Two and Three-Dimensional Systems

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1 0 Two and Three-Dimensional Systems Separation of variables; degeneracy theorem; group of invariance of the two-dimensional isotropic oscillator. 0. Consider the Hamiltonian of a two-dimensional anisotropic harmonic oscillator: ( p 2 H = 2m + ) ( p 2 mω2 q m + ) 2 mω2 2 q2 2 ; ω ω 2. a) Exploit the fact that the Schrödinger eigenvalue equation can be solved by separating the variables and find a complete set of eigenfunctions of H and the corresponding eigenvalues. b) Assume that ω /ω 2 =3/4. Find the first two degenerate energy levels. What can one say about the degeneracy of energy levels when the ratio between ω and ω 2 is not a rational number? c) Write the eigenfunctions of the Hamiltonian in the case ω 2 =0. Consider now a particle of mass m in two dimensions subject to the potential: V (q,q 2 )=mω 2( q 2 q q 2 + q 2 2). d) Say whether the problem of finding the eigenvalues of the Hamiltonian H =(p 2 + p 2 2)/2m + V (q,q 2 )canbesolvedbythemethodofseparation of variables. 0.2 A particle of mass m in two dimensions is constrained inside a square whose edge is 2a: x a, y a. a) Write the Schrödinger equation, separate the variables and find a complete set of eigenfunctions of the Hamiltonian. b) Find the energy levels of the system and say whether there is degeneracy. c) Say whether there exist operators (i.e. transformations) that commute with the Hamiltonian but do not commute among themselves. In the affirmative case, give one or more examples. E. d Emilio and L.E. Picasso, Problems in Quantum Mechanics: with solutions, UNITEXT, DOI 0.007/ _0, Springer-Verlag Italia 20 99

2 200 0 Two and Three-Dimensional Systems Assume now that within the square the potential: V a (x, y) =V 0a cos(πx/2a) cos(πy/2a) is present. d) Is it still possible to separate the variables in the Schrödinger equation? Do degenerate energy levels exist? e) Say whether it is possible to guarantee the existence of degenerate energy levels if, instead, the potential is V b (x, y) =V 0b sin(πx/a) sin(πy/a). Is any relationship between the eigenfunctions of the Hamiltonian ψ E (x, y) and ψ E (y, x) expected?(namely,aretheyequal?aretheydifferent?...) 0.3 A particle of mass m in two dimensions is constrained inside the triangle whose vertices have the coordinates (x = 0, y =0); (x = a, y =0); (x = a, y = a) (a half of the square with edge a ). a) Find eigenvectors and eigenvalues of the energy. b) For the same system and exploiting the results of the previous question, find a complete set of eigenvectors of the operator that implements the reflection through the straight line x + y = a (the dotted line in the figure). 0.4 A particle of mass m in three dimensions is confined within an infinite rectilinear guide with a cross section that is a square of edge a. y a) Find eigenfunctions and eigenvalues of the Hamiltonian. What is the minimum energy (threshold energy) the particle must have in order to propagate along the guide? Consider the wavefunctions: ψ (x, y, z) =A sin(2πx/a)sin(πy/a)e i k z ψ 2 (x, y, z) =B sin(πx/a)sin(πy/a)e i k2 z. b) Determine the normalization coefficients A and B in such a way that the integral of the densities ρ,2 over a slice of the guide of unit volume equals (normalization oneparticleperunitvolume ). c) Calculate the probability current densities: ȷ,2 (x, y, z) = h m Im ( ψ,2(x, y, z) ψ,2 (x, y, z) ) x

