5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:

Transcription

1 5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ ξ ξ 3, ξ 1 ξ 3 ; η 0 γ 0 γ η 3. a) Verify that the two observables are compatible with each other and find the basis with respect to which they are simultaneously diagonal. b) Are the two observables a complete set of compatible observables? 5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis: a) from e 1, e 2 to cos ϑ e 1 +sinϑ e 2, sin ϑ e 1 +cosϑ e 2 ; ( 1 b) from e 1, e 2 to 2 e1 ± i e 2 ). 5.5 Adiatomicmoleculecancaptureanelectronandweassumethestate space of the electron is two-dimensional and generated by the orthogonal vectors 1, 2 that respectively represent the state of the electron captured by either the first or the second atom. a) Write the matrix representing, with respect to the basis 1, 2, the most general electron Hamiltonian H and determine its eigenvalues. b) Show that it is always possible to choose the phases of the basis vectors 1 =e i ϕ1 1, 2 =e i ϕ2 2 so that the matrix representing H in the new basis 1, 2 is real. Assume that the molecule consists of two identical atoms. The equality of the atoms entails the invariance of H under the unitary transformation that exchanges the states (not necessarily the vectors) represented by 1 and 2 : Π 1 = α 2, Π 2 = β 1, α = β =1.

2 Problems 65 c) The requirement that Π HΠ = H (or equivalently, since Π = Π 1, that ΠH = HΠ)impliessomerestrictionsbothonthematrixthat represents H in the basis 1, 2 and on α, β. Find these restrictions. Show that, provided Π is redefined by a multiplicative factor ( Π e i χ Π ), it is possible to have α = β =1. Assume now that the molecule is triatomic (the three atoms not necessarily being equal) and that it can capture the electron in the three orthogonal states 1, 2, 3. d) By suitably choosing the phases of the basis vectors, how many of the elements of the matrix representing H can be made real? 5.6 Consider the scale transformation q q = λq, p p = λ 1 p where λ is an arbitrary real parameter. a) Verify that the transformation is a canonical transformation. The Hamiltonian of a harmonic oscillator of mass m and angular frequency ω is H = p2 2m mω2 q 2. b) Exploit the von Neumann theorem (there exists a unitary operator U implementing the transformation q q λq = UqU 1, p p λ 1 p = UpU 1 )toshowthatthetwohamiltonians H 1 and H 2 : H 1 = p m 1 2 m 1 ω 2 q 2, H 2 = p m 2 2 m 2 ω 2 q 2 have the same eigenvalues. (As a consequence the eigenvalues of the Hamiltonian of a harmonic oscillator of given ω are independent of its mass.) Are the eigenvectors of H 1 and H 2 the same? The Hamiltonian of the hydrogen atom is H = p 2 2m e2 r c) Show that the discrete eigenvalues of H depend on the charge e and the mass m only through the product me 4.Verifythisistruefortheenergy levels given by the Bohr theory. 5.7 Consider a harmonic oscillator of mass m and angular frequency ω and the canonical transformation q q = Λp, p p = Λ 1 q,where Λ is an arbitrary real parameter (with dimensions [Λ] =TM 1 ). a) Show that there exist values of Λ such that the transformation q q, p p is an invariance transformation for the Hamiltonian H, and find them.

