exercises and tutorials for
Mathematical Quantum Mechanics

  winter term 2012/2013 (WiSe 2012/2013)

(This page is not being updated)

course web page

weekly schedule:


tutorial A



tutorial B



homework session B B-005


homework session A B-005

office hours:

László Erdős Tuesday
2-4 pm
office B-329
Alessandro Michelangeli Tuesday
4-6 pm
office B-335
Benedikt Staffler
by appointment office B-331
Tobias Ried

by appointment office B-204
Maximilian Jeblick

by appointment
office B-217
Sebastian Gottwald

by appointment

homework assignments:

  • Each new homework assignment is posted by Tuesday evening.
  • Hand-in deadline (Abgabe der Hausaufgaben): each Tuesday by 6pm, in the designated drop box located on the first floor.
  • Pick up your marked worksheets from the designated return box on the first floor.
  • Markers (Korrektoren): Maximilian Jeblick, Sebastian Gottwald, Tobias Ried (contact them for clarifications on marking). By the way: markers are not graders...
  • A selection of the homework exercises will be discussed in detail in the homework sessions. (Die Lösungen werden in den Übungen besprochen.)
  • The two weekly homework sessions are identical.
  • A diary of homework sessions will be kept updated at the bottom of this page.
  • Please register through the LMU Maths Institute HOMEWORK WEBPAGE. When you're asked the year of your Studienhauptfach (=your major), that actually refers to the year when the Prüfungsordnung (=the exam regulations) was issued, NOT the year when you take the course. TMP students should select "TMP 2010".
week 1 assignment 01

week 2 assignment 02 solution

week 3 assignment 03 solution

week 4 assignment 04 solution

week 5 assignment 05 solution

week 6 assignment 06 solution

week 7 mid-term exam

week 8
assignment 07-08

week 9 assignment 09 solution

week 10 assignment 10

week 11-12 assignment 11-12 solution

week 13-14
assignment 13-14

  • Tutorials are meant for catching up with missing background material and for more details left open in the lectures or in the homework.
  • They will be in the form of a frontal presentation, or on-the-fly problems, or an interactive discussion.
  • The two weekly tutorials have the same content.
  • Tutors (Tutoren): Benedikt Staffler, Claudio Llosa Isenrich, Tobias Reid.
  • A diary of tutorials will be kept updated at the bottom of this page.

  • Mid-term exam (Zwischenklausur): Thu 13 Dec 2012, 12:15-14:00, lecture room B-005
    exam              solution
  • Final exam (Endklausur): Sat 9 Feb 2013, 9:00-11:30, lecture room B-139
    exam              solution

        final results

weekly diary of homework and tutorials:


  • E49 - The operator domain of -Laplacian+V can collapse to {0} even if the form domain is dense.
  • E50 - A positive self-adjoint operator with a gap from 0 is invertible. The square root of positive self-adjoint operator is monotone. This is false for powers higher than 1. Also the product of two positive operators is not necessarily positive.
  • E51 - Preparatory estimates for relative compactness.
  • E52 - Estimating spectral eigenspaces.


  • E41 - A bounded operator on a H-space is compact iff it maps weakly convergent into norm convergent sequences.
  • E42 - Wave operators: closure of domain and range, invariance properties, intertwining property.
  • E43 - Cook's criterion for the existence of the wave operator. Application to -Laplacian+V with a square integrable V.
  • E44 - The one-paramenter strongly continuous unitary group of dilations and the computation of its generato

The f(x)g(\nabla) theorem -- see Problem 30:
  • f(x)g(\nabla) is defined via F-transform as an operator on L^2(R^d). Its boundedness depends on the integrability properties of f and g.
  • In particular if both f and g are square-integrable, then f(x)g(\nabla) is Hilbert-Schmidt.
  • If f and g are bounded and decay at infinity, then f(x)g(\nabla) is compact.

Self-adjointess, spectral theorem, functional calculus, and all that.
  • Unitary transformations preserve self-adjointness, spectrum, point spectrum. (See solution to Exercise 40)
  • Adjoint of a sum and of a product of operators in the unbounded case. (See Problem 15.)
  • Step-by-step computation of the adjoint of the momentum operator on [0,2pi] with boundary conditions f(0)=f(2pi)=0. (See Problem 14.) In fact, if one knows already the definition and the properties of H^1, then the computation of the adjoint is immediate.
  • You may want to practise with some of these problems on self-adjointness, spectral theorem, functional calculus.
  • E37 - The Volterra integral operator on L^2[0,1]. Boundedness, compactness, absence of eigenvalues, spectrum collapses to {0} only.
  • E38 - Hilbert-Schmidt operators on L^2 are realised as integral operators with square-integrable kernel.
  • E39 - Multiplication operator by a measurable function. Maximal domain. Adjoint. The operator is self-adjoint iff the function it multiplies by is real-valued. Its spectrum coincides with the essential range of that function. Measure characterisation of the eigenvalues.
  • E40 - Position and momentum operators: self-adjointness, absence of eigenvalues, spectrum. In fact, P and Q are unitarily equivalent, via F-transform.

