Chapter Questions
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 integrateby 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) $\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]$
Evaluate the commutators (a) $\left[H, p_{x}\right]$ and(b) $[H, x]$ where $H=p_{x}^{2} / 2 m+V(x) .$ Choose(i) $V(x)=V,$ a constant,(ii) $V(x)=\frac{1}{2} k_{t} x^{2},$ (iii) $V(x) \rightarrow V(r)=e^{2} / 4 \pi \varepsilon_{0} r .$ Hint. For part(b), case (iii), use $\left(\partial r^{-1} / \partial x\right)=-x / r^{3}$
Use the momentum representation and a general function $f\left(p_{x}\right)$ of the linear momentum to confirm that the position and momentum operators in this representation do not commute, and find the value of their commutator.
Construct quantum mechanical operators in the position representation for the following observables: (a) kineticenergy in one and in three dimensions,(b) the inverse separation, $1 / x,$ (c) electric dipole moment $\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.$ is the position of a charge $Q_{i}$ ),(d) $z$ -component of angular momentum $\left(x p_{y}-y p_{x}\right),$ (e) the mean square deviations of the position and momentum of a particle from the mean values.
Repeat Problem 1.13, but find operators in the momentum representation. Hint. The observable $1 / x$ should be regarded as $x^{-1} ;$ hence the operator required is the inverse of the operator for $x$
In relativistic mechanics, energy and momentum are related by the expression $E^{2}=p^{2} c^{2}+m^{2} c^{4}$. (a) Show that when $p^{2} c^{2} \ll m^{2} c^{4}$ this expression reduces to $E=p^{2} / 2 m+m c^{2} \cdot$ (b) Construct the relativistic analogue of the Schrödinger equation from the relativistic expression(c) What can be said about the conservation of probability? Hint: For part (b), use if ( $\partial / \partial t$ ) for the energy operator. For part $(\mathrm{c}),$ see Problem 1.16
Show that if the Schrödinger equation had the form of a true wave equation, then the integrated probability would be time dependent. Hint. A wave equation has $\kappa \partial^{2} / \partial t^{2}$ in place of it a chere $\kappa$ is a constant with the appropriate dimensions (what are they?). Solve the time component of the separable equation and investigate the behaviour of $\int \Psi^{*} \Psi d \tau$
The operator $e^{A}$ has a meaning if it is expanded as a power scrics: $\mathrm{e}^{A}=\Sigma_{n}(1 / n !) A^{n} .$ Show that if $|a\rangle$ is an eigenstate of $A$ with eigenvalue $a,$ then it is also an eigenstate of $\mathrm{e}^{A} .$ Find the latter's eigenvalue.
Evaluate the expectation value of elite. for a particle in a square well of length $L$ and described by the wavefunction $(2 / L)^{1 / 2} \sin (\pi x / L)$ in the range 0 to$L \cdot H i n t: e^{i A}=\cos A+$\[\begin{array}{l}\mathrm{i} \sin A, \cos \theta=1-(1 / 2 !) \theta^{2}+(1 / 4 !) \theta^{4}-\cdots, \sin \theta=\theta- \\(1 / 3 !) \theta^{3}+(1 / 5 !) \theta^{5}-\cdots\end{array}\]
(a) Show that $\mathrm{e}^{A} \mathrm{e}^{B}=\mathrm{e}^{A+B}$ only if $[A, B]=0 .$ (b) If $[A, B] \neq 0$ but $[A,[A, B]]=[B,[A, B]]=0,$ show that $\mathrm{e}^{A} \mathrm{e}^{B}=$$\mathrm{e}^{A+B} \mathrm{e}^{f},$ where $f$ is a simple function of $[A, B] .$ Hint. This is another example of the differences between operators $(q-\text { numbers })$ and ordinary numbers (c-numbers). The simplest approach is to expand the exponentials and to collect and compare terms on both sides of the equality. Note that $\mathrm{e}^{A} \mathrm{e}^{B}$ will give terms like $2 A B$ while $\mathrm{e}^{A+B}$ will give $A B+B A .$ Be careful with order.