3 66 5 Representations b) Exploit the above invariance transformation to show that, for any eigenstate of H, themeanvalueofthekineticenergyequalsthemeanvalueof the potential energy. c) Calculate the product q p of the uncertainties of q and p in the eigenstates of H. d) Show that the mean value of qp+pq in any eigenstate of H is vanishing. 5.8 Let U(a) = e i pa/ h be the operator that translates coordinates: U(a) qu 1 (a) =q a, and V (b) =e i qb/ h be the operator that translates momenta: V (b) pv 1 (b) =p b ( a and b being real numbers). a) Show that p ap bq and q 1 2( a 1 q + b 1 p ) are canonically conjugate variables. b) Exploit the preceding result to show that the operators U(a)V (b) = e i pa/ h e i qb/ h and W (a, b) e i(pa qb)/ h induce the same canonical transformation and that, as a consequence, U(a) V (b) = e i ϕ W (a, b) (von Neumann). Use the Baker Campbell Hausdorff identity (see Problem 4.13) to calculate the phase factor e i ϕ. c) Find the unitary operator G(v) that implements the Galilei transformation for a particle of mass m: q q G(v) qg 1 (v) =q vt, p p G(v) pg 1 (v) =p mv. 5.9 Given any one-dimensional system, consider the operator: η λ = 1 2λ h ( p i λq ), λ > 0. From the theory of the harmonic oscillator we know that there exists a unique state 0 λ such that η λ 0 λ = 0. From now on we shall simply write 0, η instead of 0 λ,η λ. Let α V (b) U(a) 0, α = b 2λ h i λ 2 h a α 1 +iα 2 where U(a) = e i pa/ h and V (b) = e i qb/ h (see Problem 5.8) are the translation operators for coordinates and momenta respectively. (The states α are named coherent states.) a) Show that for any α C, η α = α α. b) Show that q = α q α = a, p = α p α = b and that the states α are minimum uncertainty states. (In the solution we will show that the converse also holds, namely that all minimum uncertainty states are coherent states.) c) Determine the representatives n α of the vector α in the basis n = 1 n! (η ) n 0 and the scalar product α β of two coherent states. d) Show that there exists no vector orthogonal to all the α vectors.

4 Problems Consider a one-dimensional harmonic oscillator of mass m and angular frequency ω in the coherent state relative to the oscillator (i.e., see Problem 5.9, with λ = mω): b mω α = V (b) U(a) 0, α = i 2mω h 2 h a where 0 is the ground state of the oscillator. a) Calculate the mean value H of the energy in the state α, the degree of excitation of the oscillator, defined as n H/ hω 1 2, and the dispersion n = H/ hω. b) Calculate the mean values of kinetic and potential energy in the state α. c) Calculate the uncertainties q and p in the state α and verify that α = p/(2 p) i q/(2 q) Consider a one-dimensional harmonic oscillator endowed with a charge e and subject to a uniform and constant electric field E oriented in the direction of the motion of the oscillator. a) Write the Hamiltonian and find the energy levels of the system. Assume the system is in the ground state. At a given instant the electric field is turned off and afterwards the energy of the oscillator is measured. b) Calculate the probability of finding the eigenvalue E n =(n ) hω of the Hamiltonian that is now the Hamiltonian of the free oscillator, i.e. of the oscillator in the absence of the electric field Consider a harmonic oscillator of mass m and angular frequency ω. a) Calculate the mean value of q 6 in the ground state. (It may help to consider it as the squared norm of the vector q 3 0. ) Show that the result implies that the probability of finding the oscillator out of the region accessible to a classical oscillator with the same energy is nonvanishing. As for momentum, a similar result holds. b) Find the interval of the allowed values of the momentum for a classical oscillator with energy E 0 = 1 2 hω.byawiseuseoftheresultoftheprevious question, show that for a quantum oscillator in the ground state the probability of finding the momentum out of that interval is nonvanishing Consider a particle in n dimensions. a) Say what the dimensions (in terms of length L, mass M, time T) of normalized wavefunctions are in the Schrödinger representation.

5 68 5 Representations Consider a particle in one dimension in the state A whose wavefunction is ψ A (x) =N e ax4, a > 0 where N is the normalization coefficient. b) Show that N = Ca γ,where C is a constant. Find the value of γ. c) Show that, even not knowing the value of C, themeanvalueof q 4 in the state A can be explicitly calculated. d) For what other values of n can the mean value of q n in the state A be explicitly calculated? 5.14 Consider a harmonic oscillator of mass m and angular frequency ω. a) Start from the equation η 0 =0 andfindthenormalizedwavefunction ϕ 0 (k) for the ground state 0 in the momentum representation. Given the normalized eigenfunctions of the Hamiltonian H in the Schrödinger representation: 1 ( mω ) 1/4 ψ n (x) = Hn ( (mω/2 h) x2 mω/ hx)e 2n n! π h it is possible to find the eigenfunctions of H in the momentum representation without resorting to the Fourier transform. To this end it is convenient to exploit the invariance of H under the transformation: UqU 1 = p mω, UpU 1 = mωq. b) Let x and k be the improper eigenvectors respectively of q and p normalized according to x x = δ(x x ), k k = δ(k k ). Show that: U k = 1 x = k/m ω. mω In calculating the normalization factor 1/ mω it may help to recall the property δ(x/a) = a δ(x) of the Dirac delta function. c) Find the normalized eigenfunctions ϕ n (k) k n of the Hamiltonian in the momentum representation a) Calculate the mean value of qp+ pq in the coherent states α defined in Problem 5.9. b) Find the wavefunctions of coherent states in both the momentum and Schrödinger representations.