    Note: Vito Volterra, the mathematical physicist, after whom the integral operator of E37 is named. Volterra, the old capital of Etruscans.

Revision on Spectral Theorem
(from Sections 7.1, 7.2, 7.3 of this handout):
  • Functional calculus from the Spectral Theorem. The general idea is in Section 7.3 of the handout. The complete statement, i.e., construction of bounded Borel functions of a (possibly unbounded) self-adjoint operator, is Theorem VIII.5 of Reed-Simon.
  • IMPORTANT MESSAGE 1: it is part of (the proof of) the Spectral Theorem that the functional calculus produces a p.v.m. (when one constructs functions of A starting from characteristic functions of Borel sets) that is *precisely* the p.v.m. associated with A in the statement of the Spectral Theorem.
  • IMPORTANT MESSAGE 2: in practice one just constructs polynomials of A in the usual way; the general f(A) is given by a strong limit (note part (d) of Theorem VIII.5 of Reed-Simon) using the uniform density of polynomials in (bounded) continuous functions and the pointwise density of the latter in bounded Borel functions.
  • Mesaure types: decomposition of a Borel measure in pp, sc, ac part. Reference: Reed-Simon, Section I.4, pag 19-22, and Teschl, Section A.7 of the appendix.
  • E33 - The bound states of a Schrödinger Hamiltonian in dimension 3 or higher with potential in L^{d/2}+L^{\infty} are exponentially localised wave functions.
  • E34 - System of two nuclei very much far apart with two electrons: ground state energy.
  • E35 - Schur's test for kernel (integral) operators.
  • E36 - Kernel (integral) operators with square integrable kernels. The operator norm is dominated by the L^2-norm of the kernel. Approximation in operator norm with finite-rank operators. The Hilbert-Schmidt norm is the same irrespectively of the orthonormal basis considered.

    Schur's test, discussed in E35, was introduced by Schur in Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen (1911)

Revision on Spectral Theorem
(from Sections 7.1, 7.2, 7.3 of this handout):
  • Projection-valued measures (equivalently, spectral resolution).
  • Integration w.r.t. a projection-valued measure in complete analogy to the integration w.r.t. Lebesgue.
  • Construction of a (possibly unbounded) self-adjoint operator and characterisation of its domain. Statement of the Spectral Theorem in p.v.m. form (Theorem 7.3 of the handout, Theorem VIII.6 of Reed-Simon.)
  • Example: the case of a symmetric matrix (working out explicitly the Example in pag. 29 of the handout).
  • E25 - If the solution f to (-D-V)f=0 in 3dim (V non-zero, non-negative, and locally integrable) is smooth and non-negative, then either f>0 or f=0.
  • E26 - The ground state of a single well potential has only one single peak.
  • E27 - Helium Hamiltonian: variational upper bounds to the ground state energy.
  • E28 - Anti-symmetric wave functions: L2-scalar product of wedge products.
  • Tensor product of Hilbert spaces. Rigorous definitions and properties. (References: Teschl, chapter 1.4, Reed-Simon, chapter II.4.)
  • Revision on bounded linear operators on a Hilbert space, first part:
    • Definition of the algebra B(H) of bounded linear operators on H. The operator norm makes it a C*-algebra. The abstract definitions of resolvent set, spectrum, resolvent operator, as well as the properties discussed in abstract in the tutorial of week 03, carry over to this concrete case.
    • Definition of the (Hilbert) adjoint of an operator in B(H).
    • Definition of self-adjoint operator and of orthogonal projection.
    (Reference: Reed-Simon, chapter VI.)

