Solving the 1D Schrodinger Equation

Morgan Hawker, California State University-Fresno, Chemistry

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Summary

In this activity, students will explore 1-dimensional solutions to the time-independent Schrodinger equation.

Students will utilize a provided MATLAB live script to determine exact energy values for two different models (particle in a box and particle in a parabolic well), and explore how these values are impacted by key variables within two different models. Students will compare and contrast these two models, and apply the models to describe real chemical systems.

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Learning Goals

Learning objectives:
- Utilize a provided MATLAB script to determine exact energy values of the time-independent Schrodinger equation
- For the particle in a box model, describe the impact of n and box size on energy level.
- Relate the number of nodes in a wavefunction to the quantum number n.
- Apply the particle in a box model to conjugated pi bond systems to approximate their energy levels
- Explain the assumptions that must be made in the prior objective.
- Compare/contrast the particle in a box and particle in a parabolic well models in terms of energy level spacing and wavefunction.

MATLAB is utilized in this activity by providing students with a MATLAB live script that they will input and manipulate as they solve the provided problems. MATLAB improves student learning in this case because it allows students to perform complex mathematical calculations quickly. MATLAB also allows students to easily manipulate relevant variables to determine how those variables impact the solutions.

Context for Use

This activity is designed for a one-semester physical chemistry course for BA chemistry majors at a primarily undergraduate institution.

The students will begin this problem set during a synchronous class session in Zoom (50 min class period), and will complete the activity on their own outside of class (approximately 2 hours in total).

Students will have completed the 2 hour MATLAB on-ramp prior to this activity.

In terms of disciplinary skills, students will have been introduced to the underlying concepts of quantum mechanics. A conceptual understanding of the time-independent Schrodinger equation, including relevant terminology related to 1D solutions (e.g.,particle in a box model), is essential to this activity. Knowledge of the classical description of translational motion is also important.

This activity is situated mid-way through the one semester course at the start of a discussion of quantum theory. It could be easily translated into a quantum mechanics-focused course.

Students must be familiar with importing and manipulating a provided MATLAB live script.

Description and Teaching Materials

Provide the students with the attached problem set comprising 3 sections: background, particle in a box, and particle in a parabolic well. Background contains a short overview of the time-independent Schrodinger equation, particle in a box includes 10 questions, and particle in a parabolic well includes 4 questions.

Provide students with the attached MATLAB live script (QM_1D problems)


Student handout for solving the 1D Schrodinger equation problem set (Microsoft Word 2007 (.docx) 73kB Sep7 20)
MATLAB live script for solving the 1D Schrodinger equation problem set (MATLAB Live Script 396kB Oct13 20)


Teaching Notes and Tips

Assessment

Students will receive feedback on this problem set from the instructor. They will have an opportunity to revise if they do not meet the criteria established for a satisfactory score (minimum 12 of 14 questions correct).

Students will also be assessed on the learning objective via a Canvas quiz using combination of multiple choice questions and short answer questions where they must show their work.

References and Resources

Link original source material on which this activity is based (Developed by M. Lopez del Puerto - Published May 16, 2017): https://www.compadre.org/PICUP/exercises/exercise.cfm?I=250&A=1DQM