Symbolic Mth pproch to Solve Prticle-in-the-Box n H-tom Problems, Ph.D Deprtment of Chemistry Georgetown College Georgetown, KY To_Hmilton@georgetowncollege.eu Copyright 6 by the Division of Chemicl Euction, Inc., mericn Chemicl Society. ll rights reserve. For clssroom use by techers, one copy per stuent in the clss my be me free of chrge. Write to JCE Online, jceonline@chem.wisc.eu, for permission to plce ocument, free of chrge, on clss Intrnet. Gol This exercise is esigne to give unergrute physicl chemistry stuents "hns-on" experience in solving the hyrogen tom. The stuents explore step-by-step solution to the Schröinger eqution in symbolic wy n explore properties of the ril wvefunction. Objectives Upon completion of the exercise stuents shoul be ble to:. solve for the energy of the prticle in box symboliclly. solve the Schröinger eqution symboliclly for the energy of the hyrogen tom. explin how wvefunction is normlize. iscuss the fetures of the s ril wvefunction for the hyrogen tom 5. explin how the size of the s orbitl is etermine 6. solve for the energy of the one-imensionl hrmonic oscilltor symboliclly Prerequisites This exercise is intene for junior-level stuents who hve been introuce to quntum mechnics, prticulrly the prticle-in--box problem. Two semesters of clculus n semester of physics is esire. Stuents shoul lso hve some experience working with erivtives n integrls in Mthc. Instructors notes n solutions re highlighte in yellow. Plese remove these sections prior to istribution to stuents. References. Levine, I. N. Quntum Chemistry, 5th Eition, Prentice Hll,.. Levine, I. N. Physicl Chemistry, 5th Eition, McGrw-Hill,.. Hnson, D. M. n ielinski, T. J. Quntum Sttes of toms n Molecules, CD-ROM,.. Newhouse, P. F. n McGill, K. C. J. Chem. Euc. 8, (). 5. Mk, T. C. W. n Li, W. J. Chem. Euc. 77, 9 (). 6. Turner, D. E. J. Chem. Euc. 7, 85 (99). 7. Lin, L., Toree,., n lvrino, J. M. J. Chem. Euc. 58, 67 (98). Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
8. Peterson, C. J. Chem. Euc. 5, 9 (975). Introuction The Schröinger eqution cn be solve exctly for the hyrogen tom. fter some prctice with the prticle in box problem, you will solve the H tom symboliclly using the erivtives feture in MthC. The result is n expression for the energy levels of the hyrogen tom. You will lso prove tht the ril prt of the wvefunction is normlize n explore the concept of n orbitl. The Schröinger eqution in sphericl coorintes for the H tom is s follows: h µ r r r r ψ + sinθ θ sinθ θ ψ + sin θ ψ φ e r ψ = Eψ where h is ctully h-br (h ivie by ), ψ is the wvefunction, µ is the reuce mss, is the tomic number, e is ctully e-prime (e ivie by sqrt( ε )), n E is the function for the llowe energy levels. The e term is the Coulomb potentil energy function. fter seprtion of vribles, the ril prt of the Schröinger eqution cn be set equl to constnt, λ: h Rr () r r r Rr () µ r e + E + Rr () r Rr () = λ where R(r) is the ril wvefunction n λ = ( +). For n s orbitl, = n λ=. fter setting λ = n little lgebr, the Schröinger eqution becomes: r r r Rr () µ r e + h Rr () = r µ r h ERr () fter efining the ril wvefunction R(r), you will perform the erivtives in the first prt of the eqution. Then you will in the potentil energy term. Finlly, you will isolte the energy function. In this exercise, we re focusing only on the ril wvefunction for the s orbitl (n=) in orer to simplify the mthemtics. Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
. Prctice with the Prticle in Box Before you tckle the H tom, you will obtin some prctice solving the prticle in box (PIB). The wvefunction for the PIB is s follows: ψ x := sin n The Schröinger eqution for the PIB is very simple (with the potentil energy V=): ψ x E = m h ψ where m is the mss of the prticle, h is Plnck's constnt, n E is the llowe energies of the prticle. Begin by tking the erivtive of the wvefunction (select the expression below n hit Ctrl+perio, then enter). You shoul get cosine function s the result. x ψ x ψ cos n x n Now try the secon erivtive of the wvefunction below. You shoul get the originl wvefunction bck (in fct this requirement for solving the Schröinger eqution) times some constnts. ψ ψ sinn x x x n Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