3 Problems 20 for the states represented by the wavefunctions ψ and ψ 2 normalized as above, and verify that div ȷ,2 (x, y, z) =0. d) Say for which values of k 2 the probability current associated to the state represented by ψ(x, y, z, t) withψ(x, y, z, 0) = ψ (x, y, z)+ψ 2 (x, y, z), is divergenceless. 0.5 Aparticleissubjecttothepotential V = V (q 2 + q 2 2,q 3 ). a) Show that the Hamiltonian H 0 = p 2 /2m + V commutes with the angular momentum operator L z = q p 2 q 2 p. b) Use the degeneracy theorem to show that there exist degenerate energy levels. c) Say whether and how the degeneracy is removed if the system is on a platform rotating around the z axis with constant angular velocity ω. 0.6 The Hamiltonian of a two-dimensional isotropic harmonic oscillator of mass m and angular frequency ω is H = ( p 2 2m + p 2 2) + 2 mω2( q 2 + q2 2 ) = H (q,p )+H 2 (q 2,p 2 ). a) Exploit the separation of variables ( H = H +H 2 )andfindtheeigenvalues of H and their degeneracies. b) Write the eigenfunctions of the Hamiltonian in the Schrödinger representation, in the basis in which both H and H 2 are diagonal. c) Is the degeneracy found in a) in agreement with the result established in Problem 0.5? Find the maximum and the minimum of the eigenvalues m of L 3 = q p 2 q 2 p within each energy level. Do all its possible values ranging between m max and m min occur? d) For each of the first three energy levels, say which eigenvalues of L 3 do occur and explicitly write the wavefunctions relative to the states E,m (simultaneous eigenstates of H and L 3 ). 0.7 This problem is devoted to establish a priori the degeneracies of the two-dimensional isotropic harmonic oscillator found in Problem 0.6. Set η a = (p a i mωq a ), a =, 2. 2mω h a) Write the Hamiltonian H of the two-dimensional oscillator in terms of the operators η a and η a and the commutation rules [η a,η b ], a, b =, 2. b) Show that the four operators η a η b commute with the Hamiltonian H. Consider the operators: j = 2 (η η 2 + η 2 η ), j 2 = 2i (η η 2 η 2 η ), j 3 = 2 (η η η 2 η 2 ).

4 202 0 Two and Three-Dimensional Systems c) Show that the operators j a have the same commutation rules as the angular momentum (divided by h). Write j 2 and j 3 in terms of the q s and p s and show that the angular momentum operators j a have both integer and half-integer eigenvalues. Setting h 0 = H/ hω, the identity j 2 j 2 + j2 2 + j3 2 = h ( 0 2 h0 ) 2 + holds (it may be verified using the commutation rules). d) Exploit the theory of angular momentum (all the properties of the angular momentum follow uniquely from the commutation relations) and the above identity to find the eigenvalues of H and the relative degeneracies. Say which eigenvalues of L 3 do occur in each energy level. 0.8 In Problem 0.7 the energy levels of a two-dimensional isotropic harmonic oscillator and their degeneracies have been found starting from the commutation rules of the three constants of motion j,j 2,j 3 given from the outside. We now want to establish both the existence and the form of such constants of motion starting from the invariance group of the Hamiltonian. Adopting the notation of Problem 0.7 one has: ( 2 ) H = hω η a η a + ; [η a,η b ]=0, [η a,η b ]=δ ab; a, b =, 2. a= Consider the linear transformation: η a = b u ab η b. () a) Show that () is an invariance transformation both for the Hamiltonian and for the commutation rules if and only if u is a unitary 2 2matrix. We shall consider only the transformations that fulfill det u =. b) Show that all the unitary 2 2matrices,whosedeterminantis,maybe written as: ( ) z z 2 z2 z, z 2 + z 2 2 =, z,z 2 C. They, therefore, form a continuous, 3 parameter group the group SU(2). The transformation () in a neighborhood of the identity takes the form: η a = η a +iϵ b g ab η b, u l+iϵg, ϵ. (2) c) Show that the matrix g is Hermitian and traceless.

5 Problems 203 Thanks to the von Neumann theorem, for any transformation () there exists aunitaryoperatorthatimplementsit: η a = U(u) η a U (u). d) Let U(g, ϵ) =e i ϵgg be the unitary operator that implements the infinitesimal transformation (2) (the operators G g = G g are the generators of the group). Compare η a = U(g, ϵ) η a U (g, ϵ), expanded to the first order in ϵ, and(2)andshowthat [G g,η a ]= b g ab η b.findtheexpression for G g and show that [G g,h]=0. e) Show that any traceless Hermitian 2 2matrix g may be written in the form (the factor 2 is there only for the sake of convenience): g = a 2 σ + a 2 2 σ 2 + a 3 2 σ 3, a i R where σ i are the Pauli matrices ( ) ( 0 0 i σ = ; σ 0 2 = i 0 ) ; σ 3 = ( ) 0. 0 f) Write the expressions for G g in the three particular cases when only one of the a i equals and the other two are vanishing: compare the generators G,G 2,G 3 so obtained with the operators j,j 2,j 3 of Problem 0.7. Show that [G g,g g ]= ab η a [g,g ] ab η b and make use of the commutation relations of the Pauli matrices: [ 2 σ a, 2 σ b]=iϵ abc 2 σ c to find the commutation rules of the generators: [G a,g b ]=iϵ abc G c.

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