phy3113s22009
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phy3113s22009

Course Number: P 3113, Fall 2009

College/University: Lake City CC

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1 PHY3113: Quantum mechanics I Problem set 2 1. (10 points) Find the probability current for the following wave functions: a) (x, t) = Aeipx/ + Be-ipx/ e-ip 2 Winter 2009 due January 30th, 2009 t/2m , where A and B are complex numbers, b) (x, t) = (A0 (x, t) + B1 (x, t)), where k (x, t) is the k'th harmonic oscillator wavefunction, and where A and B are complex numbers so that |A|2 + |B|2 = 1. 2. (40 points)...

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Quantum 1 PHY3113: mechanics I Problem set 2 1. (10 points) Find the probability current for the following wave functions: a) (x, t) = Aeipx/ + Be-ipx/ e-ip 2 Winter 2009 due January 30th, 2009 t/2m , where A and B are complex numbers, b) (x, t) = (A0 (x, t) + B1 (x, t)), where k (x, t) is the k'th harmonic oscillator wavefunction, and where A and B are complex numbers so that |A|2 + |B|2 = 1. 2. (40 points) A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (x, 0) = A (1 (x) + 2 (x)) . a) (5 points) Normalize (x, 0). b) (5 points) Find (x, t) and |(x, t)|2 , and express the result in terms of sine and cosine functions of time, using = 2 /2ma2 . c) (5 points) Compute x . What is the frequency of oscillation of x ? What is the amplitude of oscillation? d) (5 points) Compute p e) (3 points) Find the expectation value of H. How does it compare with E1 and E2 ? f) (2 points) Find E. Although the overall phase constant of the wave function is of no physical significance, the relative phase of the coefficients in Eqn. (1) does matter. For example, suppose we change the relative phase of 1 (x) and 2 (x) to (x, 0) = A 1 (x) + ei 2 (x) , where is a constant. g) (10 points) Find (x, t), |(x, t)|2 , x and p , and compare your results with the results obtained using Eqn.(1), where was set to 0. 3. (35 points) The "radial" part of Schrdinger's equation in three dimensions is given by: o 1 d R(r) dr r2 dR dr - 2mr2 2 (1) (2) (V (r) - E) = ( + 1), (3) where R(r) is the radial wave function and r 0 is the radius. a) (5 points) Show that the change of variable (r) = rR(r) will transform the radial equation into an equivalent onedimensional problem subject to the condition r 0. What is the equivalent onedimensional potential for this problem? The nuclear potential can be modeled as a well having a finite depth of 57MeV and radius r0 1.25 A1/3 with A the total number of nucleons. b) (15 points) How many states with angular momentum these states? = 0 can you find for 56 26 F e? What are the energies of c) (10 points) Write down explicitly the normalized wave function for the lowest energy state of 56 F e. Find r for 26 this state if the radial probability density is given by P (r) = r2 |R(r)|2 . d) (5 points) What is the probability of finding a nucleon "outside" the nucleus if it is described by an wavefunction having the lowest possible energy in the above well? (4) = 0 2 4. (35 points) The Morse potential. To investigate the vibrational spectrum of a diatomic molecule, Morse has introduced the potential V (x) = D e-2x - 2e-x , x= r - r0 . r0 (5) a) (5 points) Sketch this potential, and show that it has a minimum at r = r0 . This is the equilibrium point. b) (5 points) By expanding V in series about r0 , write an approximate expression for V up to and including terms in (r - r0 )2 : 1 V (x) V0 + 2 m 2 (r - r0 )2 . (6) c) (10 points) Using the reduced mass formula: 1/m = 1/mCl + 1/mH and the parameters given below, evaluate for an electron in potential specified by the parameters describing the HCl molecule if: D = 37244 cm-1 , 2 2 /2mr0 = 10.5930 cm-1 , = 2.380. To convert from cm-1 to eV, use E(eV ) = E(cm-1 ) 1.2398 10-4 ). What is the equilibrium separation distance r0 between the two atoms in the molecule? d) (5 points) By comparing with the harmonic oscillator, obtain the (approximate) values for the lowest two energy levels of HCl. e) (10 points) By comparing with the wave functions of the harmonic oscillator, write the (approximate) radial wave function R(r) for the state of lowest energy. Compute r using Eqn.(4) ') .1 pz* F4",r-rfi: 2- \ilu-d'r,tzeq J L I ,+r I -* ^ d /-r1 ,rt"^ $rt Ua_(. ,,.4 r(',ila) R/ ^, _Ea/y r l-'* '\ l dr ol- o ) r l 1 ,. ')- - ,l , 'Le , ( n/ t ,L - ?' / v - ?t / t , nr ^ q,,(,.; 0r '-ptAc +- -BL )e, dX t. P'lo*Bn ,p u'il"S1A.'P'/"-u;'0") ^I*s^p ya 1g*-- (o-" t. t- b)-UJ.rnood (:'-,1) & .;. :.-:,14-. ^&a:t a -+$s). ! /I \4 h-. . E .to -j]}iDfL a, " -^..,.L/ _-mutx/a _ -r L.f e' h ItU-t -0 0 n ) ,t,rr a+ .r IlrE_ Ax ^A -mdf/n e .r rI It Ia':(,0.J-t--rlr- A dl,,r,u , . , , ' t l L,: \ ' t^x ,i; --i -,. -l e ntl xil_ /r [ t L(n ': n e +)r,w ,\-,(a i i ".n B' ) ar o+ 1 r y q 'h \ ( t I / , I i^'li ir 0^^d'\-|,^r&) ,'Lr h r.l'o -r.l 1: - ) :l n ) \ jr u- , , J ,- J, nu' agl = O f+. 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