(McQuarrie 4-22) Show that the following functions are orthonormal over the interval -x < x < ∞. These functions are the lowest eigenfunctions of the harmonic oscillator problem, which we will study in Chapter 5.) πT 1/4 40(x) = e-x²/2 4 1/4 41(x) = -x²/2 xe πT 42(x) = (+17) 14 (2x² - 1)e-²²/2 Hint: See the Table of Integrals handout for tips on integrating these even and odd functions. (McQuarrie 4-26) Prove that if 8nm is the Kronecker delta, then, 1 n = m бит = 0 nm Σιδη = Cm and Σanbmdnm = Σ ambn n=1 n=1 m=1 n These results will be used in various derivations. Hint: If you find this confusing, pick some particular value for m (e.g. m = 2) and then expand out the sums. Then see what the Kronecker delta does to each term in the sum.

(McQuarrie 4-22) Show that the following functions are orthonormal over the interval -x < x < ∞. These functions are the lowest eigenfunctions of the harmonic oscillator problem, which we will study in Chapter 5.) πT 1/4 40(x) = e-x²/2 4 1/4 41(x) = -x²/2 xe πT 42(x) = (+17) 14 (2x² - 1)e-²²/2 Hint: See the Table of Integrals handout for tips on integrating these even and odd functions. (McQuarrie 4-26) Prove that if 8nm is the Kronecker delta, then, 1 n = m бит = 0 nm Σιδη = Cm and Σanbmdnm = Σ ambn n=1 n=1 m=1 n These results will be used in various derivations. Hint: If you find this confusing, pick some particular value for m (e.g. m = 2) and then expand out the sums. Then see what the Kronecker delta does to each term in the sum.

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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1. (McQuarrie 8-12) Use the variational method to calculate the ground-state energy of a particle constrained to move within the region 0 ≤ x ≤ a in a potential given by: V(x) = Vox Vo(ax) ≤ x ≤ a where Vo is a constant. Use a linear combination of the first two particle-in-a-box wavefunc- tions as a trial function: (x) = c1|1) + C2|42) = C₁ Пх Σπα sin + C₂ sin a a a a Hint: Use the following integrals to simplify the calculus: H11 = (V₁| Ĥ| V₁) = h² 8ma² + aVo 1 -+ H12 = H21 = (V1|Ĥ|42) = 0 (V2|Ĥ|41) = 0 H22 = (V2|Ĥ| V2) h² a Vo + 2ma² 4 Note: These first two problems highlight how one can use choose to solve a given problem with either the variational method or perturbation theory. In this class, you'll be told which method to use, but in the real world, the quantum chemist needs to evaluate which method is most appropriate for a given problem. That decision might be based on an understanding of how big or small the perturbation is and how much effort the person is willing to apply…
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(McQuarrie 5-30) The Hamiltonian for a 3-D harmonic oscillator is given by, მ2 + მ2 + V(x, y, z) 2u \ მ2 მყ2 Əz² ħ2 მ2 Ĥ + where the potential energy is: 1 1 V (x, y, z) == ½ k²x² + ½ kyy² + 1½ k₂ z² 글++ Use separation of variables to argue that that the wave functions are given by: Vnx,ny,nz (x, y, z) = Vnz (x) Vny (y) Vnz (2) where the (u) are the 1-D harmonic oscillator wave functions (u= x, y, or z) and that the energies are: Enz,ny,nz = = hvx (nx = ( n + 1 ) + hvy (ny + 1 ) + hvz (m² + ½ ) nz ku for v₁ = 1½ (k) 1/2 μ