Quantum Mechanics Problems, Assignments of Quantum Mechanics

Download Quantum Mechanics Problems and more Assignments Quantum Mechanics in PDF only on Docsity! on . Using Planck’s radiation formula show that the maximum of energy density occurs at Amaz = b/T with T being the temperature and 64 is a constant. . According to the classical model of the hydrogen atom (Rutherford model), an electron moving in a circular orbit of radius 0.053 nm around a proton fixed at the center is unstable, and the electron should eventually collapse into the proton. From classical electromagnetic theory, an accelerating charge e radiates energy E given by dE 22 4 dad 3 dmege? where a is the magnitude of the acceleration of the charge and c is the speed of light. Estimate how long it. would take for the electron in the classical model of the hydrogen atom to collapse to the proton. . In Rutherford scattering, the distance of closest approach ryyin is given by 2Ze? Thin = 7—p 7 4regE where F is the incident energy of the a particles and Z is the atomic number of the target. Taking E = 5.3 MeV and Z = 29 for copper, find rmin- . (a) Consider a quantum oscillator inside a cavity wall (blackbody radiation). If it is vibrating with a frequency of 5.0 x 10'4 Hz then calculate the spacing between the energy levels of the oscillator. Note that the spacing is uniform for the quantum oscillator. What about the spacing of energy levels of hydrogen atom? (b) Next consider the vibration of a classical oscillator consisting of mass m = | kg and a spring of spring constant & = 1000 N/m with amplitude 0.1 m. Find the energy of the classical oscillator and compare it with the energy spacing obtained for the classical oscillator assuming energy quantization as in (a). . Consider a simple one-dimensional harmonic oscillator of frequency w. Classically, the energy of the oscillator can be equal to zero. Using uncertainty principle estimate the minimum energy of the quantum mechanical oscillator. . Inthe Bohr model of atom the electron is moving in a circle of radius r around the nucleus. (i) Use the uncertainty principle to estimate (a) the radius r and (b) the ground state energy of the hydrogen atom. . Consider a free particle with a Gaussian wave packet 4)(2,¢) given by 1 ee ‘i (x,t) = —— h)etlk—“0 dk ven= = | dhe where @(k) is the amplitude of the wave packet and is given by ok) = Aelia" (k-ko)?*1/4 with A being a normalization constant. (a) Find the normalization constant A. (b) Show that the width of the Gaussian wave packet A(t) grows with time as a Ane? A(t) = Sy1+ 5 Does it satisfy the uncertainty principle? 1. Consider a particle which is exposed to the one-dimensional potential V(z) = —ad(x) where a > 0 is a constant with suitable dimensions. Assuming that the energy of the particle is £ < 0, and the wave function is bound, obtain the possible values of £, and the corresponding wave functions. (a) Plot the wave functions and estimate their width Az. (b) What is the probability dP(p) that a measurement of the momentum of the particle in one of the normalized stationary states calculated above will give a result included between p and p+ dp. For what value of p is this probability maximum.? 2. A particle of mass m moving in one dimension is confined to a space 0 < 2 < L by an infinite well potential. In addition, the particle experiences a delta function potential of strength a given by ad(2 — L/2) located at the center of the well. Find a transcendental equation for the energy eigenvalues F in terms of the mass m, the potential strength a, and the size of the well L. 3. Consider a particle of mass m whose potential energy is V(x) = —ad(x) — ad(x — 1) where | is a constant length. (a) Calculate the bound states of the particle, setting E = —42". Show that the possible energies are given by the relation —pl (1- 2) he where p is defined by 4 = 2%". Give a graphic solution of this equation. (i) Ground state. Show that th this state is even and that its energy Es is less than the energy —E, = ma?