Now, ivie the result from the secon erivtive by -m ψ/(h/) (see the Schröinger eqution bove) to isolte the E function. To o this, copy n pste your ltest result into the little blck box below (or retype the result). Click outsie of the mth re, then bck insie, hit Ctrl +perio, then enter: m ψ h x sinn n ψ m h The nswer shoul be E=n h /(8m ) 8 n m h Since the prticle hs % probbility of being foun somewhere in the box, tht is with n x vlue somewhere between n, the integrl of the probbility ensity function ψ between x= n x = must be equl to. To ssure this we write the wve function for the prticle s ψ(x) = N sin(nx/), where N is clle the normliztion constnt, n then etermine the vlue of N tht will mke ψ ( x) x = We will use Mthc's bility to solve equtions symboliclly to to etermine the vlue of the normliztion constnt. Below is the eqution for the integrl of ψ with ψ written out s N sin(nx/). N sin nx x = Copy this eqution into the spce below. Importnt: Since n must be positive integer, replce n with ny positive integer before completing the next steps. exmple : Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
Nsin x x = Nsin x x = Next:. Click in the spce between the left prenthesis n the N so tht N n only N is enclose in the blue selection L.. Go to the symbolics menu on the tool br, click on vrible, then click on solve. This tells Mthc to solve for the vlue of N tht will mke the integrl equl to. Notice tht you get two solutions, one positive n one negtive. Only the positive vlue is meningful. Replce your originl choice for n with nother integer n repet steps n. Notice tht you get exctly the sme solution(s) for N. For the prticle in box the normliztion constnt is the sme for ll sttes (ll vlues of n). Cn you simplify the expression Mthc gives for N into more fmilir form? Note: for some reson the result tht the preceing proceure gives if if n is simply set equl to n integer with the n:= commn is not simple constnt but function contining n s vrible. The result cn be shown to be correct if n is n integer, but tht requires bit of lgebric mnipultion n evlution of trigonometric functions. Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 5
The PIB wvefunction is normlize. In other wors, the integrl of ψ over the box length is equl to (see below). Chnge the upper limit of integrtion by chnging the vrible "fctor" below (try for hlf the box, for one-thir of the box, etc.). fter chnging the upper limit, try ifferent n vlues by chnging the n vlue below (try,,, etc.) lso, notice how the grphs below chnge for ifferent n vlues. fctor := := n := c := n fctor x sin( cx) sin( cx) = ψ = sin( cx).5 x ψ = sin( cx) sin( cx).5 x Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 6
B. The Solution to the Hyrogen tom The generl form of the ril wvefunction is by-prouct of solving the Schröinger eqution. We will restrict ourselves to the s orbitl to simplify the mthemtics. The ril wvefunction for the s orbitl in the hyrogen tom is R(r) = (/) / e-r/. where is constnt with imensions of length ( =.595 Å).. Type in the wvefunction in the spce below. R := R := exp r. Tke the first erivtive of the wvefunction with respect to r (copy n pste in the wvefunction tht you type in bove): r r R r e. Next, multiply the result by r n tke the erivtive gin. ( ) r r Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 7
r r r R r r e + r r e. Now you re rey to in the potentil energy term. The full term is: µ r h e r Rr () where e /r is the Coulomb potentil energy function. However, the symbolic solution is esier to fin if we combine some terms. For exmple, if we note tht the constnt = h /µe, we cn rewrite the bove term s rr. this term to the ltest result n evlute. If you hve trouble copying n psting the result to the right of the evlution rrow, copy the terms to the left of the rrow inste. + rr r r exp + r r exp rr + r r e Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 8
5. s shown bove in the Introuction, the current result is equl to µ r h E Rr () Divie your result by ( µ r Rr ()) h to obtin E, the energy function for the s orbitl in the hyrogen tom. µ r R h r r exp µ r R h h µ You cn hve stuents confirm tht this is the sme formul for the energy levels s you fin in your physicl chemistry textbook. Notice tht this function contins only one prmeter - the tomic number, - n is pplicble to hyrogen-like toms s well. For the H tom, the groun stte energy level is -.6 ev. The n= level requires ifferent (more sophisticte) wve function to solve the Schröinger eqution. Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 9
In the clcultion of the s energy level below, chnge the vlue from (for H) to higher tomic number to clculte the energy levels for hyrogen-like toms. Try urnium! := JpereV :=.677 9 J/eV (conversion fctor from Joules to electron volts) :=.5966 ngstroms h := 6.668.6 J µ := 9. kg E = h =.598 JpereV µ ev Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
