Quantum Mechanics I

Abstract

From the beginning of the book, but in particular in this chapter, we stress the inherent uncertainty (e.g., of an initial state). After discussing the usual formalism of the (im)proper Hilbert space with the Dirac bra and ket notation and the operator formalism, the Pauli principle is introduced at an early stage. A key topic is the uncertainty principle applied to the wave–particle duality, which is not without intricacies, and therefore rarely found elsewhere. Further topics are the Pauli and von Neumann equations, the Wigner function, the collision-free Boltzmann equation, parametric oscillators, (weak) coupling to the environment and the Markov approximation, the Liouville equation, absorption and emission, dissipation, and decoherence (a hot topic in modern science). There is a list of 45 problems.

Appendices

Problems

Problem 4.1

Which probability amplitude $\psi (x)$ fits a Gauss distribution $|\psi (x)|^2$ with $\overline{x}\!=\!0$ and $\Delta x\!\ne \!0$? What does its Fourier transform

$$ \psi (k)=\frac{1}{\sqrt{2\pi }}\int ^\infty _{-\infty }\exp (-\mathrm{i}kx)\,\psi (x)\,\mathrm {d}x $$

look like? Show that the factor $1/\sqrt{2\pi }$ here ensures $\int ^\infty _{-\infty }|\psi (k)|^2\,\mathrm {d}k=1$. Determine $\Delta x\cdot \Delta k$ for this example. (6 P)

Problem 4.2

Given a slit of width of 2a, assume that the probability amplitude $\psi (x)=1/\sqrt{2a}$ for $|x|\le a$, otherwise zero. How large is $\Delta x$? Determine the Fourier transform. Where are the maximum and the neighboring minima of $|\psi (k)|^2$, and how large are they? Show that the “interference pattern” $|\psi (k)|^2$ becomes more extended with decreasing slit width, but that the product $\Delta x\cdot \Delta k$ is problematic. (6 P)

Problem 4.3

Consider the Lorentz distribution

$$ |\psi (\omega )|^2\propto 1/\{(\omega -\omega _0)^2+(\frac{1}{2}\gamma )^2\}^{-1}\;. $$

How large is the uncertainty $\Delta \omega$, and how large is its half-width, i.e., the distance at which $|\psi (\omega )|^2$ has decayed to half the maximum value? Show that $\psi (\omega )$ is the Fourier transform of $\psi (t)\propto \exp \{-\mathrm{i}(\omega _0-\mathrm{i}\frac{1}{2}\gamma )\,t\}$ for $t\ge 0$, zero for $t<0$. Can we describe decays with it? How large is the time uncertainty $\Delta t$? (8 P)

Problem 4.4

The transition from the initial state $|i\rangle$ to the final state $|f\rangle$ should be possible via any of the states $|a\rangle$, $|b\rangle$, and $|c\rangle$. How large is the transition probability $|\langle f|i\rangle |^2$ if the states $|a\rangle$ and $|b\rangle$ may interfere, but $|c\rangle$ has to be superposed incoherently? (2 P)

Problem 4.5

Prove $\int ^\infty _{-\infty }f(x)\;\delta ^{(n)}(x-x')\;\mathrm {d}x =(-)^n\;f^{(n)}(x')$ for square-integrable functions using integration by parts. Deduce from this that the equation $x\;\delta '(x)=-\delta (x)$ is true for the integrand. Prove $\delta (ax)=\frac{1}{|a|}\;\delta (x)$. (6 P)

Problem 4.6

A series of functions $\{g_n(x)\}$ forms a complete orthonormal set in the interval from a to b, if $\int ^b_ag_n{}^*(x)g_{n'}(x)\,\mathrm {d}x =\delta _{nn'}$ and $f(x)=\sum _ng_n(x)f_n$ for all (square-integrable) functions f(x). How can the expansion coefficients $f_n$ be determined? Expand the delta-function $\delta (x-x')$ (with $x'\in [a,b]$) with respect to this basis. Does the sequence $g_n(x)=(2a)^{-1/2}\;\exp (\mathrm{i}\pi nx/a)$ form a complete orthonormal system in the interval $-a\le x\le a$? (6 P)

Problem 4.7

The system of Legendre polynomials $P_n(x)$ is complete in the interval $|x|\le 1$. The generating function is $1/\sqrt{1-2sx+s^2}=\sum ^\infty _{n=0}P_n(x)\;s^n$ for $|s|<1$. How does the associated orthonormal system read? Show that the Legendre polynomials may also be represented by

$$ P_n(x)=1/\{2^n\;n!\}\;d^n(x^2-1)^n/\mathrm {d}x ^n\quad \text {(Rodrigues' formula)}. $$