6 Problems a) Show that the mean value of p in any state with real wavefunction ψ(x) vanishes. b) Calculate the mean value of p in the state described by the wavefunction ψ(x) =e i ϕ(x) χ(x) withϕ(x) and χ(x) realfunctions. c) Calculate the mean value of p in the state whose wavefunction is ψ(x) = χ(x)e i kx Afreeparticleinonedimension(H = p 2 /2m) isinthestate 1 whose normalized wavefunction is ψ 1 (x) =(α/π) 1/4 e αx2 /2. a) Calculate the mean values of q 2, p 2 and p 4 and show that the odd powers of p have vanishing mean values. b) Exploit the preceding results and calculate the mean values of p 2 and p 4 in the state 2 whose wavefunction is ψ 2 (x) =(α/π) 1/4 e αx2 /2 e i kx. c) Calculate the energy uncertainty E in the state 2 and show that when α k 2, E/E 2 p/p (as in classical physics, due to E p 2 ) The Schrödinger representation of an operator ξ is given by the function of two variables (actually it is a distribution) ξ(x, y) x ξ y, where x, y are the improper eigenvectors of position. a) Given A SR ψ A (x), find the wavefunction ψ B (x) ofthestate B = ξ A. Given ξ SR ξ(x, y), what is the Schrödinger representation of the operator ξ? b) Find the Schrödinger representation of the projection operator E A = A A.Showthatif χ(x) isanarbitrarynormalizedfunction,theoperator E χ E χ (x, y) =χ(x) χ (y) isaprojectionoperatorthatprojects SR onto a one-dimensional manifold. c) If n SR ψ n (x) isanorthonormalbasis,writethecompletenessrelation (or decomposition of the identity) n n n =1l in the Schrödinger representation. The trace of an operator ξ (when it exists) is defined as (see Problem 4.8): Tr ξ n n ξ n where n is an arbitrary orthonormal basis. d) Show that Tr ξ = + ξ(x, x)dx. e) Given the projection operator E V onto the manifold V, showthat TrE V equals the dimension of V.

7 70 5 Representations Consider the operator E whose Schrödinger representation is given by: λ E(x, y) = 2 π e (λ/2)(x +y 2 ) ( 1+2λxy ). f) Show that E is a projection operator: E = E, E 2 = E, calculatethe dimension of the manifold onto which it projects and characterize it Aparticleinonedimensionisinthestate: A = A 0 +e i ϕ U(a) A 0 where U(a) =e i pa/ h is the translation operator and A 0 is the state with wavefunction ψ 0 (x) =(2π 2 ) 1/4 e x2 /4 2, A 0 A 0 =1. a) What condition must a and satisfy in order that A 0 U(a) A 0 be negligible? Calculate A 0 U(a) A 0 for a =10. From now on we shall assume that A 0 U(a) A 0 is negligible. b) Determine the probability density ρ(x) for the position of the particle. Within the approximation A 0 U(a) A 0 0, is it possible to determine the phase ϕ by means of position measurements? c) Determine the probability density ρ(k) for the momentum of the particle. d) Say what is the required precision for momentum measurements in order to distinguish the state A from the statistical mixture { A 0,ν 1 = 1 2 ; U(a) A 0, ν 2 = 1 2 } Let A and B be two states whose wavefunctions in the Schrödinger representation are ψ A (x, y, z) andψ B (x, y, z) =ψa (x, y, z). (Assume that ψ A (x, y, z) and ψa (x, y, z) are not proportional to each other.) a) Which, among the following observables, may have different mean value in the two states A and B : f( q ); p i (i =1, 2, 3) ; p 2 ; L i ( q p ) i? b) Given the wavefunction ϕ A ( k )of A in the momentum representation, find the wavefunction ϕ B ( k ) of B Aparticleisinastatewhoseprobabilitydensityforthepositionis ρ(x) = ψ(x) 2 = N 2 (x 2 + a 2 ) 2 a) Say whether the state of the particle is uniquely determined. b) Is it possible to calculate the mean values of p and q in such a state?

