Questions and Answers of Modern Quantum Mechanics

A one-dimensional simple harmonic oscillator of angular frequency ω is acted upon by a spatially uniform but time-dependent force (not potential)At t = -∞, the oscillator is known to be in the
Derive the neutrino oscillation probability and use it, along with the data in Figure 2.2, to estimate the values of Δm2c4 (in units of eV2) and θ. Survival
A particle in one dimension is trapped between two rigid walls:At t = 0 it is known to b e exactly at x = L /2 with certainty. What are the relative probabilities for the particle to be found in
An electron is subject to a uniform, time-independent magnetic field of strength B in the positive z-direction. At t = 0 the electron is known to be in an eigenstate of S • n̂ with eigenvalue
Consider a particle in one dimension bound to a fixed center by a δ-function potential of the formFind the wave function and the binding energy of the ground state. Are there excited bound states?
A particle of mass m in one dimension is bound to a fixed center by an attractive δ-function potential:At t = 0, the potential is suddenly switched off (that is, V = 0 for t > 0). Find the wave
Find, by explicit construction using Pauli matrices, the eigenvalues for the Hamiltonianfor a spin 1/2 particle in the presence of a magnetic field H = 2μ ħ S.B
The spin-dependent Hamiltonian of an electron positron system in the presence of a uniform magnetic field in the z-direction can be written asSuppose the spin function of the system is given by(a) Is
Consider the 2 x 2 matrix defined bywhere a0 is a real number and a is a three dimensional vector with real components.(a) Prove that U is unitary and unimodular.(b) In general, a 2 x 2 unitary
Consider a spin 1 particle. Evaluate the matrix elements of S₂(S₂+h)(S₂-ħ) and Sx(Sx+ħ)(Sx -ħ).
Let the Hamiltonian of a rigid body bewhere K is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for K, and then find Euler's equation of motion
Let U = eiG3αeiG2β eiG3γ, where (a ,β, γ) are the Eulerian angles. In order that U represent a rotation (a ,β, γ), what are the commutation rules that must be satisfied by the Gk? Relate G to
Follow these steps to show that solutions to Kummer's Equation (3.7.46) can be written in terms of Laguerre polynomials Ln (x ), which are defined according to a generating function aswhere 0 (a)
Show that the orbital angular-momentum operator L commutes with both the operators p2 and x2; that is, prove (3.7.2). [L,p²] = [L,x²] = 0 (3.7.2)
Suppose a half-integer l-value, say 1/2, were allowed for orbital angular momentum. Fromwe may deduce, as usual,Now try to construct Y1/2,-1/2(θ,ϕ) by (a) applying L_ to Y1/2,-1/2(θ,ϕ); and (b)
The wave function of a particle subjected to a spherically symmetrical potential V (r) is given by(a) Is Ψ an eigenfunction of L2? If so, what is the ! value? If not, what are the possible values of
(a) Consider a system with j = 1. Explicitly writein 3 x 3 matrix form.(b) Show that for j = 1 only, it is legitimate to replace e-iJyβ/h̄ by(c) Using (b), prove (j = 1,m'Jy|j = 1,m)
The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of L2 and Lz, in terms of the Cartesian eigenstates
We are to add angular momenta j1 = 1 and j2 = 1 to form j = 2, 1 , and 0 states. Using either the ladder operator method or the recursion relation, express all (nine) {j,m} eigenkets in terms of
Express the matrix elementin terms of a series in (α2β2γ2\J3|αιβιγι)
Consider a spherical tensor of rank 1 (that is, a vector)Using the expression for d(J=l) given in Problem 3.26, evaluateand show that your results are just what you expect from the transformation
Consider a spinless particle bound to a fixed center by a central force potential.(a) Relate, as much as possible, the matrix elementsusing only the Wigner-Eckart theorem. Make sure to state under
(a) Write xy, xz, and (x2 - y2) as components of a spherical (irreducible) tensor of rank 2.(b) The expectation valueis known as the quadrupole moment. Evaluatewhere m′ = j, j - 1, j - 2, . . . ,
(a) Construct a spherical tensor of rank 1 out of two different vectors U = (Ux, Uy, Uz) and V = (Vx, Vy, Vz). Explicitly write T±(1)1,0 in terms of Ux,y,z and Vx,y,z.(b) Construct a spherical
Consider an isotropic harmonic oscillator in two dimensions. The Hamiltonian is given by(a) What are the energies of the three lowest-lying states? Is there any degeneracy? (b) We now apply a
Consider a symmetric rectangular double-well potential:Assuming that V0 is very high compared to the quantized energies of low-lying states, obtain an approximate expression for the energy splitting
Consider a particle in a two-dimensional potentialWrite the energy eigenfunctions for the ground state and the first excited state. We now add a time-independent perturbation of the formObtain the
Establish (5.1.54) for the one-dimensional harmonic oscillator given by (5.1.50) with an additional perturbation Show that all other matrix elements Vk0 vanish. 2 V = 1/εmw²x².
Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other) arguments.[In (a) and (b), |nlm〉 stands for the energy
A slightly anisotropic three-dimensional harmonic oscillator has ωz ≈ ωx = ωy. A charged particle moves in the field of this oscillator and is at the same time exposed to a uniform magnetic
The Hamiltonian matrix for a two-state system can be written asClearly, the energy eigenfunctions for the unperturbed problems (λ = 0) are given by(a) Solve this problem exactly to find the energy
A one-electron atom whose ground state is nondegenerate is placed in a uniform electric field in the z-direction. Obtain an approximate expression for the induced electric dipole moment of the ground
Ap-orbital electron characterized by |n, l = l, m = ±1,0) (ignore spin) is subjected to a potential(a) Obtain the "correct" zeroth-order energy eigenstates that diagonalize the perturbation. You
Consider a spinless particle in a two-dimensional infinite square well:(a) What are the energy eigenvalues for the three lowest states? Is there any degeneracy?(b) We now add a potentialTaking this
Compute the Stark effect for the 2S1/2 and 2P1/2 levels of hydrogen for a field ε sufficiently weak that eεa0 is small compared to the fine structure, but take the Lamb shift δ (δ = 1,057 MHz)
A system that has three unperturbed states can be represented by the perturbed Hamiltonian matrixwhere E2 > E1· The quantities a and b are to be regarded as perturbations that are of the same
Consider a particle bound to a fixed center by a spherically symmetrical potential V(r).(a) Provefor all s-states, ground and excited.(b) Check this relation for the ground state of a
(a) Suppose the Hamiltonian of a rigid rotator in a magnetic field perpendicular to the axis is of the formif terms quadratic in the field are neglected . Assuming B ≫ C, use perturbation theory to
Work out the Stark effect to lowest nonvanishing order for the n = 3 level of the hydrogen atom. Ignoring the spin-orbit force and relativistic correction (Lamb shift), obtain not only the energy
Work out the quadratic Zeeman effect for the ground-state hydrogen atom [(x|0) = (1/√πa03) e-r/a0] due to the usually neglected e2A2/2mec2- -term in the Hamiltonian taken to first order. Write the
Suppose the electron had a very small intrinsic electric dipole moment analogous to the spin-magnetic moment (that is, μel proportional to σ). Treating the hypothetical -μel. E interaction as a
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator usingas a trial function with β to be varied. You may use (x|0) = e-Blx|
For the He wave function, usewith Zeff = 2 - 5/16, as obtained by the variational method. The measured value of the diamagnetic susceptibility is 1.88 x 10-6 cm3/mole.Using the Hamiltonian for an
Estimate the lowest eigenvalue (λ) of the differential equationusing the variational method withas a trial function. Numerical data that may be useful for this problem areThe exact value of the
Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is wo. For t 0 there is also a time-dependent potentialwhere F0 is constant in both space and time. Obtain an
A one-dimensional harmonic oscillator is in its ground state for t (a) Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited
The unperturbed Hamiltonian of a two-state system is represented byThere is, in addition, a time-dependent perturbation(a) At t = 0 the system is known to be in the first state, repre sented byUsing
Consider a particle bound in a simple harmonic-oscillator potential. Initially (t < 0), it is in the ground state. At t = 0 a perturbation of the formis switched on. Using time-dependent
(a) Consider a pure ensemble of identically prepared spin 1/2 systems. Suppose the expectation values 〈Sx〉 and 〈Sz〉 and the sign of 〈Sy〉 are known. Show how we may determine the state
