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physics
modern physics

Questions and Answers of Modern Physics

Look again at the Hamiltonian suppose the typist made an error and wrote H as H = H11|1> < 1| + H22 | 2> <2| + H12 |1> < 2|. What principle is now violated 7 Illustrate your point
Let x(t) be the coordinate operator for a free particle in one dimension in the Heisenberg picture. Calculate [x (t), x(0)].
Consider a particle in one dimension whose Hamiltonian is given by l*(E. - E.) = 2m " alt="Consider a particle in one dimension whose Hamiltonian is given" class="fr-fic fr-dii"> By calculating
Consider a particle in three dimensions whose Hamiltonian is given by By calculating [x ? p, H] obtain to identify the preceding relation with the quantum-mechanical analogue of the virial theorem it
Consider a free-particle wave packet in one dimension. At 1= 0 it satisfies the minimum uncertainty relation in addition, we know Using the Heisenberg picture, obtain , as a function of t(t ? 0) when
Let |a’> and |a”> be eigenstates of a Hermitian operator A with eigenvalues α’ and α”, respectively (α’ ≠ α”). The Hamiltonian operator is given by H = | α’> δ <
A box containing a particle is divided into a right and left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the
Using the one-dimensional simple harmonic oscillator as an example, illustrate the difference between the Heisenberg picture and the Schrodinger picture. Discuss in particular how(a) The dynamic
Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at t = 0 the state vector is given by where p is the momentum operator and ? is some number with
a. Write down the wave function (in coordinate space) for the state specified in Problem 11 at t = 0. You may use b. Obtain a simple expression for the probability that the state is found in the
Consider a one-dimensional simple harmonic oscillator. a. Using evaluate , . 2 | n), and 2 | n>. b. Check that the virial theorem holds for (he expectation values of the kinetic and the potential
a. Using prove b. Consider a one-dimensional simple harmonic oscillator. Starting with the Schrodinger equation for the state vector, derive the Schrodinger equation for the momentum-space wave
Consider a function, known as the correlation function, defined by C(t) = < x (t) x (0)>, where x(t) is the position operator in the Heisenberg picture. Evaluate the correlation function
Consider a function, known as the correlation function, defined by C(t) = < x (t) x (0)>, where x(t) is the position operator in the Heisenberg picture. Evaluate the correlation function
Show for the one-dimensional simple harmonic oscillator where x is the position operator.
Let where ? ? and ??? are the annihilation and creation operators of two independent simple harmonic oscillator satisfying the usual simple harmonic oscillator commutation relations. Prove
Consider a particle of mass in subject to a one-dimensional potential of the following form: a. What is the ground-state energy? b. What is the expectation value (x2) for the ground state?
Consider a particle moving in one dimension under the influence of a potential V(x). Suppose its wave function can be written as exp [iS (x, t) / h]. Prove that S(x, t) satisfies the classical
Using spherical coordinates, obtain an expression for j for the ground and excited states of the hydrogen atom. Show, in particular, that for m1 ≠ 0 states, there is a circulating flux in the sense
Derive (2.5.16) and obtain the three-dimensional generalization of (2.5.16).
Define the partition function as in (2.5.20)?(2.5.22). Show that the ground-state energy is obtained by taking Illustrate this for a particle in a one-dimensional box.
The propagator in momentum space analogous to (2.5.26) is given by . Derive an explicit expression for for the free- particle case?
Show that the wave-mechanics approach to the gravity-induced problem discussed in Section 2.6 also leads to phase-difference expression (2.6.17).
What is the meaning of the following equation? Where the three components of A are matrices? From this equation show that matrix elements transform likevectors.
