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 to solving the problem. Using a long, complicated trial function with many variational param- eters will work well in most cases, but that can take a lot of effort to solve! In many cases, using perturbation theory will be much easier, and it hopefully gives a reasonable result.

Evaluate the linear momentum and kinetic energy of a free electron described by the wavefunction e-ikx with k = 5 nm-¹

A Morse oscillator, V(r) = D[1 − e−a(r—ro)]², has D = 164 kJ/mole and w→ ve 1250 cm-1. Compute the energy level (in cm-1) corresponding to the eigenstate shown below. = (Note that the energy units, 11.9627 J/mole = 1 cm-1. It might be helpful to convert the value of D to units of cm-1 before you evaluate the energy level.) www

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