8 Problems It is known (see Problem 5.21) that the knowledge of either the probability density ρ(x) for the position or, analogously, ρ(k) for momentum is not sufficient to determine the state A of the particle, i.e. its wavefunction ψ A (x) and/or ϕ A (k). Establishing whether the knowledge of both probability densities is sufficient to uniquely identify the state of the particle is the purpose of this problem. For a particle in one dimension, consider the states A and B of definite parity described by the wavefunctions (not proportional to each other): A ψ A (x) =± ψ A ( x), B ψ B (x) =ψ A(x) =± ψ B ( x). a) Show that both ρ A (x) =ρ B (x) and ρ A (k) = ρ B (k). Since A and B are different (by assumption ψ A (x) and ψ B (x) are linearly independent) there must exist observables whose mean values in the states A and B are different. b) Say which, among the following observables, may have different mean values in the two states: f(q),g(p),h= p 2 /2m + V (q),pq+ qp. be the normalized wave- Let ψ A (x) =(a/π) 1/4 e (a+i b) x2 /2, a > 0, b R function of A. c) Calculate the mean value of the observable pq + qp in the states A and B Consider a particle in one dimension and the canonical transformations generated by the family of unitary operators ( α R ): U(α) =e i α(qp+pq)/2 h : q(α) U(α) qu 1 (α), p(α) U(α) pu 1 (α). a) Show that: d q(α) dα = q(α), d p(α) dα = p(α) and, by taking into account that q(0) = q, p(0) = p, explicitlydetermine q(α) and p(α). b) Denoting by x and k the improper eigenvectors of q and p normalized according to x x = δ(x x ), k k = δ(k k ), show that: U (α) x =e α/2 e α x, U (α) k =e α/2 e α k. c) If ψ A (x) x A and ϕ A (k) k A stand for the normalized wavefunctions of the state A respectively in the Schrödinger and momentum representations, determine the wavefunctions of à U(α) A in the two representations.

9 72 5 Representations 5.24 Consider a triatomic molecule consisting of three identical atoms placed at the vertices of an equilateral triangle. The molecule is twice ionized and the Hamiltonian of the third electron in the field of the three ions is H = p 2 2m + V ( q a )+V ( q b )+V ( q c ) a 3 c b 1 2 where the vectors a, b, c stand for the position of the three atoms with respect to the centre of the molecule (see figure). a) Taking the origin of the coordinates in the centre of the molecule, write the canonical transformation of the variables q, p corresponding to a rotation of 120 around the axis orthogonal to the plane containing the atoms and show that this transformation leaves H invariant. b) Show that H is invariant also under the reflection with respect to the plane orthogonal to the molecule and containing c (x x, y y, z z). Do the 120 rotation and the reflection commute with each other? c) May H have only nondegenerate eigenvalues? Let 1, 2, 3, with wavefunctions respectively ψ 1 ( r ) = ψ 0 ( r a ), ψ 2 ( r ) = ψ 0 ( r b ), ψ 3 ( r ) = ψ 0 ( r c ), the three particular states in which the third electron is bound to each of the three atoms. We assume (it is an approximation) that 1, 2, 3 are orthogonal to one another. The operator that implements the 120 rotation, induces the following transformation: U 1 = 2, U 2 = 3, U 3 = 1 and the operator that implements the reflection x x, y y, z z induces: I x 1 = 2, I x 2 = 1, I x 3 = 3. d) Restrict to the subspace generated by the orthogonal vectors 1, 2, 3. Write, with respect to this basis, the matrix representing the most general electron Hamiltonian invariant under the transformations induced by U and I x. e) Find the eigenvalues of such a Hamiltonian. f) Still restricting to the subspace generated by 1, 2, 3, findthe simultaneous eigenvectors of H and I x and those of H and U.