Consider a sequence of Euler rotations represented by Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an
a. Prove that the time evolution of the density operator ? (in the Schrodinger picture) is given by b. Suppose we have a pure ensemble at t = 0. Prove that it cannot evolve into a mixed ensemble as
Consider an ensemble of spin 1 systems. The density matrix is now a 3 x 3 matrix. How many independent (real) parameters are needed to characterize the density matrix? What must we know in addition
An angular-momentum Eigen state | j, m = m max= j> is rotated by an infinitesimal angle ε about the y-axis. Without using the explicit form of the d(j)m’m function,
A particle in a spherically symmetrical potential is known to be in an Eigen state of L2 and Lz with Eigen values h2l (l + 1) and mh, respectively. Prove that the expectation values between |
Consider an orbital angular-momentum Eigen state | l = 2, m = 0>. Suppose this state is rotated by an angle β about the y-axis. Find the probability for the new state to be found in m = 0, ± 1,
What is the physical significance of the operators K+ ≡ α† + α† – and K – ≡ α + α – in Schwinger’s scheme for angular momentum? Give the non-vanishing matrix elements of K±.
Consider a system made up of two spin ½ particles. Observer A specializes in measuring the spin components of one of the particles (s1z, s1x and so on), while observer B measures the spin components
A spin 3/2 nucleus situated at the origin is subjected to an external inhomogeneous electric field. The basic electric quadrupole interaction may be taken to be where ? is the electrostatic potential
Show that the differential cross section for the elastic scattering of a fast electron by the ground state of the hydrogen atom is given by (Ignore the effect of identity).
Calculate the three lowest energy levels, together with their degeneracies, for the following systems (assume equal mass distinguishable particles):a. Three non-interacting spin particles in a box of
Let Td denote the translation operator (displacement vector d); D(n, Φ), the rotation operator (n and Φ are the axis and angle of rotation, respectively); and r the parity operator. Which, if any,
A quantum-mechanical state ψ is known to be a simultaneous eigenstate of two Hermitian operators A and B which anti commute, AB + BA = 0. What can you say about the eigenvalues of A and B for state
A spin 1/2 particle is bound to a fixed center by a spherically symmetrical potential. c. Show that your result in (b) is understandable in view of the transformation properties of the operator S?x
Because of weak (neutral-current) interactions there is a parity-violating potential between the atomic electron and the nucleus as follows: v = ? [?(3)(x) S ? p + S ? p ?(3)(x)], where S and p are
a. Let ψ(x, t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that ψ*(x, – t) is the wave function for the plane wave with the momentum
a. Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless non-degenerate system at any given instant of time can always be chosen to be real.b.
Let Φ(p’) be the momentum-space wave function for state | α >, that is, Φ(p’) = < p’ | α >. Is the momentum-space wave function for the time- reversed state θ | α > given by
A simple harmonic oscillator (in one dimension) is subjected to a perturbation ?H1 = bx where b is a real constant. a. Calculate the energy shift of the ground state to lowest non-vanishing order. b.
In non degenerate time-independent perturbation theory, what is the probability of finding in a perturbed energy eigenstate (|k>) the corresponding unperturbed eigenstate (|k(0)>)? Solve this
A one-electron atom whose ground state is non degenerate is placed in a uniform electric field in the z-direction. Obtain an approximate expression for the induced electric dipole moment of the
A hydrogen atom in its ground state [(n, l, m) = (1, 0, 0)] is placed between the plates of a capacitor. A time-dependent but spatial uniform electric field (not potential!) is applied as follows:
Derive an expression for the density of free particle states in two dimensions, that is, the two-dimensional analog of your answer should be written as a function of k (or E) times dE d ?, where ? is
A particle of mass m constrained to move in one dimension is confined within 0 Obtain an expression for the density of states (that is, the number of states per unit energy interval) for high
Linearly polarized light of angular frequency w is incident on a one-electron ?atom? whose wave function can be approximated by the ground state of a three-dimensional isotropic harmonic oscillator
Find the probability | Φ (p’)| 2d3 p’ of the particular momentum p’ for the ground-state hydrogen atom. (This is a nice exercise in three-dimensional Fourier transforms. To perform the angular
Obtain an expression for τ(2p → 1s) for the hydrogen atom. Verify that it is equal to 1.6 x 10– 9s.
