Physics 402 Homework: Linear Operators, Quantum Mechanics - Prof. Thomas D. Cohen, Assignments of Quantum Physics

Download Physics 402 Homework: Linear Operators, Quantum Mechanics - Prof. Thomas D. Cohen and more Assignments Quantum Physics in PDF only on Docsity! PHYS 402 Homework---Due February 7 1. Consider the construction used in class to combine two Hilbert spaces with the basis states given by the set of states of the form )2()1(, jiji = and operators sums of operators of the form )2()1( ˆˆˆ BAC = with the operation of Ĉ on the basis states given by ( )( ))2()2()1()1( ˆˆ,ˆ jBiAjiC = . a. Show that if )1( and )2(B̂ are linear operators in their respective spaces then Ĉ is a linear operator in the combined space. b. Show that if )1( and )2(B̂ are Hermitian operators in their respective spaces then Ĉ is a Hermitian operator in the combined space. 2. The Hamiltonian for a particle moving in three dimensions, the Hamiltonian in abstract form is )ˆ,ˆ,ˆ( 2 ˆˆˆˆ 222 zyxV M ppp H zyx + ++ = . The time-dependent Schrödinger equation in abstract form is ψψ tiH ∂∂=
ˆ . By multiplying the abstract time-dependent Schrödinger equation on the left by the position space bra vector xyx ,, , inserting the unit operator in the form 1= zyxzyxdzdydx ,,,,∫ , and using standard delta function results and matrix elements given in class show that the in the position basis the dependent Schrödinger equation may be written as );,,();,,(),,( 2 22 tzyxitzyxzyxV M t ψψ ∂∂=      +∇−

. 3. Consider a particle constrained to stay inside a cubic box with sides L. One corner of the box is at the origin and the box is oriented along the positive x, y and z axes. The energy eigenstates are labeled by the quantum numbers xn , yn and zn with wave functions as given in class. Suppose that the state of the system at some time is given by 3,2,21,1,1 2 1 2 1 ===+==== zyxzyx nnnnnnψ . a. Find the probability that if the position of the particle were to be measured that it would be found at x<1/2. b. Find the probability that if the position of the particle were to be measured that it would be found at x<1/2 and y<1/2 4. Griffiths Problem 4.9