/2h?. Represent graphically the corresponding wave function. (ii) Excited state. Show that, when / is greater than a value which you are to specify, there exists an odd excited state, of energy E4 greater than —£,. Find the corresponding wave function. 4, Consider a particle of mass m and energy FE subject to a potential defined by Viz) = —V. O<2 s a 0, otherwise where Vo > 0. Obtain the solutions of the time-independent Schrodinger equation for this system if0 < E < —Vg. 5. Consider a square well potential of width a and depth Vo. We intend to study the properties of the bound state of a particle in this well when its width a approaches zero. (a) Show that there indeed exists only one bound state and calculate its energy E. (6) Let 9 — 0. Show that in the bound state the probability of finding the particle outside the well approaches a How can the preceding considerations be applied to a particle placed in the potential V(a) = —ad(2x)? 6. Consider a particle of mass m moving in the potential co, a2<0 V(a)=¢ -V. O<x<a 0, z>a where Vo > 0. (a) Find the wave function. (b) Show how to obtain the energy eigenvalues from a graph. (c) Calculate the minimum value of Vo so that. the particle will have one bound state; then calculate it for two bound states. Express S, and S_ in the {|+) ,|-)} basis and show that SL = S_ (lt) Show tliat S+ly=|t, S-I=lh, S-=0 , SEI =0 and find T1S4, W154, (tS, 1S . Let A be the Hamiltonian operator of a physical system. Denote by |@,,) the eigenvectors of H, with eigenvalues En, A\dn} = En |@n) (a) For an arbitrary operator A, prove the relation (@n| [As 4] lon} = 0 (b) Consider a one-dimensional problem, where the physical system is a particle of mass m and of potential energy V(X). In this case, HW is written p2 H= 5 +V(X) (i) In terms of P, X and V(X), find the commutators: [H, P], [H,X], and [H, XP]. (ii) Show that the matrix element (¢,| P |d,,) is zero. (iii) Establish a relation between Ex = (dn| £ |dn) and (bn| X¥¢ |en)- . Using the relation (ap) = (27h)-!/2e!?*/4, find the expressions (#|.X Ply) and (a|PX |y) in terms of (a). Can these results be found directly by using the fact that in the {|x}} representation, P acts like 44. 1. Consider a free particle. (a) Show, applying Ehrenfest’s theorem, that (X) is a linear function of time, the mean value (P) remaining constant. (b) Write the equations of motion for the mean values (X?) and (X P+ PX). Integrate these equations. (c) Show that, with a suitable choice of the time origin, the root-mean-square deviation AX is given by : (AX)? = (AP RP +(AX)2 where (AX)? and (AP)2 are the root-mean-square deviations at. the initial time. How does the width of the wave packet vary as a function of time? 2. Let J* be the probability current associated with a wave function y)(r) describing the state of a particle of mass m. (a) Show that: m / dy J(F) = (P) where (P) is the mean value of the momentum. (b) Consider the orbital angular momentum operator E defined by [ — A x P. Are the three components of FE Hermitian operators? Establish the relation m far [Fx Je) =(B) 3. Consider a particle of mass m which moves under the influence of gravity; the particle’s Hamiltonian is P? H= 5 ~mgzZ where g is the acceleration due to gravity. d(Z) dP. d(H (a) Calculate 42), 472) ang 42 (b) Solve the equation “?) and obtain (Z) (t), such that (Z) (0) = h and (Pz) (0) = 0. Compare the result with the classical relation z(t) = —tgt? +h. 4. (a) In a one-dimensional problem, consider a particle with the Hamiltonian: Pp? H => +V(X) where V(X) =AXx" Calculate the commutator [H, XP]. If there exists one or several stationary states |y)in the potential V, show that the mean values (7) and (V) of the kinetic and potential energies in these states satisfy the relation: 2(T) =nlV). (b) In a three-dimensional problem, H is written: P H= mt V(R) Calculate the commutator [H, R- P]. Assume that V(R) is a homogeneous function of nth order in the variables X,Y, Z. What relation necessarily exists between the mean kinetic energy and the mean potential energy of the particle in a stationary state? Apply this to a particle moving in the potential V(r) = —e?