a. N identical spin ½ particles are subjected to a one-dimensional simple harmonic oscillator potential. What is the ground-state energy? What is the Fermi energy?b. What are the ground state and
It is obvious that two non-identical spin 1 particles with no orbital angular moment a (that is, s-states for both) can form j = 0, j = 1, and j = 2. Suppose, however, that the two particles are
Consider three weakly interacting, identical spin 1 particles.a. Suppose the space part of the state vector is known to be symmetric under interchange of any pair. Using notation | + > | 0 > |
Suppose the electron were a spin particle obeying Fermi-Dirac statistics. Write the configuration of a hypothetical Ne (Z = 10) atom made up of such “electrons” [that is, the analog of (1s) 2
Two identical spin ½ fermions move in one dimension under the influence of the infinite-wall potential V = ∞ for x < 0, x > L, and V = 0 for 0 ≤ x ≤ L.a. Write the ground-state wave
The Lippmann-Schwinger formalism can also be applied to a one- dimensional transmission-reflection problem with a finite-range potential, V(x) ? 0 for 0 a. Suppose we have an incident wave coming
Prove in each of the following ways. a. By integrating the differential cross section computed using the first-order born approximation. b. By applying the optical theorem to the forward-scattering
Consider the scattering of a particle by an impenetrable sphere (a) Derive an expression for the s-wave (l = 0) phase shift. (You nee not know the detailed properties of the spherical Bessel
Use δ1 = ∆(b) | b – 1/k to obtain the phase shift δ1 for scattering at high energies by (a) The Gaussian potential, V = V0 exp (– r2 / a2), and(b) The Yukawa potential, V = V0 exp (–
Two meter sticks, A and B, move past each other as shown in Figure. Stick A has paint brushes at its ends. Use this setup to show that in the frame of one stick, the other stick still has a length of
Two planets, A and B, are at rest with respect to each other, a distance L apart, with synchronized clocks. A spaceship flies at speed v past planet A and synchronizes its clock with A’s (they both
Two bombs lie on a train platform, a distance L apart. As a train passes by at speed v, the bombs explode simultaneously (in the platform frame) and leave marks on the train. Due to the length
A stick of length L moves past you at speed v. There is a time interval between the front end coinciding with you and the back end coinciding with you. What is this time interval in?(a) Your frame?
A square with side L flies past you at speed v, in a direction parallel to two of its sides. You stand in the plane of the square. When you see the square at its nearest point to you, show that it
A train and a tunnel both have proper lengths L. The train speeds toward the tunnel, with speed v. A bomb is located at the front of the train. The bomb is designed to explode when the front of the
A ruler is positioned perpendicular to a wall. A stick of length L flies by at speed v. It travels in front of the ruler, so that it obscures part of the ruler from your view. When the stick hits the
Cookie dough (chocolate chip, of course) lies on a conveyor belt which moves along at speed v. A circular stamp stamps out cookies as the dough rushes by beneath it. When you buy these cookies in a
Consider the usual twin paradox: Person A stays on the earth, while person B flies quickly to a distant star and back. B is younger than A when they meet up again. The paradox is that one might argue
A ball moves at speed v1 with respect to a train. The train moves at speed v2 with respect to the ground. What is the speed of the ball with respect to the ground? Solve this problem (that is, derive
Verify that the values of ?x and ?t in the table in the example in Section 10.4 satisfy the Lorentz transformations between the six pairs of frames, namely AB, AC, AD, BC, BD, and CD. (See Figure)
An object moves at speed v1/c ? ?1 with respect to S1, which moves at speed ?2 with respect to S2, which moves at speed ?3 with respect to S3, and so on, until finally SN??1 moves at speed ?N with
A and B travel at 4c/5 and 3c/5, respectively, as shown in Figure. How fast should C travel between them, so that he sees A and B approaching him at the same speed? What is thisspeed?