(6 P)

Problem 4.8

The normalized state $|\psi \rangle =|\alpha \rangle \,a+|\beta \rangle \,b$ is constructed from the orthonormalized states $|\alpha \rangle$ and $|\beta \rangle$. What constraint do the coefficients $a\ne 0$ and $b\ne 0$ satisfy? How do they depend on $|\psi \rangle$? Determine which of the following normalized states $|\psi _i\rangle$ are physically equivalent to $|\psi \rangle$ (disregarding the phase factor): $|\psi _1\rangle =-|\alpha \rangle \,a-|\beta \rangle \,b$, $|\psi _2\rangle =|\alpha \rangle \,a-|\beta \rangle \,b$, $|\psi _3\rangle =|\alpha \rangle \,a\text {e}^{\mathrm{i}\varphi }+|\beta \rangle \,b\text {e}^{-\mathrm{i}\varphi }$, $|\psi _4\rangle =|\alpha \rangle \,\cos \varphi \pm |\beta \rangle \,\sin \varphi$. (6 P)

Problem 4.9

Does the sequence of Hilbert space vectors

$$\begin{aligned} \left( \begin{array}{c}1\\ 0\\ 0\\ \vdots \end{array}\right) ,\; \left( \begin{array}{c}0\\ 1\\ 0\\ \vdots \end{array}\right) ,\; \left( \begin{array}{c}0\\ 0\\ 1\\ \vdots \end{array}\right) ,\;\ldots \end{aligned}$$

converge strongly, weakly, or not at all? If so, give the vector to which the sequence converges. (4 P)

Problem 4.10

Consider the function $\psi (x)=x$ for $-\pi \le x\le \pi$. How does it read as a Hilbert vector in the sequence space if we take the basis $\{g_n(x)\}$ of Problem 4.6 (with $a=\pi$)? How does the Hilbert vector in the function space read if it has the components $\psi _n=\delta _{n,1}+\delta _{n,-1}$ in this basis of the sequence space? (4 P)

Problem 4.11

Are the functions $f_0(x)\propto 1$ and $f_1(x)\propto x$ orthogonal to each other for $-\pi \le x\le \pi$? Determine their normalization factors. Extend the orthonormalized basis $\{f_0,f_1\}$ so that it is complete for all second-order functions $f(x)=a_0+a_1\,x+a_2\,x^2$ in $-\pi \le x\le \pi$. (6 P)

Problem 4.12

Determine $[A,[B,C]_\pm ]+[B,[C,A]_\pm ]+[C,[A,B]_\pm ]$ and simplify the expression $[C,[A,B]_\pm ]_+-[B,[C,A]_\pm ]_+$. Is

$$ (A[B,C]_\pm -[C,A]_\pm B)D+C(A[B,D]_\pm -[D,A]_\pm B) $$

a simple commutator? (6 P)

Problem 4.13

Let the unit operator 1 be decomposed into a projection operator P and its complement Q, viz., $1=\text {P}+\text {Q}$. Is Q also idempotent? Are P and Q orthogonal to each other, i.e., is $\text {tr}(\text {PQ})=0$ true? What are the eigenvalues of P and Q? (4 P)

Problem 4.14

Is the inverse of a unitary operator also unitary? Is the product of two unitary operators unitary? Is $(1-\mathrm{i}A)(1+\mathrm{i}A)^{-1}$ unitary if A is Hermitian? Justify all answers! (4 P)

Problem 4.15

Suppose $(A-a_11)(A-a_21)=0$ and let $|\psi \rangle$ be arbitrary, but not an eigenvector of A. Show that $(A-a_1)|\psi \rangle$ and $(A-a_2)|\psi \rangle$ are eigenvectors of A, and determine the eigenvalues. Determine the eigenvalues of the 2 $\times$ 2 matrix A with elements $A_{ik}$. If the matrix is Hermitian, show that no degeneracy can occur if the matrix is not diagonal. (6 P)

Problem 4.16

Do orthogonal operators remain orthogonal under a unitary transformation? (2 P)

Problem 4.17

Why is the determinant of the matrix elements of the operator A equal to the product of its eigenvalues? (4 P)