/r (hydrogen atom). Recall that a homogeneous function V of nth degree in the variable x, y and z by definition satisfies the relation: V(axr,ay,az) = a"V(z,y, 2) and satisfies Buler’s identity: av av av a +y— +2— =nV(z,y, 2). Ox oy az 5. Consider a physical system whose state space, which is three-dimensional, is spanned by the orthonormal basis formed by the three kets |ui), |u2)and |ug}. In this basis, the Hamiltonian operator H of the system and the tayo observables A and B are written: 100 100 010 H=fw| 020); A=al001]; B=b/100 002 010 001 wherewo,a and 6 are positive real constants. The physical system at time ¢ = 0 is in the state: W(0)) = Ina) +5 Ia) + 5 ua) (a) At time ¢ = 0, the energy of the system is measured. What values can be found, and with what probabilities? Calculate, for the system in the state |y(0)), the mean value (H) and the root-mean-square deviation AH. (b) Instead of measuring H at time t = 0, one measures A; what results can be found, and with what probabilities? What is the state vector immediately after the measurement? (c) Calculate the state vector |1)(t)) of the system at time t. (d) Calculate the mean values (A) (¢) and (8) (¢) of A and B at time ¢. What. comments can be made? (e) What results are obtained if the observable A is measured at time ¢? Same question for the observable B. Interpret. 1. Prove the following identities involving the Pauli spin matrices (a) ojon = OT + ity E jer (b) (7 - a)(o - b) = (a- b)I +i - (a x b) (c) exp[$ 6-0] = I cos(0/2) — iu-o sin(#/2) where J is the 2 x 2 identity matrix, a and b are ordinary vectors and w is a unit vector. In part (a), the indices j and & can take on the values 1,2 and 3, which refer respectively to the z,y and = components of the vector o = (o7,0y,¢z). The Levi-Civita tensor ¢;,; is defined such that +1, ifj, k, lis an even permutation of 1,2,3 yk = -l ifj, k,lis an odd permutation of 1,2,3 0 if j,k, lare not distinct integers 2. Consider a spin 1/2 particle of magnetic moment Md = +S.The spin state space is spanned by the basis of the |+) and |—) vectors, eigenvectors of 5, with eigenvalues +/fi/2 and —h/2. At time ¢ = 0, the state of the system is [b(t = 0)) = |+) (i) If the observable 5, is measured at time t = 0, what results can be found, and with what probabilities? (ii) Instead of performing the preceding measurement, we let the system evolve under the influence of a magnetic field parallel to Oy, of modulus Bp. Calculate, in the {|-++) ,|—}} basis, the state of the system at time t. (iii) At this time ¢,we measure the observables Sz, 5,,5;.What values can we find, and with what probabilities? 3. Consider a spin 1/2 particle placed in a magnetic field Bg with components 1 Br = yy Bo By = 0 B,= a Bo The notation is the same as that of Problem 1. (i) Calculate the matrix representing, in the {|+),|—)} basis, the operator H, the Hamiltonian of the system. (ii) calculate the eigenvalues and the eigenvectors of H. (iii) The system at time ¢ = 0 is in the state |—) .What values can be found if the energy is measured, and with what probabilities? (iv) Calculate the state vector at time t. At this instant 5, is measured; what is the mean value of the results that can be obtained? 4. Consider a spin 1/2, of magnetic moment M = 7S, placed in a magnetic field Boof components B, = —we/y, By = —wy/y, Bz = —we/7. We set wo = 7|Bo| (i) Show that the evolution operator of this spin is: U(t,0) = ei where M is the operator: 1 1 M= i [we Sr + wy Sy + weSz] = 3 [wror + Wydy + weoz] where ¢,,0, and o, are the three Pauli matrices. Calculate the matrix which represents M in the {|+) ,|—)} basis of eigenvectors of . Show that: ne er Se M = 5 [ok +3 +02] = (2) (ii) Put the evolution operator into the form: U(t,0) = cos (=) — 2 vtsin (*) o (iii) Consider a spin which at time t = 0 is in the state |1)(0)) = |+). Show that the probability P,, (é) of finding it in the state |+)at time ¢ is: Py (t) =| (4 (é,0) [+) |? 