C. Normliztion of the H tom Wvefunction You will now prove tht the ril wvefunction for the s orbitl in the hyrogen tom is normlize. First, squre the ril wvefunction. The wvefunction is provie below to i in the process; type in the expression for the squre wvefunction. := R := exp r R := R := exp r Next, multiply the squre wvefunction by r n integrte over r. The fctor of r tkes into ccount the fct tht you re integrting the volume insie surfce locte t istnce r from the nucleus. Note tht you re integrting from to infinity. gin, if you hve trouble with copying n psting, simply enter "R " into the integrl below. r r r exp r r.9999999999999999999 Exmine the results. Is the ril wvefunction for the H tom normlize? Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
Yes, the vlue of the integrl is essentilly. normlize function will integrte to. D. Plotting the H tom Wvefunction Plot the ril wvefunction R versus r for = (copy n pste the ril wvefunction R into the box on the y-xis). Set the y-xis limits to (,) n the x-xis limits to (,5). := := r e 5 r Now plot the squre of the ril wvefunction R, times r, versus r (put the r fctor t the en of the R eqution). Set the y-xis limits to (,.6) n the x-xis limits to (,5). Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
r e r.5 5 r E. Orbitls The integrl of the function tht you just plotte is equl to the probbility of fining the electron t prticulr istnce from the nucleus. Orbitls illustrte in textbooks usully efine the surfce of the orbitl t the istnce r which contins 9% of the electron probbility. In the ctive region below, the upper limit is in multiples of the constnt (remember is pproximtely.5 Å). You see tht sphere of rius equl to the constnt contins bout % of the probblility. Chnge the upper limit of integrtion until the sphere contins 9% of the probbility (until the integrl evlutes to exctly.9). := := upperlimit := upperlimit r e r r =. vlue of.66() for the upper limit integrtes to exctly.9 (9%). The upper limit vlue tht you just foun is the rius (s multiple of ) tht contins 9% of the electron probbility. This is the size of the s orbitl usully isplye in textbooks. Fin pproximtely where this rius is on the plots bove (in Section D). You hve efine the surfce of the s orbitl of the H tom. How fr out must you integrte to get essentilly % of the probbility (until the integrl evlutes to exctly )? Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
vlue of 6.() for the upper limit integrtes to essentilly (%). F. Mstery Exercise The Schröinger eqution for the One-Dimensionl Hrmonic Oscilltor is s follows: ( ) ψ ψ + m Eh α x = x where α = νm/h, h is ctully h-br, n m is the mss of single prticle ttrcte towr the origin by force proportionl to the prticle's isplcement from the origin: F x = -kx The force constnt, k, is relte to the vibrtionl frequency, ν, s follows: ν = (/)(k/m) / The first two wvefunctions tht solve the Schröinger eqution re: Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge
g q ψ = α x α e ψ = α α xe x Show tht these wvefunctions solve the Schröinger eqution for the One-Dimensionl Hrmonic Oscilltor n stte the energies of the first two levels. You will rerrnge the Schröinger eqution to solve for the energy (similr to wht ws one for the H tom). Importnt note: open new winow to solve this section. lso, plot the wvefunctions n prove tht they re normlize (set α = for the purposes of plotting n normliztion). Remember to integrte from - to +. Instructor: It is very importnt to solve this section on brn new worksheet (solving it in the sme worksheet s the PIB n H tom cuses problems uring evlution, possibly ue to hving vribles in common with erlier results). The energy results re equl to E =(/)hν n E =(/)hν ( little lgebr is require to prove this). ψ := α x α e α ψ x α x αe + α α x α x e Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 5
α α := αexp α x α α α x + exp α x α x ψ αexp ψ m h α x α h m α α x αe Note tht evlution is isble for this region (see note t the top of this pge). α y α.5 e α e α x x y α α ψ := xe x ψ x x xe + x e x α α xexp α x α x α + exp α x α x ψ xe x Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 6
α α xexp α x α x α + exp α x α x α ψ m h exp α x Note tht evlution is isble for this region (see note t the top of this pge). 8 α h m α := α α ye y y α x e α x x Crete Mrch 5 Upte ugust 5 Upte Februry 6 H tom Instructor Version.mc Pge 7
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