Problem 4.18

Let the vectors $\mathbf {a}$ and $\mathbf {b}$ commute with the Pauli operator $\mathbf {\upsigma }$. How can $(\mathbf {a}\cdot \mathbf {\upsigma })(\mathbf {b}\cdot \mathbf {\upsigma })$ then be expressed in the basis $\{1,\mathbf {\upsigma }\}$? What follows for $(\mathbf {a}\cdot \mathbf {\upsigma })^2$ and what for the anti-commutator $\{\mathbf {a}\cdot \mathbf {\upsigma },\mathbf {b}\cdot \mathbf {\upsigma }\}$? Expand the unitary operator $U=\exp (\mathrm{i}\,\mathbf {a}\cdot \mathbf {\upsigma })$ in terms of the basis $\{1,\mathbf {\upsigma }\,\}$. (6 P)

Problem 4.19

The boson annihilation operator $\Psi$ is in fact not Hermitian and therefore does not necessarily have real eigenvalues, but any complex number $\psi$ may be an eigenvalue of $\Psi$. Determine (up to the normalization factor) the associated eigenvector in the particle-number basis, and hence the coefficients $\langle n|\psi \rangle$ in $|\psi \rangle =\sum ^\infty _{n=0}|n\rangle \;\langle n|\psi \rangle$. Why is this not possible for the creation operator $\Psi ^\dagger$? For arbitrary complex numbers $\alpha$ and $\beta$, consider the scalar product $\langle \alpha |\beta \rangle$ and determine the unknown normalization factor. (8 P)

Problem 4.20

Show using the method of induction that

$$\begin{aligned} \Bigl .\begin{array}c \Psi ^m\;\Psi ^{\dagger \;n}\\ \Psi ^{\dagger \;n}\;\Psi ^m\end{array} \Bigr \}=\sum _l\frac{(\pm )^lm!\;n!}{l!\;(m-l)!\;(n-l)!}\; \Bigl \{\begin{array}c \Psi ^{\dagger \;n-l}\;\Psi ^{m-l}\;,\\ \Psi ^{m-l}\;\Psi ^{\dagger \;n-l}\;. \end{array}\Bigr . \end{aligned}$$

(7 P)

Problem 4.21

Which 2 $\times$ 2 matrices correspond to the Pegg–Barnett operators $\widetilde{\Psi }$, $\widetilde{\Psi }^\dagger$, and $\widetilde{\Psi }\widetilde{\Psi }^\dagger \!\pm \widetilde{\Psi }^\dagger \widetilde{\Psi }$, if the basis has only two eigenvalues ($s=1$)? Do these operators behave like field operators for fermions? (4 P)

Problem 4.22

From $\upsigma _x\upsigma _y=\mathrm{i}\upsigma _z=-\upsigma _y\upsigma _x$ and $\upsigma _x{}^2=1$ (and cyclic permutations), and also $\upsigma _\pm =\frac{1}{2}\,(\upsigma _x\pm \,\mathrm{i}\upsigma _y)$, determine $\upsigma _z\upsigma _\pm$, $\upsigma _\pm \upsigma _z$, $\upsigma _\pm \upsigma _\mp$ and $\upsigma _\pm {}^2$. What do we obtain therefore for $U\upsigma _\pm U^\dagger$ with $U=\exp (\mathrm{i}\alpha \upsigma _z)$, according to the Hausdorff series? Simplify the Hermitian operators $\upsigma _z\mathbf {\upsigma }\upsigma _z$, $\upsigma _\pm \mathbf {\upsigma }\upsigma _\mp$, and $\mathbf {\upsigma }\upsigma _\pm \upsigma _\mp +\upsigma _\pm \upsigma _\mp \mathbf {\upsigma }$. (9 P)

Problem 4.23

As is well known, the position and momentum coordinates of a particle span its phase space. Show that a classical linear oscillation with angular frequency $\omega$ traces an ellipse in phase space, and determine its area as a function of the energy. How large is the probability density for finding the oscillator at the displacement x for oscillations with amplitude $\widehat{x}$, if all phase angles are initially equally probable? (Here we thus consider a statistical ensemble.) (6 P)

Problem 4.24

Since $\Delta X\cdot \Delta P\ge \frac{1}{2}\hbar$, the phase-space cells may not be chosen arbitrarily small (more finely divided cells would be meaningless). How large is the area if the energy increases by $\hbar \omega$ from cell to cell? Is it possible to associate particles at rest with the cell of lowest energy, which would start oscillating only after gaining energy? What is the mean value of the energy in this cell? (4 P)