    Suggested problems in class:

  • P11 - Orthogonal projections on a Hilbert space. Kernel and range give an ortogonal decomposition of H. Spectrum of the projection. Resolvent operator of the projection.
  • P12 - Explicit computation of the norm and the adjoint of an operator on L^2(R).
  • P13 - Important formulas for a bounded operator T on a Hilbert space. KerT* is the orthogonal complement to RanT. The special case of normal operators.

     mid-term exam
  • Integral of sinc(x)^2: F-transform, Parseval.
  • The distributional solution to xu'+u=0 is a linear combination of the delta and the principal value.
  • Weak convergence in H^1(Omega) implies weak convergence in L^2(Omega). This is always true (it's a general Banach space fact), here you were asked to prove it for Omega=R, thus by means of Fourier transform.
  • Cyclic vectors for multiplication by x on L^2[-1,1]. The function 1 is, the Heaviside function is not. Multiplication by x^2 on L^2[-1.1] has no cyclic vectors.
  • In 3 dimensions the H^2 norm controls the L^infinity norm.
  • -D+tV in d>2 dimensions with a V that is non-positive, vanishing at infinity, and d/2-integrable, and with negative ground state energy for some t_0, has a ground state energy E(t) that is strictly decreasing in t for t>t_0.

    Despite what claimed in some solution sheets, the answer to the cyclic vector problem is NOT Spongebob, sorry.

     problems in class
  • P07 - In a H-space convergence against elements of an ONB + boundedness of norms is the same as weak convergence.
  • P08 - Useful identities and inequalities involving resolvents.
  • P09 - Integral of sinc(x)^4: F-transform, Parseval.
  • P10 - Higher order Sobolev norms control the L^p norm.

     assignment 06
  • E21 - Construction of cut-off functions. IMS localisation formula.
  • E22 - Potentials vanishing at infinity in dim=3 (or higher) give rise to non-positive ground states. 
  • E23 - Variational bound from above and localisation estimate from below of the ground state of a system made of an electron and two fixed nuclei.
  • E24 - Potentials that scale under dilations give rise to a virial theorem for the ground state energy E, and E turns out to be non positive.

     problems in class
  • P01 - Variational characterisation of ||f|| as sup of the duality products against elements of the unit ball of the dual. Convenient fact: it suffices to use only a norm-dense subspace of the dual in order to compute ||f|| with this variational characterisation
  • P02 - L2-norm of gradient of f is controlled by the L2-norm of f and the L2-norm of Laplacian of f. L2-norm of Laplacian of f controls the L2 norm of any second derivative of f. L2-norm of f and of some higher derivative of f controls Lp-norm of f.
  • P03 - In 3 dim or more, if the potential V is sufficiently small in the L^{d/2} sense then the ground state of -Laplace+V is non-negative.
  • P04 - Practise with Young's inequality. The convolution of 1/|x| times a density rho.
  • P05 - Examples of spaces that are / are not C*-algebras.
  • P06 - Simple properties on the spectrum of A^n and a-A. The spectra of A and of A* coincide

     assignment 05
  • E17 - The free Schrödinger evolution is a strongly continuous (not norm continuous) unitary group on L^2. It converges weakly to zero as time goes to infinity.
  • E18 - The spectrum of an element of a C*-algebra is a compact subset of complex number contained in the disk of radius ||A||. The C*-condition implies that for a normal element A one has ||A^m||=||A||^m for any integer m. In a non-commutative C*-algebra the only scalar commutator possible is 0. (Consequence: observables P and Q cannot be simultaneously bounded.)
  • E19 - If the commutant of the range of a representation of a C*-algebra consists only of multiples of the identity, then the representation is irreducible. (This is in fact <=>.) If the representation is irreducible, then any vector of the representation space is cyclic. (This is in fact <=>.) Pure states can be interpreted as (positive) bounded linear functionals on the C*-algebra of observables.
  • E20 - Polarisation observable for an EPR pair of transverse photons flying apart in opposite directions. Angles for which the corresponding Bell's inequality is quantum-mechanically violated.

Problem 20 is easy because there are two entangled photons only... Look at this October 2012 PRL article where they entangled more then 10000 photons!

  • Retrospective and concluding remarks on the C*-algebraic formulation of Quantum Mechanics.
    (Informal notes of the tutorial.)

