*Write down the x, y and z components of the angular momentum operator in terms of these canonical variables. Given our interpretation of Lz as of rotations around the z axis, can you interpret your result in terms of the transformation of the vector L under the coordinate transformation generated Lz? Then follow the explicit procedure (Legendre transformation) in the lecture to find the corresponding Hamiltonian. The Hamiltonian for this case in cylindrical coordinates z) with canonical momenta , , Pz ) is given 2 2 2 r ( Pz 2m writing down equations of motion, give an interpretation (in terms of the momenta or velocities) of , , Pz ) . Answ.: Lx ypz zpy , Ly zpx xpz , Lz xpy ypx L x , Lz 0 0 pz x zpx 0 0 Similarly, , Lz .Problem 2) Write down the Lagrangian for two equal masses m at positions x1 and x2 (each measured relative to the equilibrium position), coupled to each other and (on their other sides) to two fixed walls with springs with constant k but otherwise free to move along the If the system is in equilibrium, all three springs are relaxed is exactly the set up in Example 1.8.6 in book, p. Since Lz is the generator of rotations around the z axis, any change of a variable under such a rotation an infinitesimally small angle is given Lz . PHYSICS 621 Fall Semester 2013 ODU q r 2 b ) 2 ( which is true if either of the two expressions in parantheses is zero. This means that if the charge is momentarily moving only in radial and (no tangential motion), than instantaneously its radial momentum is conserved.Quantum Mechanics cannot predict with certainty the result of any particular measurement on a single particle. A 32 4 2 Then, we determine the orthogonal part of the 2nd vector: 6 j) (1.2 0.8 j) 3.12 B ( B A) As the final step, we have to normalize this vector: B 3.12 0.6 5.2 and form an orthonormal basis. PHYSICS 621 Fall Semester 2013 ODU Graduate Quantum Mechanics Problem Set 3 Solution Problem 1) 0 0 1 Consider the matrix 0 0 0 . Answ.: Yes it is identical to its adjoint (swapping rows and columns). 1 0 1 2 2 Verify that U is diagonal, U being the matrix formed using each normalized eigenvector as one of its columns. ) Answ.: 1 1 1 1 0 0 2 2 2 2 U 0 1 0 0 1 0 U, 1 0 1 1 0 1 2 2 2 2 q.e.d 1 1 1 1 0 0 2 2 2 1 0 0 2 U 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 2 2 2 2 iv) 1 n (where, for any matrix, M0 1). Collecting all terms, we find PHYSICS 621 Fall Semester 2013 ODU 1 1 1 0 16 120 2 24 0 1 0 1 1 0 1 1 6 120 2 24 which clearly is a unitary matrix.*

Immediately afterwards, the physical observable corresponding to B is measured, and again immediately after that, the one corresponding to A is remeasured (independently from the result of the 2nd measurement).

What is the probability of obtaining a1 a second time?

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With what eigenvalues (called PHYSICS 621 Fall Semester 2013 ODU Graduate Quantum Mechanics Problem Set 5 Solution Problem 1) An operator A, corresponding to a physical observable, has two normalized eigenstates and with eigenvalues a1 and a2, respectively.

Immediately afterwards, the physical observable corresponding to B is measured, and again immediately after that, the one corresponding to A is remeasured.

If all possible information on a system is given, Quantum Mechanics can predict the outcome of any future measurement on the system accurately. Quantum Mechanics cannot predict anything precisely c.

Quantum Mechanics cannot predict with certainty the result of any particular measurement on a single particle d.

CORRECT Problem 5) The most general solution is y(x) A exp(mx) B which can be shown plugging it in (as a 2nd order differential equation, there must be two integration constants, A and B). Problem 5) Assume the two operators and are Hermitian. 1 b 1 f (ax b)dx f (x) a (x a )dx a b f( ) a where we integrated the r.h.s.

Since and we can solve for A and B in terms of the initial conditions at Problem 6) z exp(c) exp(Re(c) i Im(c)) exp(Re(c)) ( cos(Im(c)) exp(Re(c)) ( cos(Im(c)) Im(c)) Problem 7) See next recitation PHYSICS 621 Fall Semester 2013 ODU Graduate Quantum Mechanics Problem Set 2 Problem 1) Do continuous functions defined on the interval and that vanish at the end points x 0 and x L form a vector space? How about all functions with If the functions do not qualify, list the things that go wrong. What can you say about i) Answ.: The product is not necessarily Hermitian since the 2 operators necessarily commute: PHYSICS 621 Fall Semester 2013 ODU ii) iv) Answ.: This is Hermitian: , Answ.: This is not Hermitian (unless the commutator is zero). following the standard rules for the For the left hand side, we make a variable substitution: u ax, du adx.

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