Evaluate (by considering cqn 1.43a) the limitation on the simultancous specification of the following obscrvables:(a) the position and momentum of a particle, (b) the three components of linear momentum of a particle,(c) the kinetic energy and potential energy of a particle, (d) the electric dipole moment $(-e x)$ and the total energy of a one-dimensional system, (e) the kinetic energy and the position of a particle in one dimension.
Evaluate the quantity $\Delta_{4} x \Delta_{4} p_{x}$ for the ground state $(n=1)$ of a particle of mass $m$ in a box of length $L,$ where $\Delta_{4} \Omega=\left\langle(\Omega-\langle\Omega\rangle)^{4}\right\rangle^{1 / 4}$
Use eqn 1.44 to find expressions for the rate of change of the expectation values of position and momentum of a harmonic oscillator; solve the pair of differential equations, and show that the expectation values change in time in the same way as for a classical oscillator. Hint. Use the results of Problem 1.11 part (ii).
The only non-zero matrix elements of $x$ and $p_{x}$ for a harmonic oscillator are $$\begin{array}{l}\langle v+1|x| v\rangle=\left(\frac{\hbar}{2 m \omega}\right)^{1 / 2}(v+1)^{1 / 2} \\\langle v-1|x| v\rangle=\left(\frac{\hbar}{2 m \omega}\right)^{1 / 2} v^{1 / 2} \\\left\langle v+1\left|p_{x}\right| v\right\rangle=\mathrm{i}\left(\frac{\hbar m \omega}{2}\right)^{1 / 2}(v+1)^{1 / 2} \\\left\langle v-1\left|p_{x}\right| v\right\rangle=-\mathrm{i}\left(\frac{\hbar m \omega}{2}\right)^{1 / 2} v^{1 / 2}\end{array}$$See Section $2.15 .$ Use the completeness relation, eqn 1.25 to deduce the value of the matrix element $\left\langle v\left|x p_{x}^{2} x\right| v\right\rangle$
Write the time-independent Schrödinger equations for (a) the hydrogen atom, (b) the helium atom, (c) the hydrogen molecule, (d) a free particle, (e) a particle subjected to a constant, uniform force. Hint. Identify the appropriate potential energy terms and express them as operators in the position representation.
The time-dependent Schrödinger equation is separable when $V$ is independent of time.(a) Show that it is also separable when $V$ is a function only of time and is uniform in space.(b) Solve the pair of equations. Let $V(t)=V \cos \omega t$ find an expression for $\Psi(x, t)$ in terms of $\Psi(x, 0) .(\mathrm{c})$ Is $\Psi(x, t)$ stationary in the sense specified in Section $1.14 ?$
(a) Calculate the probability of the electron being found anywhere within a sphere of radius $a_{0}$ for the atom defined in Problem 1.7 . (b) If the radius of the atom is defined as the radius of the sphere inside which there is a 90 per cent probability of finding the electron, what is the atom's radius? Hint. For part (b), find the solution numerically (e.g. by successive approximation, using software, or graphically).
A particle is moving in a circle in the $x y$ plane. The only coordinate of importance is the angle $\varphi$ which can vary from 0 to $2 \pi$ as the particle goes around the circle. We are interested in measurements of the angular momentum $l_{z}$ of the particle. The angular momentum operator for such a system is given by $(\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} \varphi$(a) Suppose that the state of the particle is described by the wavefunction $\psi(\varphi)=N \mathrm{e}^{-\mathrm{i} \varphi}$ where $N$ is the normalization constant. What values will we find when we measure the angular momentum of the particle? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum?(b) Now suppose that the state of the particle is described by the normalized wavefunction $\psi(\varphi)=N\left\{(3 / 4)^{1 / 2} \mathrm{e}^{-i \varphi}-(\mathrm{i} / 2) \mathrm{e}^{2 \text { ip }}\right\} .$ When we measure theangular momentum of the particle, what values will we find? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum?
Provide a proof of eqn $1.47 ;$ eqns 1.47 and $1.46 b$ jointly form Ehrenfest's theorem.