Test3solution - Name I.D Test#3(Closed Book and Notes...

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Name: I.D.: Test #3 (Closed Book and Notes, Calculators Allowed) Physics 1138 Quantum Mechanics Monday, November 14,20Il (Check to see that there are 9 pages, including this cover page.) Problem 1: Problem 2: Problem 3: Total: t50 lls 115
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l. (15 points) Consider a harmonic oscillator with the potential energy V(x) : (1/2)kx2. fhe allowed energies are En: (n+1)hrrl (n:0, 1,2,3,...), and r:1L/-)t", k is the force constant. Suppose the force constant is changed by a small amount e, i.e. k' : (1+r)k. (a) (3 points) What is H'? / t+'= +Hx._ tkr- =L(r+e)kx--*kxo (b) (6 points) What is the first order correction to the energy, i.e. E,,1? Hint use the virial theorem: <T>: <V>. E:: <9:{n'lp*"> €<*:tvlh'> : t1v7 1v7 = %p<cfn|u- Valr,-e-- V f41<cf+fftru Valu,-t* "( {* w"y+yla{be A s-fi4J€s * 1v7 1t7 + 1r> Eo= 1v7 = 21v7 4 1v > 4trt+.)t" t *En"
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(c) (6 points) Obtain the exact energy for the new harmonic oscillator with the new force constant k'. Expand this energy to first order in e. Eb= h+t) t; (,* ,)/' .Y, t+ L) tftt I tJ= LJ LJ ( ( J tt btu h' m ( rt t) k, 'ft| fr'rtrt-#€?* ) :("rt)hu((+r)/' 0* 41) *(v*L)l* t: lt E* E* x E: + +("+!)f, d Fll t t )r" \
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4"ry 2.
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