2 2 Py4(t)=1- ee sin? (3*) ‘O and derive the relation: . Consider an electron of a linear triatomic molecule formed by three equidistant atoms. We use |) ,|¢p) ,|¢c) to denote three orthononnal states of this electron, corresponding respectively to three wavefunctions localized about the nuclei of atoms A,8,C. We shall confine ourselves to the subspace of the state space spanned by loa) lop), |e). When we neglect. the possibility of the electron jumping from one nucleus to another, its energy is described by the Hamiltonian Hp whose eigenstates are the three states |¢4).|¢p),|¢c) with the same eigenvalue Eo. The coupling between the states |64),|¢p),|¢c) is described by an additional Hamiltonian W defined by W |@a) = —a\dp) W\|ép)= -al¢a) — alec) W |oc) = —al|dp) where a is a real constant. (i) Calculate the energies and the stationary states of the Hamiltonian Ho+W. (ii) The electron at time ¢ = 0 is in the state |¢4). Discuss qualitatively the localization of the electron at subsequent times ¢. Are there any values of t for which it is perfectly localized about atom A,B or C? (ili) Let D be the observable whose eigenstates are |d4),|¢a).|¢c) with respective eigenvalues —d,0,d.D is measured at. time t;what values can be found, and with what probabilities? 1. Consider a system of angular momentum j = 1, whose state space is spanned by the basis {|+-1),|0),|—1)} of three eigenvectors common to J? (eigenvalue 2h”) and j. (respective eigenvalues +f, 0 and-h). The state of the system is |) = a|+1) + BO) +7|-1)} where a, 8,7 are three given complex parameters. (a) Calculate the mean value (J) of the angular momentum in terms of a, 3 and ¥. (b) Give the expression for the three mean values (J2), (J?) and (J?) in terms of the same quantities. 2. Consider an arbitrary physical system whose four dimensional state space is spanned by a basis of four eigen- vectors |j,mz) common to J? and j2(j = 0 or 1; —j < mz < +,), of eigenvalues j(j + LA? and mh such that Jz = hy /jG +1) —mz(m, +1|j,m, +1) Jx|j,5) = J-|j,-7) = 0 (a) Express in terms of the kets |j,m.), the eigenstates common to J? and j,, to be denoted by |j, mz). (b) Consider a system in the normalized state: \p) =alj = Lm, = 1) +8 |j =1,m, =9) +y|j = Lm, = -1) +6]j =0,m, =0)} (i) What is the probability of finding 24? and fh if J? and jz are measured simultaneously? (ii) Calculate the mean value of J, when the system is in the state |y}, and the probabilities of the various possible results of a measurement. bearing only on this observable. (iii) Same questions for the observables J? and jz. (iv) J? is now measured; what are the possible results, their probabilities, and their mean values? 3. Let D= Rx P be the angular momentum of a system whose state is E,. Prove the commutation relations: [Li, Ry] = then Re [Li, Pj] = the ijn Pe (Li, P?] = [Li, R?) = (Li, R- P] =0 where Li,Rj,P; denote arbitrary components of L,R, P in an orthonormal system, and <ijx is the Levi-Civita ‘tensor. 4, A system whose state space is £,. has for its wavefunction: B(2,y.2) = N(w@tyt sje /™ where a,which is real, is given and N is a normalization constant. (a) The observables L, and L? are measured; what are the probabilities of finding 0 and 24? Recall that: [3 YP (6,2) = amon ¥"(0,y) = #4/ 2 sinde*” 1 , 87 is it possible to predict directly the probabilities of all possible results of measurements of LE? and L, in the system of wavefunction (2, y, 2) (b) Lf one also uses the fact that: 5. The state vector of the hydrogen atom at time t = 0 is given by W(r, 8,2) = 201,0,0,(7, 8,2) + Yar0(7, 8, 2) where wn.Jt.m,(7, 9, y) are the eigenfunctions of the hydrogen atom. (a) Normalize U(r, 0, y).