In the Classroom edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Intuitive Solution to Quantum Harmonic Oscillator at Infinity Cory C. Pye Department of Chemistry, Saint Mary’s University, Halifax, NS Canada B3H 3C3;
[email protected] Over the course of the last few years, the author has expended considerable effort into developing the laboratory component of a one-semester quantum chemistry course at Saint Mary’s University. The approach the author has taken is to allow the students to solve the “big” problems in quantum chemistry by subdivision into smaller problems that can be independently tackled by a student with a two-year calculus background. Some examples of the big problems solved by the students are the derivation of the Rayleigh–Jeans and Planck’s law, the harmonic oscillator (classical and quantum), the radial equation of the hydrogen atom, the derivation of the Laplacian in spherical polar coordinates, the solution of the associated Legendre equation, and a particle in a semiinfinite box. In the process of creating these labs, the author has combined intuition and rigor in order to build on the students’ mathematical skills and has found intuitive demonstrations of previously “assumed” results. The solution to the quantum harmonic oscillator problem is widely discussed in elementary quantum mechanics texts (1–8) and in this Journal (9). The Schrödinger equation applied to the harmonic oscillator is, −
1 h 2 d2 ψ + kx 2 ψ ( x ) = E ψ( x ) 2µ dx 2 2
(1)
where ប is Planck’s constant divided by 2π, µ the reduced mass, k the spring constant, x the displacement, E the energy, and ψ the wavefunction. After suitable algebraic manipulation, we obtain d2 ψ + λ − α2 x 2 ψ ( x ) = 0 dx 2
(
)
(2)
At this point, it is traditional to examine the behavior at infinity of this differential equation by examining the slightly simpler differential equation: d 2ψ 2 − ξ ψ(ξ) = 0 2 dξ
Approximate solutions to eq 7 are usually given without derivation or plausibility arguments as: 2
ψ ( ξ ) = exp ±
2
ψ ( ξ ) = H ( ξ ) exp −
kµ 2 h
( D 2 − ξ2 ) ψ
830
(7a)
but this is not the same thing as, (7b)
because D and ξ do not commute (they are operators corresponding to essentially momentum and position). Equation 7b actually expands into,
= 0
(7c)
2
(4)
where the middle two operators do not cancel. However, the solutions to eq 7b are the approximate solutions of eq 7. What follows is a simple, intuitive approximate solution to eq 7.
αx
(5)
Solutions at Infinity
(6)
Equations 6 and 7 state that taking the second derivative of a function gives the same function back times another (rather simple) function. Although the exponential function itself does not satisfy eq 7, it suggests that we try a related
gives: d 2ψ + dξ 2
(9)
= 0
( D 2 + ξD − Dξ − ξ2 ) ψ
Linear scaling of the independent variable by, ξ =
ξ 2
one then proceeds to show that the power series H(ξ) must truncate to give the Hermite polynomials. The solutions, eq 8, once given, are relatively easy to verify as approximate solutions to eq 7. However, it is not immediately obvious why they have the form that they do. The approach taken in ref 9 is actually incorrect. One can recast eq 7 in operator form,
and: α =
(8)
(D + ξ)(D − ξ)ψ = 0 (3)
1
ξ 2
After eliminating the positive exponent and assuming the solution to eq 6 as having the form,
where
2 µE λ = 2 h
(7)
λ − ξ2 ψ (ξ) = 0 α
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Vol. 81 No. 6 June 2004
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n = 2, to get:
function as a trial solution, namely:
ψ ( ξ ) = exp f ( ξ )
(11)
The second order term will disappear if 4a2 − 1 = 0, that is, a = ±1兾2. This can be shown by dividing eq 18 by ξ2 and taking the limit as ξ → ∞. Substituting the values for a and n back into eq 15, and this result back into eq 10, gives the desired result, eq 8. In principle, one can apply the same argument to eq 6 directly, which gives:
and d 2ψ df = exp f ( ξ ) dξ dξ 2
2
+ exp f ( ξ )
d2 f (12) dξ 2
Substituting eq 12 into eq 7 gives:
df exp f ( ξ ) d ξ
2
+ exp f ( ξ )
dξ 2
)
(18a)
= 0
(14)
We have converted a linear differential equation to a nonlinear one, for which the theory is more complicated and general techniques for solving them are lacking (10). However, the simple form of eq 14, in spite of its nonlinearity, suggests that a polynomial may work. We use a trial function, f ( ξ ) = a ξn
(15)
n ( n − 1) a ξn − 2 + n2 a 2 ξ2 n − 2 − ξ 2 = 0
(16)
to obtain:
The order of the highest term, and thus the one dominating as ξ → ∞, is given by: 2n − 2 if max ( 2n − 2, n − 2, 2 ) = if 2
n ≥ 2 n ≤ 2 (17)
We note that if n = 2, the first term of eq 16 is constant and the second and third terms are both second order, which implies that they could cancel (i.e., the differential eq 14 can have an approximate solution for large ξ). We therefore take
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1 2
+
λ = 0 α
(18b)
without the need to discuss approximate solutions at infinity.
2 2
2 −
(13)
The exponential factor multiplying all terms is never zero, and therefore we may divide through to get:
− ξ
(
d2 f
− ξ exp f ( ξ ) = 0
df dξ
λ + 4a 2 − 1 ξ2 = 0 α
2a +
(18)
The same result follows and, in addition, one can obtain the energy of the n = 0 level by solving,
2
d2 f + dξ 2
)
2a + 4a 2 − 1 ξ2 = 0
The first two derivatives of this trial function are
dψ df = exp f ( ξ ) dξ dξ
(
(10)
Acknowledgments I would like to thank the Chemistry 412 (quantum chemistry) students of 1999–2002 at Saint Mary’s University for working through the labs described above and correcting many errors in the early versions. Literature Cited 1. Eyring, H.; Walter, J.; Kimball, G. E. Quantum Chemistry; Wiley: New York, 1944; pp 75–79. 2. McQuarrie, D. A. Quantum Chemistry; University Science Books: Mill Valley, CA, 1983; pp 175–177. 3. Eisberg, R.; Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed.; Wiley: New York, 1985, p I2. 4. Dykstra, C. E. Introduction to Quantum Chemistry; PrenticeHall: Englewood Cliffs, NJ, 1994; p 35. 5. Griffiths, D. J. Introduction to Quantum Mechanics; PrenticeHall: Upper Saddle River, NJ, 1995; pp 37–43. 6. House, J. E. Fundamentals of Quantum Mechanics; Academic Press: San Diego, CA, 1998; pp 112, 119. 7. Levine, I. N. Quantum Chemistry, 5th ed.; Prentice-Hall: Upper Saddle River, NJ, 2000; pp 67, 90. 8. Ratner, M. A.; Schatz, G. C. Introduction to Quantum Mechanics in Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001; pp 73–78. 9. Dushman, S. J. Chem. Educ. 1935, 12, 381–389. 10. Boyce, W. E.; DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed.; Wiley: New York, 1986.
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