First Order Corrections to Energy Levels in Quantum Mechanics, Assignments of Quantum Mechanics

Download First Order Corrections to Energy Levels in Quantum Mechanics and more Assignments Quantum Mechanics in PDF only on Docsity! HW Chapter 12 Problem 1- In a simple 1D harmonic oscillator let the spring constant change from k to k(1+ε) . What is the 1st order correction to the ground state energy? H =! p 2 2m + 1 2 k(1+ !)x2 =! p 2 2m + 1 2 kx 2 +! 1 2 k!x2 = T +V +V ' E (0) = T +V = n+1/2( )!" ! V ' = +! 1 2 k!x2 #E (1) =!<$ n !| 1 2 k!x2 |$ n >!= ! !<$ n !| 1 2 kx 2 |$ n >= ! <V >! #E (1) = ! <V >!!!!= ! 2 < E >!!!!!!!!!!!<V >= 1 2 < E > !for!the!harmonic!oscillator.!!!! #E (1) =!!!= ! 2 n+1/2( )!" ! Use!the!Virial!Theorem!you can show < T >!=!<V>! for!the!harmonic!oscillator. < T >= n 2 % &' ( )* <V >!!!where!n = 2! for !V = 1 2 kx 2 !!!+!< T >!=!<V >!! Problem 2- Consider a particle of mass in in an infinite square well of width Δx=a. The left half of the well 0 ! x ! a / 2 is raised by an amount ΔV = εV0 . Find the first order correction to the ground state energy . H =!H 0 + !V 0 !!!!!!!" n (x) = 2 a sin( n# x a ) $E n (1) =!< n | !V 0 | n >!=!! 0 a /2 % " n (x)!!V0 !" n (x) Problem 3- From the electron’s stationery reference frame in a hydrogen atom it sees a proton circulating about producing a small magnetic field ! B(r) = K ! L r 3 . The perturbing Hamiltonian can be written H ' = ! ! µe " ! Bp =! ge µB ! S " # $% & '( " !K ! L r 3 # $% & '( = ge µB " K r 3 ! ! S " ! L Determine the first order correction to the nth energy level of the hydrogen atom. H =! p 2 2m ! 1 4"# 0 e 2 r + ge µB ! K r 3 ! " L $ " S %En (1) =!< nlm | ge µB ! K r 3 ! " S $ " L | nlm >!=!!ge µB ! K 1 r 3 " L $ " S 0 a/2 a L S e p