     assignment 04
  • E13 - For a sequence to converge weakly in L^p it is not enough to converge when tested on a dense only of L^q. Weak convergence plus convergence of the norms imply norm convergence in L^2 (in fact, in L^p).
  • E14 - Weak convergence of a sequence in L^2 and of the sequence of the weak derivatives implies that the limit is in H^1. Differences between norm/weak convergence in L^2 and in H^1.
  • E15 - Examples of functionals on L^p that are or are not continuous in norm/weakly.
  • E16 - The dispersive estimate for the free Schroedinger evolution.
  • Revision on weak convergence in infinite-dim H-spaces and in L^p spaces, 1<p<inf. Definition: it's the convergence when tested against all dual elements. Convergence against a dense only is not enough. (See Exericse 13.) Nor it suffices to use an ONB of the H-space. But convergence against elements of an ONB + boundedness of norms is the same as weak convergence. (See Problem 7 here.)
  • The weak limit is unique. (The underlying topology is Hausdorff, it separates points. A basis of neighbourhoods at 0 are cylinders in all but finitely many dimensions.) Warning: we shall only consider sequential convergence.
  • Norm conv. => Weak conv. Opposite is false. But weak convergence + convergence of norms imply norm convergence (Exericse 13).
  • The norm closure of a subset is contained in its weak closure. If a vector suspace is norm closed, then it is also weakly closed.
  • Every norm-separable H-space is also weakly separable. Every H-space is also (sequentially!) weakly complete, i.e., every weakly Cauchy sequence converges weakly to some element of H. Proof of last fact involves Uniform Boundedness Principle + Riesz Representation Thm, that is, completeness is heavily used.
  • Pointwise convergence does not imply weak conv. Weak conv. does not imply pointwise conv. Weak conv. => distributional conv. but not vice versa. Examples.
  • The norm is norm-continuous, not weakly cont., but weakly lower semicont. In general, non-linearity destroys weak convergence (examples).
  • Scalar products (or L^p-L^q products) of two weakly convergences does not necessarily converge. They do if one sequence converges in norm.
  • General mechanisms for weak convergence in L^p(Omega): (1) rapid oscillation, (2) concentration, (3) wandering off to infinity.
  • Relevant consequence of duality L^p/L^q (or of self-duality of a H-space): variational characterisation of ||f|| as sup of the duality products against elements of the unit ball of the dual. Convenient fact: it suffices to use only a norm-dense subspace of the dual in order to compute ||f|| with this variational characterisation. (See Problem 1 here.)
  • Our main tool based upon weak convergence: the Banach-Alaoglu theorem in L^p(Omega), 1<p<inf. It is a compactness theorem: extracting weakly convergent subsequences is crucial for us to prove the existence of minimisers of energy functionals. Proof of Banach-Alaoglu is constructive in this case because L^p is separable: it boils down to a Cantor diagonal trick (see Lieb-Loss, Section 2.18). Remark: being a weakly-compactness fact, Banach-Alaouglu is NOT quantitative in the rate of convergence of the extracted subsequence.

assignment 03
  • E09 - General expression of a smooth function times the delta distribution. General distributional solution to (x^k T)=0. A first order distributional O.D.E.
  • E10 - The free evolution of a coherent state is again a coherent state whose position and momentum expectations evolve classically.
  • E11 - Three dimensional Green function of -Laplacian+m^2.
  • E12 - The principal value (PV) distribution.
  • C*-algebra theory: abstraction of the structure of bdd operators on H-space. Provides conceptually clean language for Q.M. of large systems (infinite particles) and natural language to axiomatize Q.M. when emphasis is put on observables instead of states.
  • Def of: commutative, non commutative, *-, normed-, Banach-, Banach*- algebra. The C* condition. Examples of C*-algebras.
  • At most one identity in a C*-alg. If not present, natural embedding in a larger algebra with identity.
  • Left/right/two-sided(bilateral) ideals. Closed, two-sided ideals have special role: they make the quotient algebra again a C*-algebra. Notion of simple C*-algebras. Examples.
  • Normal, self-adjoint, unitary, projection, positive elements of a C*-alg. Inverse.
  • Resolvent set, spectrum, resolvent. Spectrum of A+z, A*, A^{-1}. Spectrum is non-empty and compact in C. Typical technique: series expansion and analytic continuation. Neuman series for the resolvent.
  • Spectral radius, spectral radius formula for self-adj elements. It's the key technical ingredient towards the Spectral Theorem.
  • *-homomorphisms between C*-algebras. They turn out to map positive elements into positive elements (easy) and to be continuous with norm at most 1 (less evident). Examples.
  • Representation of a C*-alg on a Hilbert space. Faithful representations. Irreducible representations. Many examples.
  • Structure theorems: any C*-alg is *-isomorphic to a norm-closed, self-adjoint sub-algebra of bdd operators on H-space. Any commutative C*-alg is *-isomorphic to continuous functions vanishing at infinity over a locally compact Hausdorff space. (Existence of non-trivial repr., as well as structure thms, follows from Hahn-Banach thm.)
  • Characters of commutative C*-algebras. Examples.
  • References: Bratteli-Robinson, sections 2.1.1, 2.2.1, 2.2.2., 2.2.3, 2.3.1, 2.3.4, and 2.3.5, Thirring, sections I.2.2, I.2.3, and Strocchi, sections 1.4, 1.5, and 2.6.
  • Informal notes of the tutorial.

