# Chem 113B Homework 1 Due January 15 2016

Chem 113BHomework 1Due January 15 2016 in class by 1:00 PM1. When lithium is irradiated with light the kinetic energy of the ejected electron is 1.83 eV for?= 300 nm and 0.80 eV for ?= 400 nm.a) Calculate the Plank constant (in J-s)b) Calculate the threshold frequency for ejecting electrons from lithium (in s-1)c) Calculate the work function of lithium (in J)Hint: Be sure to get your units right.2. The black body radiation law of Plank can be written in terms of frequency( ) = () =8h3( h)3/ – 1a) Using the relationship ? = c convert the above expression into the one given in class interms of wavelength.( ) = () =8h1( h)5/ – 1b) Show both () and () have units of energy/vol.3. Calculate the deBroglie wavelength in pm fora) A 5oz baseball delivered by a pitcher at 90 mphb) An a-particle ejected from radium at 1.5 x 107 m/sc) An electron accelerated to 10eV 4. Heisenberg’s uncertainty principle provides a fundamental limit on how well we cansimultaneously know complimentary variables (i.e. variables where the corresponding operatorsdo not commute). The following calculations indicate how important this restriction is for atomicsystems.a) The momentum px and the position x are complimentary variables and these obey therestriction: ?p ?x = h/2A typical bond in a molecule is about 1Å = 10-8 cm = 100pm. Suppose we wanted to locateour electron in a molecule to an accuracy of 50pm. What would the uncertainty be in ?p?To get a feel for this uncertainty use this value to calculate the uncertainty in the velocityof the electron in m/s.b) We live in a high technology age where we can probe molecules using femtosecondlasers. It turns out the energy and time are complimentary variables like p and x. If weirradiate a molecules with a 100 fs laser pulse what is the uncertainty of the energy of thatpulse in J? In kcal/mol?5. The Hamiltonian operator is composed of two parts: Kinetic Energy and Potential Energy^^^Hx = KEx V (x)a) Determine the following commutators:[^ ^ ] =[^ ^ ()] =b) Determine the following commutators:[ ^ ] =^[ ^ ()] =^Where in both cases V(x) = f(x) with no derivatives.c) Under what circumstances could ^ commute with ^ and why?d) Under what circumstances would ^ commute with ^ and why? 6. Two unnormalized excited state wave functions of the hydrogen atom are:) -/21 () = (2 -2 ( ) = sin cos -/2a) Normalize 1 and 2 . Hint dt = r2 dr sin d d? where0= = 8;0= = ;0 = = 2b) Show 1 and 2 are orthogonal7. Assume a particle can freely move in the space 0 = = 8 (i.e. V (x) = 0 in this space). Itsunnormalized wavefunction is ?(x) = e-ax. If a = 2 m-1 what is the probability of finding the particlein the space x = 1m?8. Suppose we had the wavefunction ?(x) = N eand N is a normalization constant.2-x / 22a-8 = = 8 where a is a constanta) Normalize the function by evaluating it Nb) Find the most probable location of the particle with this wavefunction.Hint: Recall the probability density is * .

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