Physics 463 - Final Exam Work all ve problems on this exam. You have until 4:30PM of the day after you receive this exam to complete it and turn it in. 1. Use the variational principle to nd an appropriate bound for the rst excited state energy (n = 2) of
Physics 463 - Final Exam: This exam is to be turned in 24 hours after it is received. In solving the following problems you may use any notes, homework sets, and/or solutions accumulated during the semester, one quantum mechanics textbook (including all v
Physics 463 - Final Exam Your grade will be based upon your answers to 4 of the following 6 problems. If you turn in more than 4 solutions, your 4 highest scores will be used to compute your nal exam grade. 1. Let cfw_|nx , ny , nz i denote the eigenstate
Physics 463 - Test I: Solve 4 of the following 6 problems 1. Consider a quantum system in an eigenstate j ; j; mi.of J 2 and Jz . with (b) Use the results to compute the statistical uncertainties Jx and Jy associated p a measurement of a component of angu
Physics 463 - Final Exam Your grade will be based upon your answers to 4 of the following 6 problems. If you turn in more than 4 solutions, your 4 highest scores will be used to compute your nal exam grade. 1. Let cfw_|nx , ny , nz i denote the eigenstate
Test II - Physics 463 This is an open book, open note test. You must turn it in, in the Physics department front oce, by 4:00PM one day after you pick it up, or by 4:00PM on Friday at the latest. Work all problems in this test, starting each problem on a
24. Electron-positron pairs are emitted in a decay process (moving in opposite directions) in a spin state ~ ~ |0, 0ispin = |s, mi of zero total spin angular momentum S = S1 + S2 . (We focus here only on the spin state of the pair). (a) What is the mean v
20. If j1 j2 , then
j =j1 j2
jX2 1 +j
(2j + 1) = 2
j =j1 j2
jX2 1 +j
j+
j =j1 j2
jX2 1 +j
1.
The second sum on the right is equal to the number of terms in the sum, which is j1 +j2 (j1 j2 1) = 2j2 +1. The rst sum on the right satises 2
jX2 1 +j
j
=
j =j1
16. In the spin space of a particle of spin 1/2, suppose the particle is in a spin state which is spin-up along the ~ z direction: | s i = | 1 , 1 iz , so that Sz | s i = 1 | s i. Consider the component Su = S u, of the spin operator 22 2 ~ S along a dire
12. Consider the 3 3 rotation operators Au () which rotate vectors in R3 . For innitesimal rotations these take the form Au () = 1 + Mu . (a) A vector ~ is rotated in an innitesimal rotation Au () into the vector ~ + ( ~ ), Thus, the v v uv transformation
8. A system with Hamiltonian H (0) is sented by the following matrices 0 0 0 h i 0 0 0 H (0) = 0 0 0 0 0 0 000
subject to a perturbation H (1) , which in a certain ONB can be repre0 0 0 20 0 0 0 0 0 20 00 0 0 0 0 0 00 0 0 000 0 i 0 0 0 i 0
(a) Find the n
4. Use the variational method with wave functions of the form r a 1 and a (r) = 2 r 2 + a2
er/a a (r) = a3
to estimate the ground state energy of a particle subject to a Coulomb potential V (r) = e2 /r. After verifying that the wave function is properly
1. Consider a particle of mass m moving in 1D subject to the conning potential V (x) = k |x| , where k is a positive constant. (a) Using a trial wave function of the form (x) = exp ( |x|) , we compute Z ~2 ~2 2 1 hP 2 i = dx |0 |2 = 2m 2m 2m and hV i = k
28. Consider a harmonically bound electron in 1D with Hamiltonian H0 = 1 ~ (q 2 + p2 ), initially in its ground 2 state. A heavy particle passes through the region at high speed. This heavy particle interacts with the electron through a weak short-range i
24. Electron-positron pairs are emitted in a decay process (moving in opposite directions) in a spin state ~ ~ |0, 0ispin = |s, mi of zero total spin angular momentum S = S1 + S2 . (a) What is the mean value that will be obtained in a measurement performe
20. If j1 j2 show that
j =j1 j2
jX2 1 +j
(2j + 1) = (2j1 + 1)(2j2 + 1).
~ ~ 21. Consider three particles of spin 1 , and let S1 , S2 , and S3 denote the corresponding spin operators. In the 2 ~ ~ ~ ~ combined spin space associated with the total spin oper
16. In the spin space of a particle of spin 1/2, suppose the particle is in a spin state which is spin-up along the ~ z direction: | s i = | 1 , 1 iz , so that Sz | s i = 1 | s i. Consider the component Su = S u, of the spin operator 22 2 ~ S along a dire
12. Consider the 3 3 rotation operators Au () which rotate vectors in R3 . For innitesimal rotations these take the form Au () = 1 + Mu . (a) Using the fact that in an innitesimal rotation Au () the vector ~ is rotated into the vector ~ + ( v v u ~ ), con
8. A system with Hamiltonian H (0) is sented by the following matrices 0 0 0 i 0 0 0 h H (0) = 0 0 0 0 0 0 000
subject to a perturbation H (1) , which in a certain ONB can be repre0 0 0 20 0 0 0 0 0 20 00 0 0 0 0 0 00 0 0 000 0 i 0 0 0 i 0
Find the new e
4. Use the variational method with wave functions of the form r a 1 (r) = and 2 r2 + a2
er/a (r) = a3
to estimate the ground state energy of a particle subject to a Coulomb potential V (r) = e2 /r.Note that the second wave function above reduces, for the
1. Consider a particle of mass m moving in 1D subject to the conning potential V (x) = k |x| , where k is a positive constant. (a) Using a trial wave function of the form (x) = exp ( |x|) , use the variational method to estimate the ground state energy of
Chapter 9 SCATTERING THEORY
9.1 General Considerations
In this chapter we consider a situation of considerable experimental and theoretical interest, namely, the scattering of particles o of a medium containing some type of scattering centers, such as ato
Chapter 8 TIME DEPENDENT PERTURBATIONS: TRANSITION THEORY
8.1 General Considerations
The methods of the last chapter have as their goal expressions for the exact energy eigenstates of a system in terms of those of a closely related system to which a const
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS
~ In classical mechanics the total angular momentum L of an isolated system about any ~ xed point is conserved. The existence of a conserved vector L associated with such a system is itself a consequence of the fac
Chapter 4 MANY PARTICLE SYSTEMS
The postulates of quantum mechanics outlined in previous chapters include no restrictions as to the kind of systems to which they are intended to apply. Thus, although we have considered numerous examples drawn from the qua
Chapter 5 APPROXIMATION METHODS FOR STATIONARY STATES
As we have seen, the task of prediciting the evolution of an isolated quantum mechanical can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system. Un
Chapter 4 BOUND STATES OF A CENTRAL POTENTIAL
4.1 General Considerations
As an application of some of these ideas we briey investigate the properties of a particle of mass m subject to a force deriving from a spherically symmetric potential V (r). Obvious