     assignment 02
  • E05 - Distributional F-transform of 1/|x| and 1/|x|^2. Variational upper bound on the ground state of hydrogenic atoms using coherent states.
  • E06 - Sequences of L1_loc functions that converge or not in the sense of distributions.
  • E07 - Link between the decay of a function and the differentiability of its F-transform. Also, a function and its F-transform cannot be both compactly supported.
  • E08 - Proof of Hardy inequality (with the right constant) through the vector field method.
  • The Sobolev space H^1(Omega), our natural "energy space": defined as the space of L^2 functions with weak derivative in L^2 (recall the def. of weak derivative from class).
  • H^1 is complete and has a natural structure of Hilbert space. 
  • Smooth (C^inf) functions with L^2-derivative are dense in H^1(Omega). In particular compactly supported smooth functions are dense in H^1(R^d).
  • A bounded smooth function times an H^1 function is still H^1 and chain rule applies. Partial integration between H^1 functions applies in the form int(fg')=-inf(f'g).
  • For H^1 functions, gradient of |f| has a smaller L^2 norm than gradient of f.
  • Fourier characterisation of H^1.
  • Useful facts to know (try to prove them as an exercise for your own preparation at home, possibly future homework):
    (1) An L^2 function is in H^1 if and only if its difference quotient converges in L^2 sense to the weak derivative of the function.
    (2) Functions in H^1(R) (or of an interval) are continuous, and Hoelder continuous almost everywhere.
    (3) The characteristic function of a set in R^d with positive measure cannot belong to H^1(R^d).
  • References: Lieb-Loss, sections 7.2 to 7.9.

     assignment 01
  • E01 - Indeterminacy relation in F-transform language in d dimension. Indeterminacy is minimised by coherent states.
  • E02 - Fourier transform of a positive definite quadratic form in d dmensions.
  • E03 - Kernel of the free Schroedinger evolution.
  • E04 - Bessel kernel, the kernel of (1-Laplacian)^{-1}.
Concerning the Uncertainty Principle (E01) visit the American Physical Society online exhibit on the Uncertainty Principle.
  • Topology to have a structure in which we can say what convergence means, even without the need of a distance.
  • From metric to topological spaces. Definition of topology: a family of subsets (the "opens") with given consistency conditions. Subspace topology.
  • Hierarchy topological/metric/normed/Banach/Hilbert spaces.
  • Open and closed. Neighbourhood. Interior. Boundary. Limit point.
  • Base for a topology. Basis of neighbourhoods.
  • 1st countable / 2nd countable / separable topological spaces.
  • Convergence and continuity in a topological space.
  • Compact/bounded topological spaces.
  • References: standard textbooks in Topology; and this handout.
  • Lebesgue integral on R^n, visual picture of Riemann vs Lebesgue, why Lebesgue measure/integral is superior to Riemann.
  • Dominated and monotone convergence, Fatou's Lemma, Fubini. Examples and counterexamples.
  • References: Reed-Simon, volume 1, section I.3; Lieb-Loss, sectons 1.6, 1.7, 1.8., 1.12, 2.1; plus section 1 of this handout.
  • Fourier transform from physical point of view: convenient unitary transformation of wave-functions: implements indeterminacy principle, diagonalises energy, monitors short x with large p and vice versa.
  • F-transform in L^1: well defined, linear; has symmetry under translation/scaling/rotation; is bounded and injective from L^1 to C_inf, not surjective though; maps Gaussians in Gaussians, convolutions in point-wise products.
  • We like/need to invert F, natural space for inversion is the Schwartz class. Here F becomes a linear bijection, with explicit inversion formula. Up to factors 2pi, interchanges differentiation and multiplication by monomials. Preserves L^2 norm (Plancherel/Parseval) => it extends uniquely to a unitary operator on L^2.
  • Computationally: F-transf of f in L^1 intersected L^2 is computed with the explicit formula, if f is in L^2\L^1 then first compute F(f) on L^1 (or Schwartz-) approximants. Typical approximations: truncation out of a big ball, smoothing, suppressing the tale, etc.
  • References: Reed-Simon, volume 2, sections IX.1, IX.2; Lieb-Loss, sections 5.1 to 5.5; Teschl, section 7.1; plus section 2 of this handout.