A moves at speed v, and B is at rest, as shown in Figure. How fast must C travel, so that she sees A and B approaching her at the same rate? In the lab frame (B??s frame), what is the ratio of the
For the special case ux = 0, the transverse velocity addition formula, equation (10.35), yields uy = uy/?. Derive this in the following way. In frame S??, a particle moves with speed u?? in the
In the lab frame, an object moves with velocity (ux, uy), and you move with speed v in the x-direction. What must v be so that you also see the object moving with speed uy in your y-direction? (One
In the lab frame, two particles move with speed v along the paths shown in Figure. The angle between the trajectories is 2?. What is the speed of one particle, as viewed by the other?
In the lab frame, particles A and B move with speeds u and v along the paths shown in Figure. The angle between the trajectories is ?. What is the speed of one particle, as viewed by the other?
Consider the following variation of the twin paradox. A, B, and C each have a clock. In A’s reference frame, B flies past A with speed v to the right. When B passes A, they both set their clocks to
A train with proper length L moves at speed c/2 with respect to the ground. A ball is thrown from the back to the front, at speed c/3 with respect to the train. How much time does this take, and what
In one reference frame, Event 1 happens at x = 0, ct = 0, and Event 2 happens at x = 2, ct = 1. Find a frame where the two events are simultaneous.
Consider the Minkowski diagram in figure. In frame S, the hyperbola c2t2 ?? x2 = 1 is drawn. Also drawn are the axes of frame S??, which moves past S with speed v. Use the invariance of the interval
An object moves at speed v1 with respect to frame S’. Frame S’ moves at speed v2 with respect to frame S. (in the same direction as the motion of the object). What is the speed, u, of the object
Use a Minkowski diagram to do the following problem: Two people stand a distance d apart. They simultaneously start accelerating in the same direction (along the line between them), each with proper
Two spaceships float in space and are at rest relative to each other. They are connected by a string (see Figure). The string is strong, but it cannot withstand an arbitrary amount of stretching. At
The Lorentz transformation in equation (10.55) may be written in matrix form as Show that by applying an L.T. with v1 = tanh ?1, and then another L.T. with v2 = tanh ?2, you do indeed obtain the
A spaceship is initially at rest in the lab frame. At a given instant, it starts to accelerate. Let this happen when the lab clock reads t’ = 0 and the spaceship clock reads t0 = 0. The proper
Accepting the facts that the energy and momentum of a photon are E = hv and p = hv/c (where v is the frequency of the light wave, and h is Planck’s constant), derive the relativistic formulas for
Two photons each have energy E. They collide at an angle µ and create a particle of mass M. What is M?
A large mass M, moving with speed V, collides and sticks to a small mass m, initially at rest. What is the mass of the resulting object? Work in the approximation where M >> m.
A photon collides with a stationary electron. If the photon scatters at an angle ? (see Figure), show that the resulting wavelength, ???, is given in terms of the original wavelength, ? bywhere m is
A ball of mass M and energy E collides head-on elastically with a stationary ball of mass m. Show that the final energy of mass Mis
A mass MA decays into masses MB and MC. What are the energies of MB and MC? What are their momenta?
A particle of mass m and energy E collides with an identical stationary particle. What is the threshold energy for a final state containing N particles of mass m? (“Threshold energy” is the
A particle of mass m moves along the x-axis under a force F = ??m?2x. The amplitude is b. Show that the period is given by
Consider a dumbbell made of two equal masses, m. The dumbbell spins around, with its center pivoted at the end of a stick (see Figure). If the speed of the masses is v, then the energy of the system
Consider the relativistic rocket from Section 11.6. Let mass be converted to photons at a rate σ in the rest frame of the rocket. Find the time, t, in the ground frame as a function of v.16
A dustpan of mass M is given an initial relativistic speed. It gathers up dust with mass density λ¸ per unit length on the floor (as measured in the lab frame). At the instant the speed is v,
Consider the setup in Problem 11. If the initial speed of the dustpan is V, what are v(x), v(t), and x(t)? All quantities here are measured with respect to the lab frame.
Consider the setup in Problem 11. Calculate, in both the dustpan frame and lab frame, the force on the dustpan-plus-dust-inside system (due to the newly acquired dust particles smashing into it) as a

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