*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.In both states, the expectation value of the energy n H n is the same, E (n 1,2).*

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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PHYSICS 621 Fall Semester 2012 ODU Graduate Quantum Mechanics Problem Set 1 Problem 1) Write down the total mechanical energy (kinetic plus potential) of a mass m in free fall, expressing it in terms of the momentum p and the height x above ground: E Tkin V (x) H ( p, x) . Then derivative of the function H with respect to p and show that it is equal to the velocity, show that the negative of the partial derivative with respect to x equals the force, i.e.

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.

PHYSICS 621 Fall Semester 2012 ODU Problem 5) d 2 y(x) m 2 y(x) 0 for real m. Problem 3) Consider the two vectors A 4 and B 6 in the space of the plane. Problem 4) Prove the triangle inequality V W V W for arbitrary vectors in any vector space with an inner product. Problem 5) Assume the two operators and are Hermitian. However, they are neither normalized nor orthogonal to each other.

Make sure you find the most general 2 dx solution what are the Solve the differential equation Problem 6) Proof that for any complex numbers c, z with z exp ( c ) we have exp ( c Problem 7) Find the Fourier transform ( p) of the function 2 1 exp x : 2 1 ( p) f f (x) ( ) PHYSICS 621 Fall Semester 2012 ODU c. To turn them into an orthonormal set, first we have to normalize the first one: A 4 0.8 . 1 0 0 1 isin cos cos isin cos i cos sin sin 2 cos2 q.e.d. 0 The corresponding normalized eigenvectors are 0 , 1 , 0 . Calculate exp Answ.: 1 0 0 0 0 i 0 0 0 0 0 1 2 3 0 1 0 , 0 0 0 , 0 0 0 , 0 0 0 0 0 1 i 0 0 0 0 0 0 and from there on, it repeats.

Problem 2) The ammonia molecule NH3 has two different possible configurations: One (which we will call 1 ) where the nitrogen atom is located above the plane spanned the three H atoms, and the other one (which we will call 2 ) where it is below.

(These two states span the Hilbert space in our simple example).

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