Show that the 3 x 3 matrices Gi(i = 1, 2, 3) whose elements are given bywhere j and k are the row and column indices, satisfy the angular-momentum commutation relations. What is the physical (or
(a) Let J be angular momentum. (It may stand for orbital L, spin S, or Jtotal·) Using the fact that Jx, ly, lz(± ≡ Jx ± i Jy) satisfy the usual angular-momentum commutation relations, prove(b)
Consider a particle in one dimension moving under the influence of some timeindependent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known . We
Consider a two-level system with E1 2. There is a time-dependent potential that connects the two levels as follows:At t = 0, it is known that only the lower level is populated-that is, c1(0) = 1,
(a) In the presence of a uniform and static magnetic field B along the z-axis, the Hamiltonian is given bySolve this problem to obtain the energy levels of all four states using degenerate
Show that the slow-tum-on of perturbation V → 7 Vent can generate a contribution from the second term in (5.7.36). VRH 2μ S₂ = -S, ħ - (5.6.36)
Consider an atom made up of an electron and a singly charged (Z = 1) triton (3H). Initially the system is in its ground state (n = 1, l = 0). Suppose the system undergoes beta decay, in which the
Consider the spontaneous emission of a photon by an excited atom. The process is known to be an E1 transition. Suppose the magnetic quantum number of the atom decreases by one unit. What is the
Show that An (R) defined in (5.6.23) is a purely real quantity. An (R) i (n; t| [VR\n;t)], (5.6.23)
The ground state of a hydrogen atom (n = 1, l = 0) is subjected to a time-dependent potential as follows:Using time-dependent perturbation theory, obtain an expression for the transition rate at
A particle of mass m constrained to move in one dimension is confined within 0 < x < L by an infinite-wall potentialObtain an expression for the density of states (that is, the number of states
Consider a potentialwhere V0 may be positive or negative. Using the method of partial waves, show that for the differential cross section is isotropic and that the total cross section is given
Estimate the radius of the 40Ca nucleus from the data in Figure 6.6 and compare to that expected from the empirical value ≈ 1.4A1/3 fm, where A is the nuclear mass number. Check the validity of
A spinless particle is scattered by a weak Yukawa potentialwhere μ > 0 but V0 can be positive or negative. It was shown in the text that the first-order Born amplitude is given by(a) Using
Consider scattering by a repulsive δ-shell potential:(a) Set up an equation that determines the s-wave phase shift δ0 as a function of k(E = h̄2k2 /2m).(b) Assume now that γ is very large,Show
A spinless particle is scattered by a time-dependent potentialShow that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or
Check explicitly the x - px uncertainty relation for the ground state of a particle confined inside a hard sphere: V = ∞ for r > a, V = 0 for r < a.
Prove[AB,C D] =- AC{D, B} + A{C, B}D - C{D, A}B + {C, A}DB.
Using the rules of bra-ket algebra, prove or evaluate the following:(a) tr(XY) = tr(Y X), where X and Y are operators.(b) (XY)† = Y† X†, where X and Y are operators.(c) exp[if(A)] =? in
Using the orthonormality of |+〉 and | - 〉, provewhere [Si, Sj]=i&ijkh Sk. (Si, Sj): = 7/2 ħ² Sij,
A beam of spin 1/2 atoms goes through a series of Stem-Gerlach-type measurements as follows:(a) The first measurement accepts sz = h̄/2 atoms and rejects sz = -h̄2 atoms.(b) The second measurement
Evaluate the x-p uncertainty product 〈(Δx)2 〉 〈(Δp)2〉 for a one-dimensional particle confined between two rigid walls,Do this for both the ground and excited states. 0 v = { % V for 0 < x
Let A and B be observables. Suppose the simultaneous eigenkets of A and B {|a′, b′〉} form a complete orthonormal set of base kets. Can we always conclude that [A, B] = 0?If your answer is yes,
Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the