $\vec{R_e} = (x_e,y_e,z_e)$, what is the total kinetic energy for this system? + \part If the nucleus is at coordinates $\vec{R_n} = (x_n,y_n,z_n)$ and the electron is at $\vec{R_e} = (x_e,y_e,z_e)$, what is the total kinetic energy operator for this system? \emph{Hint: you can simplify the notation by remembering that $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$. You can give $\nabla^2$ a subscript to indicate which set of coordinates (nuclear or electronic) it is operating on.} diff --git a/includes/the-He-atom.tex b/includes/the-He-atom.tex index 6a74316..a6546ca 100644 --- a/includes/the-He-atom.tex +++ b/includes/the-He-atom.tex @@ -8,7 +8,7 @@ % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section*{Motivation: The Helium Atom\sectionmark{Exercise: The Helium Atom}} +\section*{The Helium Atom\sectionmark{Exercise: The Helium Atom}} \begin{questions} diff --git a/includes/variational-principle.tex b/includes/variational-principle.tex index 2b88f27..6d28346 100644 --- a/includes/variational-principle.tex +++ b/includes/variational-principle.tex @@ -10,10 +10,10 @@ \section*{The Variational Principle\sectionmark{Exercise: The Variational Principle}} - In this set of exercises, we'll consider the value of the expectation value of the energy for an arbitrary state. For these exercises, assume that the $\{\ket{n}\}$ are energy eigenstates, with $\hat H \ket{n} = E_n \ket{n}$. + In this set of exercises, we'll consider the expectation value of the energy for an arbitrary state, $\ket{\Psi_{trial}}$. For these exercises, assume that the $\{\ket{n}\}$ are energy eigenstates, with $\hat H \ket{n} = E_n \ket{n}$. \begin{questions} - \question Suppose we pick a trial state $\ket{\Psi_{trial}} = \sum_n c_n \ket{n}$. + \question Suppose we pick $\ket{\Psi_{trial}}$ such that $\ket{\Psi_{trial}} = \sum_n c_n \ket{n}$. \begin{parts} \part What is the expectation value of the energy, $E_{trial}$, for this state? diff --git a/quantum-exercises.pdf b/quantum-exercises.pdf new file mode 100644 index 0000000..a2321dc Binary files /dev/null and b/quantum-exercises.pdf differ