Chapter 1

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)

Are the linear combinations \(2 x-y-z, 2 y-x-z\) \(2 z-x-y\) linearly independent?

Evaluate the commutators (a) \([x, y]\) (b) \(\left[p_{x}, p_{y}\right]\) (c) \(\left[x, p_{x}\right]\) (d) \(\left[x^{2}, p_{x}\right]\) (e) \(\left[x^{n}, p_{x}\right]\)

Show that (a) \([A, B]=-[B, A],\) (b) \(\left[A^{m}, A^{n}\right]=0\) for all \(m, n,\) (c) \(\left[A^{2}, B\right]=A[A, B]+[A, B] A\) (d) \([A,[B, C]]+[B,[C, A]]+[C,[A, B]]=0\)

Evaluate the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) given that \(\left[l_{x}, l_{y}\right]=i \hbar l,\left[l_{y}, l_{z}\right]=i \hbar l_{x},\) and \(\left[l_{z}, l_{x}\right]=i \hbar l_{y}\)

A particle in an infinite one-dimensional system was described by the wavefunction \(\psi(x)=\mathrm{Ne}^{-x^{2} / 2 r^{2}}\). Normalize this function. Calculate the probability of finding the particle in the range \(-\Gamma \leq x \leq \Gamma\). Hint. The integral encountered in the second part is the error function. It is available in mathematical software.

The ground-state wavefunction of a hydrogen atom has the form \(\psi(r)=N \mathrm{e}^{-b r}, b\) being a collection of fundamental constants with the magnitude \(1 / a_{0},\) with \(a_{0}=53 \mathrm{pm}\) Normalize this spherically symmetrical function. Hint. The volume element is \(\mathrm{d} \tau=\sin \theta \mathrm{d} \theta \mathrm{d} \varphi r^{2} \mathrm{d} r,\) with \(0 \leq \theta \leq \pi\) \(0 \leq \varphi \leq 2 \pi,\) and \(0 \leq r<\infty,\) 'Normalize' always means 'normalize to 1 ' in this text.

Confirm that the operators (a) \(T=-\left(\hbar^{2} / 2 m\right)\left(d^{2} / d x^{2}\right)\) and (b) \(l_{z}=(\hbar / \mathrm{i})(\mathrm{d} / \mathrm{d} \varphi)\) are Hermitian. Hint. Consider the integrals \(\int_{0}^{L} \psi_{a}^{*} T \psi_{b} \mathrm{d} x\) and \(\int_{0}^{2 \pi} \psi_{a}^{*} l_{z} \psi_{b} \mathrm{d} \varphi\) and integrate by parts.

Find the operator for position \(x\) if the operator for momentum \(p\) is taken to be \((\hbar / 2 m)^{1 / 2}(A+B),\) with \([A, B]=1\) and all other commutators zero. Hint. Write \(x=a A+b B\) and find one set of solutions for \(a\) and \(b\)

Evaluate the commutators (a) \(\left[(1 / x), p_{x}\right]\) (b) \(\left[(1 / x), p_{x}^{2}\right]\) (c) \(\left[x p_{y}-y p_{x}, y p_{z}-z p_{y}\right]\) (d) \(\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]\)