4.1a Suppose we are using two sets of basis vectors aa, 1 ≤ a ≤ N, and ai, 1≤ i≤N, which are not necessarily Cartesian. The expansion coefficients of the new basis vectors with respect to the old basis vectors are denoted asMai,
ai =aaMai. (4.102)
Show that the new expansion coefficientsvi of a general vectorv,
v =aivi =aava, (4.103)
are related to the old expansion coefficients va through transformation with the inverse matrixM−1from the left,
vi =(M−1)iava. (4.104)
7Normalizability is important for the correctness of Eq. (4.100), because for states in an energy continuum the left-hand side of Eq. (4.98) may not vanish in the degenerate limitEψ →Eφ, see Problem4.10.
4.4 Problems 83 In general, upper indices (“contravariant indices”) transform with the inverse transformation matrixM−1from the left (or equivalently with the “contragredient matrix” M−1,T from the right), while the lower indices (“covariant indices”) transform with the transformation matrixMfrom the right.
4.1b Show that you can use the “metric”gwith components in the basisaa,
gab=aaãab, (4.105)
and the inverse metricg−1with componentsgab, to switch between covariant and contravariant indices,
va=gabvb, va=gabvb. (4.106) Show also that the scalar products of the dual basis vectorsaa =gababare given byaaãab=gab.
4.2 We consider again the rotation (4.10) of a Cartesian basis, ˆ
ea→ ˆei = ˆeaRai, (4.107) but this time we insist on keeping the expansion coefficients va of the vector v =vaeˆa. Rotation of the basis with fixed expansion coefficents{v1, . . . vN}will therefore generate a new vector
v→v≡vieˆi. (4.108)
This is the active interpretation of transformations, because the change of the reference frame is accompanied by a change of the physical objects.
In the active interpretation, transformations of the expansion coefficients are defined by the condition that the transformed expansion coefficients describe the expansion of thenewvectorvwith respect to theoldbasiseˆa,
v≡vieˆi =vaeˆa. (4.109) How are the new expansion coefficientsvarelated to the old expansion coefficients vifor an active transformation?
Equation (4.109) implies that we can describe an active transformation either through a transformation of the basis with fixed expansion coefficients, or equiv- alently through a transformation of the expansion coefficients with a fixed basis.
This is different from the passive transformation, where a transformation of the basis is always accompanied by a compensating contragredient transformation of the expansion coefficients.
4.3 Two basis vectorsa1anda2have length one and the angle between the vectors isπ/3. Construct the dual basis.
4.4 Nickel atoms form a regular triangular array with an interatomic distance of 2.49 Å on the surface of a Nickel crystal. Particles with momentump = h/λare incident on the crystal. Which conditions for coherent elastic scattering off the Nickel surface do we get for orthogonal incidence of the particle beam? Which conditions for coherent elastic scattering do we get for grazing incidence in the plane of the surface?
4.5 SupposeV (x)is an analytic function ofx. Write down thek-representation of the time-dependent and time-independent Schrửdinger equations. Why is thex- representation usually preferred for solving the Schrửdinger equation?
4.6 Sometimes we seem to violate the symmetric conventions (2.7) and (2.8) in the Fourier transformations of the Green’s functions that we will encounter later on. We will see that the asymmetric split of powers of 2π that we will encounter in these cases is actually a consequence of the symmetric conventions (2.7) and (2.8) for the Fourier transformation of wave functions.
Suppose that the operatorGhas translation invariant matrix elements,
x|G|x =G(x−x). (4.110)
Show that the Fourier transformed matrix elements k|G|k satisfyk|G|k = G(k)δ(k−k)with
G(k)=
d3xG(x)exp(−ikãx), G(x)= 1
(2π )3
d3kG(k)exp(ikãx). (4.111) 4.7 Suppose that the Hamilton operator depends on a real parameterλ,H =H (λ).
This parameter dependence will influence the energy eigenvalues and eigenstates of the Hamiltonian,
H (λ)|ψn(λ) =En(λ)|ψn(λ). (4.112) Useψm(λ)|ψn(λ) =δmn(this could also be aδfunction normalization), to show that8
δmn
dEn(λ)
dλ = ψm(λ)|dH (λ) dλ |ψn(λ) +[Em(λ)−En(λ)]ψm(λ)| d
dλ|ψn(λ). (4.113)
8Güttinger [70]. Exceptionally, there isnosummation convention used in Eq. (4.113).
4.4 Problems 85 Form=ndiscrete this is known as the Hellmann–Feynman theorem [50,79]. The theorem is important for the calculation of forces in molecules.
4.8 We consider particles of mass mwhich are bound in a potential V (x). The potential does not depend onm. How do the energy levels of the bound states change if we increase the mass of the particles?
The eigenstates for different energies will usually have different momentum uncertaintiesp. Do the energy levels with large or smallpchange more rapidly with mass?
4.9 Show that the free propagator (3.71) and (3.72) is thex representation of the one-dimensional free time evolution operator,
U (t )=exp
−it−i 2mh¯ p2
, U (x−x, t )= x|U (t )|x. (4.114) Here a small negative imaginary part was added to the time variable to ensure convergence of a Gaussian integral. Show also that the free time-evolution operator in three dimensions satisfies
U (x−x, t )= x|exp
−it−i 2mh¯ p2
|x
=
m 2πih(t¯ −i)
3
exp
i m
2h(t¯ −i)(x−x)2
. (4.115) For later reference we also note that this implies the formula
exp
ih¯t−i 2m
∂2
∂x2
δ(x−x)=
m 2πih(t¯ −i)
3
×exp
i m
2h(t¯ −i)(x−x)2
. (4.116) 4.10 Apply Eq. (4.98) in the caseV (x) = 0 to plane wave states. Show that in this case the left-hand side does not vanish in the limit E(k) → E(k). Indeed, the equation remains correct in this case onlybecausethe left-hand side does not vanish.
4.11 Use the calculation ofporxexpectation values in the wave vector represen- tation and in the momentum representation of the state|ψto show that momentum and wave vector eigenstates ind spatial dimensions are related according to|p =
|k/h¯d/2. Does this comply with properδfunction normalization of the two bases?
Formal Developments
We have to go through a few more formalities before we can resume our discussion of quantum effects in physics. In particular, we need to address minimal uncertain- ties of observables in quantum mechanics, and we have to discuss transformation and solution properties of differential operators.
I have also included an introduction to the notion of length dimensions of states, since this is useful for understanding the meaning of matrix elements in scattering theory in Chaps.11and13. Furthermore, I have included a section on frequency- time Fourier transformation, although that can only be defined in a distributional sense for time-dependent wave functions. However, it is sometimes useful to represent the decompositions of states|ψ (t )in terms of energy eigenmodes|ψα, H|ψα = Eα|ψα, in the framework of Fourier transformation to a frequency- dependent state |ψ(ω). The frequency-dependent states vanish ifhω¯ is not part of the spectrum ofH, and they containδ-functions forhω¯ in the discrete spectrum ofH.