Problem 4.25

Show that the matrix $\langle \psi _1|\,P\,|\psi _2\rangle =\int ^\infty _{-\infty } \psi _1{}^*(x)\;\frac{\hbar }{\mathrm{i}}\;\frac{\mathrm {d}}{\mathrm {d}x }\;\psi _2(x)\;\mathrm {d}x$ is Hermitian. What can be concluded from this for the expectation values $\langle P\rangle$ and $\langle P^2\rangle$ for a real wave function? (6 P)

Problem 4.26

Derive the 2 $\times$ 2 density matrix of the spin states of unpolarized electrons. Why is it not possible to represent it by a Hilbert vector? (4 P)

Problem 4.27

Why does the quantum-mechanical expression $\frac{1}{2}\,\{f(X)\,P{+}P\,f(X)\}$ correspond to the classical $f(x)\,p$ according to the Weyl correspondence?

Hint: Use $\mathrm{i}\hbar f'(X)=[f(X),P]$. (6 P)

Problem 4.28

Justify the validity of the following quantum-mechanical expressions—independent of the representation—with a homogeneous magnetic field $\mathbf {B}$ and Coulomb gauge: $\mathbf {A}=\frac{1}{2}\mathbf {B}\times \mathbf {R}$, $\mathbf {P}\cdot \mathbf {A}+\mathbf {A}\cdot \mathbf {P}=\mathbf {B}\cdot \mathbf {L}$, and $\mathbf {P}\times \mathbf {A}+\mathbf {A}\times \mathbf {P}=-\mathrm{i}\hbar \mathbf {B}$. (4 P)

Problem 4.29

In approximate calculations for motions with high orbital angular momentum, we often replace $\langle L^2\rangle /\hbar ^2$ by the square of a number (as if it were the expectation value of $L/\hbar$). Which number is better than l? How large is the relative error for $l=3$ and $l=5$? (4 P)

Problem 4.30

Is it possible to express the Poisson bracket $[\mathbf {l}\cdot \mathbf {e}_1,\mathbf {a}\cdot \mathbf {e}_2]$ in terms of the triple product $\mathbf {a}\cdot \,(\mathbf {e}_1\times \mathbf {e}_2)$ if $\mathbf {a}$ is the position or momentum vector? (4 P)

Problem 4.31

Derive the uncertainties $\Delta L_x$ and $\Delta L_y$ for the state $|l,m\rangle$. Hence, determine also $(\Delta L_x)^2+(\Delta L_y)^2+(\Delta L_z)^2$. (2 P)

Problem 4.32

Does $\mathbf {L}$ commute with $R^2$ and $P^2$? (2 P)

Problem 4.33

For classical vectors $\mathbf {r}$ and $\mathbf {p}$, the equations

$$ (\mathbf {r}\times \mathbf {p}\,)^2=r^2\,p^2-(\mathbf {r}\cdot \mathbf {p}\,)^2\;,\qquad \mathbf {p}\times (\mathbf {r}\times \mathbf {p}\,)=\mathbf {r}\,p^2-\mathbf {p}\;\mathbf {r}\cdot \mathbf {p}\;, $$

are valid. How do they read for the associated operators? (4 P)

Problem 4.34

Derive all spherical harmonics for $l=0$, 1, and 2. (4 P)

Problem 4.35

Determine the integrals over all directions $\Omega$ of $Y^{(l)\;*}_m(\Omega )$, $Y^{(l')}_{m'}(\Omega )$, and $Y^{(l)}_m(\Omega )$.

Hint: Express the integrals initially with scalar products $\langle \Omega |lm\rangle$. (2 P)

Problem 4.36

For spherically symmetric problems, the ansatz

$$ \psi _{nlm}(\mathbf {r}\,)=r^{-1}\;u_{nl}(r)\;\mathrm{i}^{\,l}\,Y^{(l)}_m(\Omega ) $$

turns out to be useful. Using this, reduce $\langle nlm|\,r\cos \theta \,|n'00\rangle$ to a simple integral, given that the integral over the directions is known.

Hint: $r\cos \theta$ corresponds to $\mathbf {R}\cdot \mathbf {e}_z$ in the position representation. (4 P)

Problem 4.37

What do we obtain for $\langle nlm|\,(r\cos \theta )^2\,|n'00\rangle$ and $\langle nlm|\,\mathbf {P}\cdot \mathbf {e}_z\,|n'00\rangle$ with the ansatz just mentioned? (4 P)

Problem 4.38

The scalar product of two angular momentum operators $\mathbf {J}_1$ and $\mathbf {J}_2$ may be expressed in terms of $J_{1z}$, $J_{1\pm }$ and $J_{2z}$, $J_{2\pm }$, viz.,

$$ \mathbf {J}_1\cdot \mathbf {J}_2=\frac{1}{2}(J_{1+}J_{2-}+J_{1-}J_{2+})+J_{1z}J_{2z}\;. $$

This helps for the uncoupled basis, but for the coupled basis, the total angular momentum $\mathbf {J}=\mathbf {J}_1+\mathbf {J}_2$ should be used. Determine the matrix elements of the operator $\mathbf {\upsigma }_1\cdot \mathbf {\upsigma }_2$ in the uncoupled basis $\{|\frac{1}{2}m_1,\frac{1}{2}m_2\rangle \}$ and in the coupled one $\{|(\frac{1}{2}\frac{1}{2})sm\rangle \}$. How can we express the projection operators ${\mathtt P}_S$ on the singlet and triplet states (with $S=0$ and $S=1$, respectively) using $\mathbf {\upsigma }_1\cdot \mathbf {\upsigma }_2$? (6 P)

Problem 4.39

Represent all $d_{3/2}$ states $|(2\frac{1}{2})\frac{3}{2}m\rangle$ in the uncoupled basis. (4 P)

Problem 4.40

How many p states are there for a spin-$\frac{1}{2}$ particle? Expand in terms of the basis of the total angular momentum. (4 P)

Problem 4.41

Which Ehrenfest equations are valid for the orbital angular momentum? In particular, is the angular momentum a constant on average for a central force? (6 P)

Problem 4.42

Let $\psi (\mathbf {r}\,)\;\approx \;f(\theta )\;r^{-1}\,\exp (\mathrm{i}kr)$ hold for large r. How large is the associated current density for large r? (2 P)

Problem 4.43

How does the position uncertainty for the Gauss wave packet

$$ \psi (k)\!= \exp \{-\frac{1}{4}(\Delta k)^{-2}(k\!-\!\overline{k})^2\} /\root 4 \of {2\pi }\sqrt{\Delta k} $$

depend upon time? In the final result, use $\Delta x(0)$ and $\Delta v$ instead of $\Delta k$. Determine $\overline{x}(t)$ for the case $\overline{x}(0)=0$. (6 P)

Problem 4.44

Write down the Schrödinger equation for the two-body hydrogen atom problem in center-of-mass and relative coordinates. Which (normalized) solution do we have in center-of-mass coordinates? (4 P)

Problem 4.45

For the generalized Laguerre polynomials $L^{(m)}_n(s)$ and for $|t|<1$, there is a generating function $(1-t)^{-m-1}\;\exp \{-st/(1-t)\}= \sum ^\infty _{n=0}L^{(m)}_n(s)\;t^n$. For $\int ^\infty _0\text {e}^{-s}s^k\;L^{(m)}_n(s)\;L^{(m')}_{n'}(s)\;\mathrm {d}s$, use this to derive the expansion

$$ (-)^{n+n'}\sum _l\left( {\begin{array}{c}k-m\\ n-l\end{array}}\right) \left( {\begin{array}{c}k-m'\\ n'-l\end{array}}\right) \;(k+l)!/l!\;. $$

It is needed for the expectation value $\langle R^k\rangle$ of the hydrogen atom, viz., $\langle R^k\rangle =\int ^\infty _0|u|^2\,r^k\;\mathrm {d}r$, with

$$\begin{aligned} u_{nl}(r)=\sqrt{\frac{(n-l-1)!}{a_0\;(n+l)!}}\; \frac{s^{l+1}}{n}\;\exp \Bigl (-\frac{s}{2}\Bigr )\;L^{(2l+1)}_{n-l-1}(s) \end{aligned}$$

and $s\equiv 2r/(na_0)$. How large is $\langle R\rangle$ as a function of n, l, and $a_0$? (8 P)

List of Symbols

We stick closely to the recommendations of the International Union of Pure and Applied Physics (IUPAP) and the Deutsches Institut für Normung (DIN). These are listed in Symbole, Einheiten und Nomenklatur in der Physik (Physik-Verlag, Weinheim 1980) and are marked here with an asterisk. However, one and the same symbol may represent different quantities in different branches of physics. Therefore, we have to divide the list of symbols into different parts (Table 4.3).

Table 4.3 Symbols used in Quantum Mechanics I